The first law of general quantum resource theories

The first law of general quantum resource theories Carlo Sparaciari1 , Lı́dia del Rio2 , Carlo Maria Scandolo3 , Philippe Faist4 , and Jonathan Oppenheim1 1 Department 2 Institute 3 Department 4 Institute of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland of Computer Science, University of Oxford, Oxford OX1 3QD, UK for Quantum Information and Matter, Caltech, Pasadena CA, 91125 USA arXiv:1806.04937v2 [quant-ph] 15 Dec 2018 December 18, 2018 We extend the tools of quantum resource theories to scenarios in which multiple quantities (or resources) are present, and their interplay governs the evolution of the physical systems. We derive conditions for the interconversion of these resources, which generalise the first law of thermodynamics. We study reversibility conditions for multi-resource theories, and find that the relative entropy distances from the invariant sets of the theory play a fundamental role in the quantification of the resources. The first law for general multiresource theories is a single relation which links the change in the properties of the system during a state transformation and the weighted sum of the resources exchanged. In fact, this law can be seen as relating the change in the relative entropy from different sets of states. In contrast to typical single-resource theories, the notion of free states and invariant sets of states become distinct in light of multiple constraints. Additionally, generalisations of the Helmholtz free energy, and of adiabatic and isothermal transformations, emerge. We thus have a set of laws for general quantum resource theories, which generalise the laws of thermodynamics. We first test this approach on thermodynamics with multiple conservation laws, and then apply it to the theory of local operations under energetic restrictions. Contents 1 Introduction 2 2 Framework for multi-resource theories 4 3 Reversible multi-resource theories 7 4 Bank states, interconversion relations, and the first law 14 5 Examples 22 6 Conclusions 28 A Reversibility and asymptotic equivalence for single-resource theories 31 B Convex boundary and bank states 34 C Energy-entanglement interconversion protocol 36 D Proofs 38 References 50 Carlo Sparaciari: [email protected] 1 1 Introduction Resource theories. Resource theories are a versatile set of tools developed in quantum information theory. They are used to describe the physical world from the perspective of an agent, whose ability to modify a quantum system is restricted by either practical or fundamental constraints. These limitations mean that while some states can still be created under the restricted class of operations (the free or invariant set of states), other state transformations can only be done with the help of additional resources. The goal of resource theories is then to quantify this cost, and to consequently assign a price to every state of the system, from the most expensive to the free ones. Because of their very general structure, which only involves the set of states describing a quantum system and a given set of allowed operations for acting on such system, resource theories can be used to study many different branches of quantum physics, from entanglement theory [1–5] to thermodynamics [6–13], from asymmetry [14–16] to the theory of magic states [17–19]. Additionally, these theories can often be formulated within more abstract, axiomatic frameworks [20–26]. Thanks to the underlying common structure present in all the theories described within this framework, one can find general results which apply to all. For example, a resource theory may be equipped with a zeroth, second, and even third law, i.e., relations that regulate the different aspects of the theory, which are reminiscent of the Laws of Thermodynamics. In fact, we have that the zeroth law for resource theories states that there exists equivalence classes of free states, and that states from one of these classes are the only ones that can be freely added to the system without trivialising the theory [27]. The second law of resource theories states that some quantities, linked to the amount of resource contained in a system, never increase under the action of the allowed operations [28], and for reversible resource theories satisfying modest assumptions, this quantity is unique [29, 30] — an example of this is the free energy, which is a monotone in thermodynamics as it decreases in any cyclic process, and the local entropy for pure state entanglement theory. Finally, one might have a generalisation of the third law which places limitations on the time needed to reach a state when starting from another one, rather than simply telling us whether such transformation is possible or not [31]. With the present work, we aim to derive the first law for resource theories, and to do so we will have to extend the framework so as to include multiple resources. Multiple resources. It is often the case that many resources are needed to perform a given task. For example, any quantum computational scheme requires the input qubits to be pure, and the gates to create coherence, and therefore these two quantities, coherence and purity, are necessary resources for performing quantum computation. Thus, a possible approach to investigate quantum computation might consist in combining the resource theories of purity [7, 32] and coherence [33–35] together. Similarly, thermodynamics can be understood as a resource theory with multiple resources [36, 37], where in order to transform the state of the system we need both energy and information, or equivalently, work and heat. Other examples of theories in which multiple resources are considered can be found in the literature [36–44]. Given the success of resource theories to describe physical situations where only one resource is involved, it seems natural to ask the question whether the framework can be extended to the case in which more resources are involved. For example, it is known that the resource theoretic approach to thermodynamics allows us to derive a second law relation even in the case in which many (commuting, non-commuting) conserved quantities are present [45–49], and one can consider trade-offs of these [50]. We are thus interested in understanding if one can extend these results to other resource theories, and whether a first law of general resource theories exists. Contribution of this work. In this paper we present a framework for resource theories with multiple resources, introduced in Sec. 2. In our framework we first consider the different constraints and conservation laws that the model needs to satisfy, and for each of these constraints, we introduce the corresponding single-resource theory. Then, we define the class of allowed operations of the multi-resource theory as the set of maps lying in the intersection of all the classes of allowed operations of the single-resource theories. Due to this construction, we find that a multi-resource theory with m resources has at least m invariant sets (i.e., sets of states that are mapped into themselves by the action of the allowed operations of the theory), each of them corresponding to the set of free states of one of the m single-resource theories. In order to make the paper self-contained, in Sec. 2 we also provide a brief review of the resource theoretic formalism (see Ref. [51] for a more detailed review on this topic). We then discuss, in Sec. 3, reversibility for multi-resource theories. In a reversible theory, we have that the resources consumed to perform a given state transformation can always be completely recovered with the reverse transformation, so that no resource is ever lost. In single-resource theories, we can rephrase this notion of reversibility in terms of rates of conversion [30, 52–54], but for general multi-resource theories this is not always possible. As a result, we focus our study on multi-resource theories that satisfy an additional property, which we refer to as the asymptotic equivalence property [23, 36], see Def. 1 below. We show that, when a multi-resource theory satisfies the asymptotic equivalence property, there is a unique measure associated with each resource present in the theory. Furthermore, when the invariant 2 Figure 1: An example of a multi-resource theory is thermodynamics, where energy and information are resources which can be inter-converted. In the figure, we represent three different systems. The main system is a Szilárd box, i.e., a box which can be divided in two partitions, here containing a single particle of gas. We can either know in which side of the box the particle is, or we might not have this information (if, for instance, the partition is removed and the particle is free to move between sides). We additionally have a thermal reservoir surrounding the box, with a well-defined temperature T , and we have an ancillary system that we use to store energy (or work), which we refer to as the battery. We can then consider the following two processes. Left. Landauer’s erasure is the process of converting some of the energy contained in the battery, ∆W , into information, ∆I, which is then used to reset the state of the particle in the box (from completely unknown, 2I , to perfectly known, |0i, in this case). The conversion is realised using the thermal environment, and energy and information are exchanged at the rate kB T , which only depends on the properties (the temperature) of the reservoir. Right. In the other direction we can convert information, ∆I, into work, ∆W , at the same exchange rate. Information is extracted form the box, and converted using the thermal bath into energy, which is then stored in the battery. Here, we generalise the function of the thermal reservoir to other multi-resource theories, and we name this system the bank, since it allows for the exchange of one resource into another. sets of the theory satisfy some natural properties, we find that the unique measures are given by the (regularised) relative entropy distances from these sets, each of those associated with a different resource. Finally, we show that when a resource theory satisfies asymptotic equivalence, it is also reversible in the sense that resources are never lost during a state transformation, and they can be recovered. This result can be seen as the extension of what has already been shown for reversible single-resource theories [28–30, 52]. In Sec. 4 we address the question of whether it is possible to exchange resources. We consider the case in which different resources are individually stored in separate systems, which we call batteries. Then, we investigate under which conditions it is possible to find an additional system, which we refer to as a bank 1 , that allows us to reduce the amount of resource contained in one battery while simultaneously increasing the amount of resource in another battery. During such conversion, we ask the bank not to change its properties – with respect to a specific measure defined in Eq. (37) – so as to be able to use this system again. For example, in thermodynamics the thermal bath plays the role of the bank, as it allows us to exchange energy for information and vice versa, see Fig. 1. In order to study interconversion, we demand the invariant sets of the theory to satisfy an ”additivity” condition, which is satisfied by some resource theories, for example by thermodynamics and purity theory. We find that a multi-resource theory needs to have an empty set of free states for a bank to exist, and when this condition is satisfied we derive an interconversion relation, see Thm. 8, which defines the rates at which resources are exchanged. We additionally show that, when the agent is allowed to use batteries and bank, they can perform any state transformation using variable amounts of resources. Indeed, since the agent can use the bank to inter-convert between resources, they can decide to invest an higher amount of one resource to save on the others. This freedom is reflected in our framework by a single relation, the first law of resource theories, which connects the different resources, each of them weighted by the corresponding exchange rate, to the change of a particular monotone between the initial and final state of the system, see Cor. 12. This equality is a generalisation of the first law of thermodynamics, where the sum of the work performed on the system and the heat absorbed from the environment is equal to the change in internal energy of the system. In fact, the first law of thermodynamics can be understood as equating various relative entropy distances which quantify different types of resources, as we discuss at the beginning of Sec. 4. Finally, in Sec. 5 we provide two examples of multi-resource theories which admit an interconversion relation between their resources. The first example concerns thermodynamics of multiple conserved quantities, for which the interconversion 1 We apologise in advance for introducing this terminology into the field of resource theories, but the banks considered here exchange resources without charging interest or fees, and are thus more akin to community cooperative banks than their more exploitative cousins. 3 of resources was shown in Ref. [45]. The second example concerns the theory of local control under energy restrictions. Here we consider a system with a non-local Hamiltonian, and we assume that the experimentalists acting on this system only have access to a portion of the system. In this scenario, the entanglement between the different portions of the system and the overall energy of the global system are the main resources of the theory, and we study under which conditions we can inter-convert energy and entanglement. For a summary of how to apply our work to an arbitrary resource theory, see the flowchart in Fig. 7. 2 Framework for multi-resource theories Let us now introduce the framework for multi-resource theories. A multi-resource theory is useful when we need to describe a physical task or process which is subjected to different constraints and conservation laws. The first step consists in associating each of these constraints with a single-resource theory, whose class of allowed operations satisfies the specific constraint or conservation law. The multi-resource theory is then obtained by defining its class of allowed operations as the intersection between the sets of allowed operations of the different single-resource theories previously defined. In this way, we are sure of acting on the quantum system with operations that do not violate the multiple constraints imposed on the task. 2.1 Single-resource theory For simplicity, we restrict ourselves to the study of finite-dimensional quantum systems. Therefore, the system under investigation is described by a Hilbert space H with dimension d. The state-space of this quantum system is given by the set of density operators acting on the Hilbert space, S (H) = {ρ ∈ B (H) | ρ ≥ 0, Tr [ρ] = 1}, where B (H) is the set of bounded operators acting on H. A single-resource theory for the quantum system under examination is defined through a class of allowed operations C, that is, a constrained set of completely positive maps acting on the state-space S (H)2 [29]. The constraints posed on the set of allowed operations are specific to the resource theory under consideration. For example, in the theories that study entanglement it is often the case that we constrain the set of allowed operations to be composed by the maps that are local, and only make use of classical communication [1]. In asymmetry theory, instead, we only allow the maps whose action is covariant with respect to the elements of a given group [14]. Furthermore, in the resource theoretic approach to thermodynamics we can, without loss of generality, constrain this set to those operations, known as Thermal Operations, which preserve the energy of a closed system, and can thermalise the system with respect to a background temperature [6, 10, 11, 55]. Once the set of allowed operations is defined, it is usually possible to identify which states in S (H) are resourceful, and which ones are not. In particular, the set of free states for a single-resource theory, F ⊂ S (H), is composed of those states that can always be prepared using the allowed operations, no matter the initial state of the system. Mathematically, this set of states is defined as F = {σ ∈ S (H) | ∀ ρ ∈ S (H) , ∃ E ∈ C : E(ρ) = σ} . (1) For example, in entanglement theory the free states are the separable states, in asymmetry theory they are the ones that commute with the elements of the considered group, and in thermodynamics they are the thermal states at the background temperature. An invariant set is a set of states that is preserved under action of any allowed operation. From the definition of free states in Eq. (1), it is easy to show that F is an invariant set, and we write this as E(F) ⊆ F for all E ∈ C. It is worth noting that while the set of free states is invariant, the opposite clearly does not need to be true. In particular, when we study multi-resource theory, we will see that several invariant sets can be found, and still there might be no free set for the theory. Due to the invariant property of free states, we can also define the class of allowed operation in a different way. Instead of considering the specific constraints defining the set of allowed operations C, we can simply assume that this set is a subset of the bigger class of completely positive and trace preserving (CPTP) maps C˜ = {E : B (H) → B (H) | E (F) ⊆ F} , (2) that is, the set of maps for which the free states F form an invariant set. It is worth noting that C is often a proper ˜ For example, in entanglement theory, we have that C might be composed by local operations and classical subset of C. 2 Although the operations we consider are endomorphisms of a given state space, our formalism is still able to describe the general case in which the agent modifies the quantum system. If the agent’s action transforms the state of the original system, associated with H1 , into the state of a final system H2 , we can model this action with a map acting on the state space of H = H1 ⊗ H2 . Suppose the operation maps ρ1 ∈ S (H1 ) into σ2 ∈ S (H2 ). Then, the map acting on S (H) takes the state ρ1 ⊗ γ2 and outputs the state γ1′ ⊗ σ2 , where γ1 and γ2′ are free states for the systems described by H2 and H1 , respectively. 4 communication (LOCC), which is a proper subset of the set of all quantum channels which preserve the separable states. Indeed, the map that swaps between the local states describing the quantum system is clearly not LOCC, but it preserves separable states [56]. We can also extend the single-resource theory to the case in which we consider n ∈ N copies of the quantum system. The class of allowed operations, which in this case we refer to as C (n) , is still defined by the same constraints, but now acts on S (H⊗n ), the state-space of n copies of the system. For example, in the resource theory of thermodynamics with Thermal Operations we have that the energy of a closed system needs to be exactly conserved. For a single system, this implies that the operations need to commute with the Hamiltonian H (1) . For n non-interacting copies of the system, Pn (1) instead, the operations commute with the global Hamiltonian Hn = i=1 Hi . Within the state-space S (H⊗n ), we can find the set of free states, F (n) ⊂ S (H⊗n ). It is worth noting that the set of free states for n copies of the system is such that F ⊗n ⊆ F (n) , that is, it contains more states than just the tensor product of n states in F. This is the case, for example, of entanglement theory, where among the free states for two copies of the system we can find states that are locally entangled, since each agent is allowed to entangle the partitions of the system they own. On the contrary, the two sets coincide for any n ∈ N for the resource theory of thermodynamics, where the free state is the Gibbs state of a given Hamiltonian. Anyway, it is still the case that F (n) is invariant under the class C (n) , and therefore we can think of the set of allowed operations acting on n copies of the system as a subset of the bigger set of CPTP maps n   o

C˜(n) = En : B H⊗n → B H⊗n | En F (n) ⊆ F (n) . (3) Thus, in order to completely define a single-resource theory that can be extended to many copies, one needs the sequence of all sets of allowed operations C (n) , where n ∈ N is the number of copies of the system the maps are acting on. It is worth noting that the allowed operations we have introduced keep the number of copies of the system fixed, see Eq. (3). Indeed, we only consider these maps because, when the number of input and output systems of a quantum channel changes, the internal structure of the channel involves the discarding (or the addition) of some of these systems. However, in a (reversible) resource theory, one can perform such operations only if the amount of resources is kept constant. This is certainly possible if we are to add or trace out some free states of the theory (which do not contain any resource), but as we will see in the next section, multi-resource theory not always have any free states. For this reason, we decide to only focus on maps that conserve the number of systems, even for single-resource theories. We can now address the problem of quantifying the amount of resource associated with different states of the quantum system. In resource theories, a resource quantifier is called monotone. This object is a function f from the state-space S (H) to the set of real numbers R, which satisfies the following property, f (E(ρ)) ≤ f (ρ) , ∀ ρ ∈ S (H) , ∀ E ∈ C. (4) The above inequality can be interpreted as a “second law” for the resource theory, since there is a quantity (the monotone) that never increases as we act on the system with allowed operations. In the thermodynamic case, in fact, we know that the Second Law of Thermodynamics imposes that the entropy of a closed system can never decrease as time goes by. We can extend the definition of monotones to the case in which we consider n copies of the system. In this case, the function f maps states in S (H⊗n ) into R, and an analogous relation to the one of Eq. (4) holds, this time for states in S (H⊗n ) and the set of allowed operations C (n) . Finally, we can also define the regularisation of a monotone f as f (ρ⊗n ) , n→∞ n f ∞ (ρ) = lim (5) where ρ ∈ S (H), and ρ⊗n ∈ S (H⊗n ). Notice that, given a generic monotone f , we need the above limit to exist and be finite in order to define its regularisation. For each resource theory there exists several monotones, and we can always build one out of a contractive distance [52]. Consider the distance C (·, ·) : S (H) × S (H) → R such that C (E(ρ), E(σ)) ≤ C (ρ, σ) , ∀ ρ, σ ∈ S (H) , ∀ E CPTP map. (6) Then, a monotone for the single-resource theory with allowed operations C and free states F is MF (ρ) = inf C (ρ, σ) , σ∈F (7) where it is easy to show that MF satisfies the property of Eq. (4), which follows from the fact that F is invariant under the set of allowed operations C, and from the contractivity of C (·, ·) under any CPTP map. A specific example of a 5 monotone obtained from a contractive distance is the relative entropy distance from the set F. Consider two states ρ, σ ∈ S (H), such that supp (ρ) ⊆ supp (σ). Then, we define the relative entropy between these two states as D(ρ k σ) = Tr [ρ (log ρ − log σ)] . (8) EF (ρ) = inf D(ρ k σ), (9) The relative entropy is contractive under CPTP maps [57], and even if it does not satisfy all the axioms to be a metric3 over S (H), we can still obtain a monotone out of this quantity, building it as in Eq. (7). This monotone is σ∈F and is known as the relative entropy distance from F. When the separable states form the set F, for example, the monotone is the relative entropy of entanglement [3]. It is worth noting that, in order for EF to be well-defined, the set F has to contain at least one full-rank state. 2.2 Multi-resource theory Let us consider the case in which we can identify in the theory a number m > 1 of resources, which can arise from some conservation laws, or from some constraints. We now introduce a multi-resource theory with these m resources. The quantum system under investigation is the same as in the previous section, described by the states in the state-space S (H). For the i-th resource of interest, where i = 1, . . . , m, we consider the corresponding single-resource theory Ri , defined by the set of allowed operations Ci acting on the state-space S (H). We denote the set of free states of this single-resource theory as Fi ⊂ S (H), and we recall that any allowed operation in Ci leaves this set invariant. Therefore, we can consider the class of allowed operation as a subset of the set of CPTP maps C˜i = {Ei : B (H) → B (H) | Ei (Fi ) ⊆ Fi } . (10) We can also extend the resource theory Ri to the case in which we consider more than one copy of the system, following (n) the same procedure used in the previous section. Then, the class of allowed operations Ci acting on n copies of the (n) system is a subset of the set of operations which leave Fi ⊂ S (H⊗n ) invariant, see Eq. (3). Once all the single-resource theories Ri ’s are defined, together with their sets of allowed operations, we can build the multi-resource theory Rmulti for the quantum system described by the Hilbert space H. The set of allowed operations for this theory is given by the maps contained in the intersection4 between the classes of allowed operations of the m single-resource theories, that is m Cmulti = ∩ Ci . (11) i=1 ˜ Notice that, alternatively, one can define the set of allowed operations Cmulti as a subset of the bigger set ∩m i=1 Ci , where C˜i is the set of all the CPTP maps for which Fi is invariant, see Eq. (10). When n copies of the system are considered, (n) the class of allowed operations for the multi-resource theory, Cmulti , is obtained by the intersection between the sets of (n) (n) (n) allowed operations Ci of the different single-resource theories, that is, Cmulti = ∩m i=1 Ci . We can now consider the invariant sets of this multi-resource theory. Clearly, each set of free states Fi associated with the single-resource theory Ri is an invariant set for the class of operations Cmulti . However, it is worth noting that the states contained in the Fi ’s might not be free when the multi-resource theory is considered, where a free state is (as we pointed out in the previous section) a state that does not contain any resource and can be realised using the allowed operations. Indeed, the states contained in the set Fi might be resourceful states for the single-resource theory Rj , and therefore we would not be able to realise such states with the class of operations Cmulti . In Fig. 2 we show the different configurations for the invariant sets of a multi-resource theory with two resources. While in the left and central panels the theory has free states, in the right panel no free states can be found, a noticeable difference from the framework for single-resource theories. The multi-resource theory Rmulti also inherits the monotones of the single-resource theories that compose it. This follows trivially from the choice we made in defining the class of allowed operations Cmulti , see Eq. (11). Furthermore, other monotones, that are only valid for the multi-resource theory, can be obtained from the ones inherited from the single-resource theories Ri ’s. For example, if fi is a monotone for the single-resource theory Ri , and fj is a monotone for 3 The relative entropy is non-negative for any two inputs, and zero only when the two inputs coincide, but it is not symmetric, nor does it satisfy the triangular inequality. 4 While other multi-resource theory constructions can be imagined, the one we use in this paper provides the certainty that no resource can be created out of free states. 6 Figure 2: The structure of the invariant sets for a multi-resource theory with two resources. For theories with m > 2 resources, the structure of the invariant sets can be obtained by composing the three fundamental scenarios presented here. Left. The invariant set F2 is a subset of F1 . This multi-resource theory has a set of free states, which coincides with F2 . An example of such a theory is that of coherence [34] and purity [32], where the invariant sets are incoherent states with respect to a given basis and the maximally-mixed state, respectively. Centre. The two invariant sets intersect each other. This theory has a set of free states which coincides with the intersection, F1 ∩ F2 . An example of multi-resource theory with this structure concerns tripartite entanglement for systems A, B, and C. The allowed operations of this theory are defined by the intersection of the operations associated with the theories of bipartite entanglement for systems AB and C, systems AC and B, and systems A and BC. Notice that this theory does not coincide with the theory of tripartite LOCC, since some of the free states are entangled [58]. Right. The two invariant sets are separated. Consequently, the theory does not have any free states. In this situation, one can find an interconversion relation between the resources, as shown in Sec. 4.3. An example of a multi-resource theory with this structure is thermodynamics of closed systems. If the agent does not have perfect control on the reversible operations they implement, and the closed system is coupled to a sink of energy (an ancillary system which can only absorb energy), then the allowed operations are given by the intersection between the set of mixtures of unitary operations, and the set of average-energy-non-increasing maps. In this case, the maximally-mixed state and the ground state of the Hamiltonian are the two invariant sets of the theory. Notice that the set of energy-preserving unitary operations, considered in Ref. [36], is a subset of this bigger set. the theory Rj , their linear combination, where the linear coefficients are positive, is a monotone for the multi-resource theory Rmulti . Interestingly, in Sec. 4 we will see that a specific linear combination of the monotones of the different single-resource theories plays an important role in the interconversion of resources. Examples of multi-resource theories that can be described within our formalism are already present in the literature. In Ref. [42], for instance, the authors study the problem of state-merging when the parties can only use local operations and classical communication (LOCC), and they restrict the local operations to be incoherent operations, that is, operations that cannot create coherence (in a given basis). This theory coincides with the multi-resource theory obtained from combining two single-resource theories, the one of entanglement, whose set of allowed operations only contains quantum channels built out of LOCC, and the one of coherence, whose set of allowed operations only contains maps which do not create coherence. In this case, the structure of the invariant sets is given by the central panel of Fig. 2. Another example is the one of Ref. [36], where thermodynamics is obtained as a multi-resource whose class of allowed operations is a subset of the one obtained by taking the intersection of energy-non-increasing maps (operations which do not increase the average energy of the quantum system, see Sec. 3.4), and mixtures of unitary operations. In this case the resources are, respectively, average energy and entropy, and the structure of the invariant sets is given by the right panel of Fig. 2, where F1 coincides with the ground state of the Hamiltonian (if the Hamiltonian is non-degenerate), while F2 coincides with the maximally-mixed state. Other examples of multi-resource theories can be found, and in future work [59] we will present the general properties of multi-resource theories with different invariant sets structures. 3 Reversible multi-resource theories In this section we study reversibility in the context of multi-resource theories. We first introduce a property, which we refer to as the asymptotic equivalence property, for multi-resource theories. We then show that, when a resource theory satisfies this property, we can (uniquely) quantify the amount of resources needed to perform an asymptotic state transformation. This allows us to introduce the notion of batteries, i.e., systems where each individual resource can be stored, and to keep track of the changes of the resources during a state transformation. Furthermore, we show that a theory which satisfies the asymptotic equivalence property is also reversible, that is, the amount of resources exchanged with the batteries during an asymptotic state transformation mapping ρ into σ is equal, with negative sign, to the 7 amount of resources exchanged when mapping σ into ρ. Finally, we show that, when the invariant sets of the theory satisfy some general properties, and the theory satisfies asymptotic equivalence, then the relative entropy distances from the different invariant sets are the unique measures of the resources. This result is a generalisation of the one obtained in single-resource theories, see Ref. [29, 30, 52]. 3.1 Asymptotic equivalence property Let us consider the multi-resource theory Rmulti introduced in Sec. 2.2. This theory has m resources, its set of allowed operations Cmulti is defined in Eq. (11), and its invariant sets are the Fi ’s, that is, the sets of free states of the different single-resource theories composing it. The multi-resource theory Rmulti is reversible if the amount of resources spent to perform an asymptotic state transformation is equal to the amount of resources gained when the inverse state transformation is performed. In this way, performing a cyclic state transformation over the system (which recovers its initial state at the end of the transformation) never consumes any of the m resources initially present in the system. For single-resource theory, the notions of reversibility and state transformation are usually associated with the rates of conversion. Suppose that we are given n ≫ 1 copies of a state ρ ∈ S (H), and we want to find out the maximum number of copies of the state σ ∈ S (H) that can be obtained by acting on the system with the allowed operations. If k is the maximum number of copies of σ achievable, then the rate of conversion is defined as R(ρ → σ) = nk , see Def. 13 in appendix A. Reversibility is then defined by asking that, for all ρ, σ ∈ S (H), the rates of conversion associated to the forward and backward state transformations are such that R(ρ → σ)R(σ → ρ) = 1, see Def. 14 in the appendix. It is worth noting that, when considering rates of conversion, one is in general allowed to trace out part of the system, or to add ancillary systems in a free state. For example, being able to map n copies of ρ into k copies of σ, with n < k, implies that we have the possibility to add k − n copies in a free state to the initial n copies of ρ, and to act globally to produce k copies of σ. This is certainly possible for single-resource theories, where free states always exists, but not always possible for multi-resource theories, see the invariant set structure of the right panel of Fig. 2. Due to the possible absence of free states in a generic multi-resource theory, we first need to introduce the following definition5 , which will then allow us to study reversibility. m Definition 1. Given the multi-resource theory Rmulti , consider a set of monotones {fi }i=1 , where each fi is a monotone for the corresponding single-resource theory Ri , and whose regularisation is not identically zero. Then, we say that Rmulti satisfies the asymptotic equivalence property if, for all ρ, σ ∈ S (H), we have that the following two statements are equivalent, • fi∞ (ρ) = fi∞ (σ) for all i = 1, . . . , m.  • There exist a sequence of maps Ẽn : S (H⊗n ) → S (H⊗n ) lim n→∞ n such that Ẽn (ρ⊗n ) − σ ⊗n 1 = 0, (12)  as well as a sequence of maps performing the reverse process. The maps Ẽn are defined as h i Ẽn (·) = TrA En (· ⊗ ηn(A) ) , (13) where A is an ancilla composed by a sub-linear number o(n) of copies of the system, and it is described by an
(A) (A) (n+o(n)) arbitrary state ηn ∈ S H⊗o(n) , such that fi (ηn ) = o(n) for all i = 1, . . . , m. The map En ∈ Cmulti is an allowed operation of the multi-resource theory. i h√ Here, fi∞ is the regularisation of the monotone fi , k · k1 is the trace norm, define as kOk1 = Tr O† O for O ∈ B (H), and we are using the little-o notation, where g(n) = o(n) means limn→∞ g(n) n = 0. An example of a multi-resource theory that satisfies the above property is thermodynamics (even in the case in which multiple conserved quantities are present), as shown in Refs. [36, 37]. In this example the monotones for which asymptotic equivalence is satisfied are the average energy and the Von Neumann entropy of the system. Notice that the above property implicitly assumes that the monotones fi ’s can be regularised, that is, that the limit involved in 5 Notice that this definition is analogous to the notion of “seed regularisation” in Ref. [23, Sec. 6], although in our case we are solely focused on reversible transformations and on equalities of monotones. 8 the regularisation is always finite. Furthermore, in this property we are allowing the agent to act over many copies of the system with more than just the set of allowed operations; we assume the agent to be able to use a small ancillary system, sub-linear in the number of copies of the main system. Roughly speaking, the role of this ancilla is to absorb the fluctuations in the monotones fi∞ ’s during the asymptotic state transformation. It is important to notice that this ancillary system only contributes to the transformation by exchanging a sub-linear amount of resources. Thus, its contribution per single copy of the system is negligible when n ≫ 1, which justifies the use of this additional tool. The asymptotic equivalence property essentially states that the multi-resource theory can reversibly map between any two states with the same values of the monotones fi ’s. In particular, transforming between such two states comes at no cost, since we can do so by using the allowed operations of the theory, Cmulti . It is worth noting that, when the number of considered resources is m = 1, that is, our theory is a single-resource theory, the notion of asymptotic equivalence given in Def. 1 corresponds to the one given in terms of rates of conversion, Def. 14. We prove this equivalence in appendix A, see Thm. 17. Finally, notice that the asymptotic equivalence property does not say anything about the state transformations which involve states with different values of the monotones fi ’s. To include these transformations in the theory, we will have to add a bit more structure to the current framework, by considering some additional systems that can store a single type of resource each, which we refer to as batteries [60]. 3.2 Quantifying resources with batteries When a multi-resource theory satisfies the asymptotic equivalence property of Def. 1, we have that states with the same values of a specific set of monotones can be inter-converted between each others. In this section, we show that these monotones actually quantify the amount of resources contained in the system. To do so, we need to introduce some additional systems, which can only store a single kind of resource each, and can be independently addressed by the agent. These additional systems are referred to as batteries. Let us suppose that the multi-resource theory Rmulti satisfies the m asymptotic equivalence property with respect to the set of monotones {fi }i=1 , and that the quantum system over which the theory acts is actually divided into m + 1 partitions. The first partition is the main system S, and the remaining ones are the batteries Bi ’s. Then, the Hilbert space under consideration is H = HS ⊗ HB1 ⊗ . . . ⊗ HBm . Let us now introduce some properties the monotones need to satisfy in order for the resources to be quantified in a meaningful way. Since each resource is associated to a different monotone, we can forbid a battery to store more that one resource by constraining the set of states describing it to those ones with a fixed value of all but one monotones. M1 Consider two states ωi , ωi′ ∈ S (HBi ) describing the battery Bi . Then, the value of the regularisation of any monotone fj (where j 6= i) over these two states is fixed, fj∞ (ωi′ ) = fj∞ (ωi ), ∀j 6= i. (14) In this way, the battery Bi is only able to store and exchange the resource associated with the monotone fi . It would be natural to extend the condition of Eq. (14) to the monotones themselves, rather than to use their regularisations. However, this stronger condition is not required in our proofs. Furthermore, in order to address each battery as an individual system, we ask the value of the monotones over the global system to be given by the sum of their values over the individual components, M2 The regularisations of the monotones fi ’s can be separated between main system and batteries, fi∞ (ρ ⊗ ω1 ⊗ . . . ⊗ ωm ) = fi∞ (ρ) + fi∞ (ω1 ) + . . . + fi∞ (ωm ), (15) where ρ ∈ S (HS ) is the state of the main system, and ωi ∈ S (HBi ) is the state of the battery Bi . The above property allows us to separate the contribution given by each subsystem to the amount of i-th resource present in the global system. We then ask the monotones to satisfy an additional property, so as to simplify the notation. Namely, we ask the zero of each monotone fi to coincide with its value over the states in Fi , (n) M3 For each n ∈ N and i ∈ {1, . . . , m}, the monotone fi is equal to 0 when computed over the states of Fi , that is fi (γi, n ) = 0, 9 (n) ∀ γi, n ∈ Fi . (16) This property serves as a way to “normalise” the monotone, setting its value to 0 over the states that were free for the specific single-resource theory the monotone is linked to. Notice that property M3 is trivially satisfied by any monotone after a translation. The next property requires that tracing out part of the system does not increase the value of the monotones fi ’s, M4 For all n, k ∈ N where k < n, the monotones fi ’s are such that fi (Trk [ρn ]) ≤ fi (ρn ),
where ρn ∈ S (H⊗n ) and Trk [ρn ] ∈ S H⊗n−k . ∀ i ∈ {1, . . . , m} . (17) This property implies that the resources contained in a system cannot increase if we discard/forget part of it. Additionally, we want the monotones to satisfy sub-additivity, namely M5 For all n, k ∈ N, the monotones fi ’s are such that fi (ρn ⊗ ρk ) ≤ fi (ρn ) + fi (ρk ),
where ρn ∈ S (H⊗n ) and ρk ∈ S H⊗k . ∀ i ∈ {1, . . . , m} . (18) That is, the amount of resources contained in two uncorrelated systems, when measured on the two systems independently, is bigger or equal to the value measured on the two systems together. This is the case, for example, of the relative entropy of entanglement [61]. Another property we require is that the monotones fi ’s scale linearly in the number of systems considered, M6 Given any sequence of states {ρn ∈ S (H⊗n )}, the monotones fi ’s are such that fi (ρn ) = O(n), ∀ i ∈ {1, . . . , m} . (19) where we are using the big-O notation. Therefore, when this property is satisfied we have that the resources scale extensively. Furthermore, the monotones that satisfy this property are the ones that can also be regularised (although their regularisation might be identically zero on the whose state space). The last property we ask concerns a particular kind of continuity the monotones need to satisfy, M7 The monotones fi ’s are asymptotic continuous, that is, for all sequences of states ρn , σn ∈ S (H⊗n ) such that kρn − σn k1 → 0 for n → ∞, where k · k1 is the trace norm, we have |fi (ρn ) − fi (σn )| → 0 for n → ∞, n ∀ i ∈ {1, . . . , m} . (20) This notion of asymptotic continuity coincides with condition (C2) given in Ref. [62]. This property implies that the monotones are physically meaningful, since their values over sequences of states converge if the sequences of states converge asymptotically. In Thm. 4 we show that, when the monotones satisfy asymptotic continuity, they are the unique quantifiers of the amount of resources contained in the main system. We can now use this formalism to discuss how resources can be quantified in a multi-resource theory, and consequently how the asymptotic equivalence property implies that the theory is reversible. Let us consider any two states ρ, σ ∈ S (HS ), that do not need to have the same values for the monotones fi ’s. Then, we choose the initial and final states of each battery Bi such that ′ fi∞ (ρ ⊗ ω1 ⊗ . . . ⊗ ωm ) = fi∞ (σ ⊗ ω1′ ⊗ . . . ⊗ ωm ), ∀ i = 1, . . . , m, (21) where ωi , ωi′ ∈ S (HBi ), for i = 1, . . . , m. Under these conditions, due to the asymptotic equivalence property of Rmulti , we have that the two global states can be asymptotically mapped one into the other in a reversible way, using the allowed operations of the theory, that is asympt ′ ρ ⊗ ω1 ⊗ . . . ⊗ ωm ←−−→ σ ⊗ ω1′ ⊗ . . . ⊗ ωm , (22) asympt where the symbol ←−−→ means that there exists two allowed operations that maps n ≫ 1 copies of the state on the lhs into the state of the rhs, and viceversa, while satisfying the condition in the second statement of Def. 1. 10 We can now properly define the notion of resources in this framework. The resource associated with the monotone fi is the one exchanged by the battery Bi during the state transformation. This quantity is defined as the difference in monotone fi∞ between the final and initial state of the battery Bi . For the transformation of Eq. (22), the amount of the i-th resource exchanged is defined as ∆Wi := fi∞ (ωi′ ) − fi∞ (ωi ), (23) where ωi , ωi′ ∈ S (HBi ) are, respectively, the initial and final state of the battery Bi . Then, the amount of the i-th resource ∆Wi needed to map the state of the main system ρ into σ can be computed. Proposition 2. Consider a theory Rmulti with m resources and allowed operations Cmulti , equipped with batteries B1 , m . . ., Bm . If the theory satisfies the asymptotic equivalence property with respect to the set of monotones {fi }i=1 , and these monotones satisfy the properties M1 and M2, then the amount of i-th resource needed to perform the state transformation ρ → σ is equal to ∆Wi = fi∞ (ρ) − fi∞ (σ). (24) ′ Proof. Due to asymptotic equivalence, a transformation mapping the global state ρ⊗ω1 ⊗. . .⊗ωm into σ ⊗ω1′ ⊗. . .⊗ωm exists iff the conditions in Eq. (21) are satisfied. For a given i, using the property M2 of the monotone fi , we can re-write the condition as ′ fi∞ (ρ) + fi∞ (ω1 ) + . . . + fi∞ (ωm ) = fi∞ (σ) + fi∞ (ω1′ ) + . . . + fi∞ (ωm ). (25) Then, we can use the property M1, which guarantees that the only systems for which fi changes are the main system and the battery Bi . Thus, we find that fi∞ (ρ) + fi∞ (ωi ) = fi∞ (σ) + fi∞ (ωi′ ) . (26) By rearranging the factors in the above equation, and using the definition of ∆Wi given in Eq. (23), we prove the proposition. It is now easy to show that, if Rmulti satisfies the asymptotic equivalence property, any state transformation on the main system S is reversible. Indeed, from Eq. (24) it follows that the amount of resources used to map the state of this system from ρ to σ is equal, but with negative sign, to the amount of resources used to perform the reverse transformation, from σ to ρ. Therefore, any cyclic state transformation over the main system leaves the amount of resources contained in the batteries unchanged. The above formalism also provides us with a way to quantify the amount of resources contained in the main system. Indeed, if the system is described by the state ρ ∈ S (HS ), the amount of i-th resource contained in the system is given by the amount of i-th resource exchanged, ∆Wi , while mapping ρ into a state contained in Fi . Using property M3 and Prop. (24) it follows that Corollary 3. Consider a theory Rmulti with m resources and allowed operations Cmulti , equipped with batteries B1 , . . ., m Bm . If the theory satisfies the asymptotic equivalence property with respect to the set of monotones {fi }i=1 , and these monotones satisfy the properties M1, M2, and M3, then the amount of the i-th resource contained in the main system, when described by the state ρ, is given by fi∞ (ρ). It is worth noting that, in general, one cannot extract all the resources contained in the main system at once. Indeed, this is only possible when the multi-resource theory contains free states, like for example in the cases depicted in the left and centre panels of Fig. 2. Being able to quantify the amount of resources contained in a given quantum state allows us to represent the whole state-space of the theory in a resource diagram [23, 36]. In fact, from the definition of asymptotic equivalence it follows that, if two states contain the same amount of resources, i.e., if they have the same values of the monotones fi∞ ’s, then we can map between them using the allowed operations Cmulti . This property implies that we can divide the entire state-space into equivalence classes, that is, sets of states with same value of the m monotones (where we recall that m is the number of resources, or batteries, in the theory). Then, we can represent each equivalence class as a point in a m-dimensional diagram, with coordinates given by the values of the monotones. By considering all the different equivalence classes, we can finally represent the state-space of the main system in the diagram, see for example Fig. 3, where the state-space of a two-resource theory is shown. 11 Figure 3: In the figure we represent the state-space S (H) of a multi-resource theory Rmulti with two resources. In order for the diagram to be a meaningful representation of this state-space, we need the theory to satisfy the asymptotic equivalence property of Def. 1 with respect to the monotones f1 and f2 . In fact, when the theory satisfies this property we can divide S (H) into equivalence classes of states with the same value of the regularised monotones f1∞ and f2∞ , which become the abscissa and ordinate of the diagram. The state-space of the theory is represented by the blue region, and the yellow segments are the invariant sets F1 and F2 . These sets are disjoint, since the two segments do not intercept each other, and the resource theory Rmulti thus corresponds to the one depicted in the right panel of Fig. 2. Two equivalence classes, respectively associated to the states ρ and σ, are represented in the diagram. The amount of resources that is exchanged when transforming from one state to the other, Eq. (24), is given in the diagram by the difference between the coordinates of these two points. 3.3 Reversibility implies a unique measure for each resource We now show that, when a multi-resource theory satisfies the asymptotic equivalence property with respect to a set of m monotones {fi }i=1 , and these monotones satisfy the properties M1 – M7, then there exists a unique quantifier for each resource contained in the main system. In particular, when the i-th resource is considered, this quantifier coincides with fi∞ (modulo a multiplicative factor which sets the scale). In the previous section, Cor. 3, we showed that a quantifier exists if the monotones satisfy the first three properties M1, M2, and M3. However, when these monotones are also asymptotic continuous, property M7, we can prove that they uniquely quantify the amount of resources contained in the main system. This means that one cannot find other monotones gi ’s that give the same equivalence classes of the fi ’s, but order them in a different way. Asymptotic continuity was used in Ref. [30] to show that the relative entropy distance from the set of free states of a reversible single-resource theory is the unique measure of resource. Thus, the following theorem (whose proof can be found in appendix D.1) can be understood as a generalisation of the above result to multi-resource theories, Theorem 4. Consider the resource theory Rmulti with m resources, equipped with the batteries Bi ’s, where i = 1, . . . , m. m Suppose the theory satisfies the asymptotic equivalence property with respect to the set of monotones {fi }i=1 . If these monotones satisfy the properties M1 – M7, then the amount of i-th resource contained in the main system S is uniquely quantified by the regularisation of the monotone fi (modulo a multiplicative constant). In particular, we now consider the case of a multi-resource theory Rmulti that satisfies the asymptotic equivalence property of Def. 1 with respect to the relative entropy distances from the invariant sets Fi ’s. We refer to the relative entropy distance from the set Fi as EFi , whose definition can be found in Eq. (9). Since the multi-resource theory we consider is equipped with batteries, and we want to be able to measure the amount of resources they contain independently of the other subsystems, we ask the invariant sets to be of the form Fi = Fi,S ⊗ Fi,B1 ⊗ . . . ⊗ Fi,Bm , (27) so that the main system S and the batteries Bi ’s all have they own independent invariant sets. We now show that, under very general assumptions over the properties of the invariant sets, the regularised relative entropy distances from these sets are the unique quantifiers of the resources, provided that these quantities are not identically zero over the whole 12 state space6 . This result follows from Thm. 4, and from the fact that these monotones satisfy the properties M1, M2, m M3, and M7 listed in the previous sections. The properties we are interested in for the invariant sets {Fi }i=1 of the theory are very general, and they are satisfied in most of the known resource theories, see Refs. [53, 63]. F1 The sets Fi ’s are closed sets. F2 The sets Fi ’s are convex sets. F3 Each set Fi contains at least one full-rank state. (k) F4 The sets Fi ’s are closed under tensor product, that is, Fi (n) (n+k) ⊗ Fi ⊆ Fi for all i = 1, . . . , m. h i (n) (n−k) F5 The sets Fi ’s are closed under partial tracing, that is, Trk Fi ⊆ Fi for all i = 1, . . . , m. Let us briefly comment on the above properties. Property F1 requires that any converging sequence in the set converges to an element in the set. This property is necessary for the continuity of the resource theory. Property F2, instead, tells us that we are allowed to forget the exact state describing the system, and therefore we can have mixture of states. Property F3 is necessary for the relative entropy distance to be physically natural, since the quantity D(ρ k σ), see Eq. (8), diverges when supp(ρ) 6⊆ supp(σ). Finally, property F4 implies that composing two systems that do not contain any amount of i-th resource is not going to increase that resource, and similarly, property F5 implies that forgetting about part of a system which does not contain resources will not create resources. When the invariant sets satisfy the above properties, the relative entropy distances EFi ’s satisfy the same properties discussed in the previous section, Proposition 5. Consider a resource theory Rmulti with m resources, equipped with the batteries Bi ’s, where i = 1, . . . , m. m Suppose the class of allowed operations is Cmulti and the invariant sets are {Fi }i=1 . If the invariant set Fi is of the form of Eq. (27), and it satisfies the properties F1 – F5, then the relative entropy distances from this set, EFi , is a regularisable monotone under the class of allowed operations, and it obeys the properties M1 – M7. This result is known in the literature, see Refs. [63, 64], but we nevertheless provide a proof in appendix D.2 to make ∞ has a positive value over the states that are the paper self-contained. By virtue of Thm. 4 it then follows that, if EF i not in Fi , then it is the unique quantifier of the amount of i-th resource contained in the system for a multi-resource theory that satisfies the asymptotic equivalence property with respect to these monotones. Furthermore, the amount of i-th resource used to map the main system from the state ρ into the state σ is then equal to ∞ ∞ ∆Wi = EF (ρ) − EF (σ), i i (28) for all i = 1, . . . , m. 3.4 Relaxing the conditions on the monotones There are situations, when we consider specific resource theories, in which some of the properties of the set of free states are not satisfied. In particular, we can have that the set of free states does not contain a full-rank state, that is, property F3 is not satisfied. An example would be the resource theory of energy-non-increasing maps for a system with Hamiltonian H, CH = {EH : B (H) → B (H) | Tr [EH (ρ)H] ≤ Tr [ρH] ∀ρ ∈ S (H)} . (29) An example of a subset of CH are unitary operations which commute with the Hamiltonian H (as in the resource theory of Thermal Operations). If the Hamiltonian H has a non-degenerate ground state |gi, then it is easy to show that this state is fixed, that is, EH (|gi hg|) = |gi hg| . (30)   † In fact, the operation Eg (·) = TrA S(· ⊗ |gi hg|A )S , where S is the unitary operation implementing the swap between the two states, belongs to CH and maps all states into the ground state. Thus, the set of free states does not contain a full-rank state, which implies that the relative entropy distance from this set is ill-defined, and it is not asymptotic continuous. Notice that the above argument holds even in the case of a degenerate ground state, with the difference that the invariant set would be composed by any state with support on this degenerate subspace. 6 An example where the regularised relative entropy from an invariant set is identically zero for all states in S (H) is the resource theory of asymmetry, see Ref. [15]. 13 We can introduce a different monotone for this kind of resource theory, that is, the average of the observable which is not increased by the allowed operations (modulo a constant factor). For the example we are considering, this monotone would be MH (ρ) = Tr [Hρ] − Eg , (31) where H is the Hamiltonian of the system, and Eg = Tr [H |gi hg|] is the Pn energy of the ground state. When n copies of the system are considered, we define the total Hamiltonian as Hn = i=1 H (i) , where H (i) is the Hamiltonian acting on the i-th copy. In this case, it is easy to show that this quantity is equal to 0 when evaluated on the fixed state |gi hg|, property M3, is monotonic under partial tracing, property M4, is additive (and therefore satisfies sub-additivity, property M5), and it scales extensively in the number of copies of the system, property M6. Furthermore, MH (·) is monotonic under the class of operations (by definition of the class itself), and it is asymptotic continuous, property M7, as shown in Prop. 21 in appendix D.2. If batteries are introduced, we can define the operator H is such a way that properties M1 and M2 are satisfied, see for example Sec. 5.1. Thus, if one (or more) of the monotones of the multi-resource theory is of the form given in Eq. (31), we have that the results of the previous section still apply, particularly Thm. 4. Furthermore, we can quantify the change in the resource associated with MH during a state transformation ρ → σ with Eq. (28), where the regularised relative entropy ∞ ∞ is replaced with the regularised monotone MH . As a side remark, we notice that the monotone MH can distance EF i be obtained as 1 MH (ρ) = lim D(ρ k τβ ), (32) β→∞ β   where τβ = e−βH /Z is the Gibbs state of the Hamiltonian H, and Z = Tr e−βH is the partition function of the system. 4 Bank states, interconversion relations, and the first law Within certain types of multi-resource theories, it is possible to inter-convert the resources stored in the batteries, i.e., to exchange one resource for another at a given exchange rate. Examples of resource interconversion can be found in thermodynamics, where Landauer’s principle [65] tells us that energy can be exchanged for information, while a Maxwell’s demon can trade information for energy [66]. In these examples, a thermal bath is necessary to perform the interconversion of resources. Indeed, in the following sections we show that in order to exchange between resources one always needs an additional system, which we refer to as a bank, that captures the necessary properties of thermal baths in thermodynamics, and abstracts them so that they can be applied to other resource theories. When such a system exists, we can pay a given amount of one resource and gain a different amount of another resource, with an exchange rate that only depends on the state describing the bank, see Thm. 8. Within the thermodynamic examples we are considering, this corresponds to exchanging one bit of information for one unit of energy, and vice versa. The exchange rate of these processes is proportional to the temperature of the thermal bath. During a resource interconversion the state of the bank should not change its main properties, so that we can keep using it indefinitely. Furthermore, we should always have to invest one resource in order to gain the other. For these reasons the bank is taken to be of infinite size, and its state to be passive, i.e., to always contain the minimum possible values of the resources. In fact, in the thermodynamic examples we are considering, the thermal bath has infinite size, and its state has maximum entropy for fixed energy, or equivalently minimum energy for fixed entropy [67]. We additionally show that the relative entropy distance from the set of bank states plays a fundamental role in quantifying the exchange rate at which resources are inter-converted, see Cor. 11. For instance, in thermodynamics this quantity is proportional to the Helmholtz free energy F = E − T S, which links together the two resources, internal energy E and information, which is proportional to −S. Through this quantity, one can define the exchange rate between energy and entropy, i.e., the temperature of the thermal bath T . Finally, we introduce a first-law-like relation for multi-resource theories. The first law consists of a single relation that regulates the state transformation of a system when the agent has access to a bank for exchanging the resources. In particular, this relation links the change in the relative entropy distance from the set of bank states over the main system to the amount of resources exchanged by the batteries during the transformations, see Cor. 12. In the example we are considering, this relation coincides with the First Law of thermodynamics, as it connects a change in the Helmholtz free energy ∆F of the system with the energy and information exchanged by the batteries, ∆F = ∆WE + T ∆WI , (33) where ∆WE is the energy exchanged by the first battery, ∆WI is the information exchanged by the second battery, and T is the background temperature, describing the state of the bank. 14 We now briefly discuss about the value that resources have in the different theories of thermodynamics, and the role of the first law in connecting these resources together. Let us first consider the single-resource theory of thermodynamics, where the system is in contact with an infinite thermal reservoir [10]. To perform a state transformation we need to provide only one kind of resource, known as athermality (∆F ), or work. Since the thermal reservoir is present, it is easy to get close to the free state, i.e. to the thermal state at temperature T , because we can simply thermalise the system with the allowed operations. However, it is difficult to go in the opposite direction, unless we use part of the athermality stored in a battery. For this reason, a positive increment in the athermality of the battery is considered valuable, while a negative change is considered a cost. Let us now move to the multi-resource theory of thermodynamics, whose allowed operations are energy-preserving unitary operations [36]. In this case, it is easy to see that negative and positive contributions of energy and information are equally valuable, since these two quantities are conserved by the set of allowed operations. As a result, the agent cannot perform state transformations in any direction without having access to the batteries. If we now allow the agent to use a thermal bath as a bank, and we keep the system decoupled from it (so that the agent cannot perform operations that thermalise the system for free), we find that changing a single resource, either energy or information, is enough to perform a generic state transformation on the system. In fact, we can always inter-convert one resource for the other with the bank, and then change the state of the system accordingly. Notice that, however, we still have that negative and positive change in one resource are equally valuable. Thus, it seems that the advantage that multi-resource theories provide over single-resource theories is that they make explicit which resources are used during a state transformation. And the link between the single resource and the multiple ones is given by the first law. In thermodynamics, for example, we have that the first law, Eq. (33), indicates that the amount of athermality ∆F needed to transform a state can be actually divided in two contributions, energy ∆WE and information ∆WI . Notice that all of these quantities can be understood in terms of the relative entropy distance to an invariant set of states. Athermality being measured by its relative entropy distance to the thermal state, information and energy being the relative entropy to the maximally mixed or ground state. As we will see, the generalised first law given in Eq. (45) also relates the relative entropy to the bank state, to the relative entropies to the invariant sets of the single resource theories. 4.1 Banks and interconversion of resources We now introduce the bank system, and show how this additional tool allows us to perform interconversion between resources. To simplify the notation, we only focus on a theory with two resources. However, the results we obtain also apply to theories with more resources, since in that case we can just select two resources and perform interconversion while keeping the others fixed. Thus, in the following we consider a resource theory Rmulti with two invariant sets F1 and F2 (each of them associated with one of the resources), and allowed operations Cmulti . We assume the theory to satisfy the asymptotic equivalence property of Def. 1 with respect to the relative entropy distances from F1 and F2 , and we ask the two invariant sets to satisfy the properties F1, F2, and F3, while we replace properties F4 and F5 with the following, more demanding, property (n) F5b The invariant sets Fi ’s are such that Fi = Fi⊗n , for all n ∈ N. The above properties implies that the relative entropy distances EF1 and EF2 are the unique quantifiers for the two resources of our theory, as we have seen in Sec. 3.3. From property F5b it follows that these two monotones are additive, ∞ i.e., EFi (ρ ⊗ σ) = EFi (ρ) + EFi (σ) for i = 1, 2, and consequently that their regularisation EF coincides with EFi . i Furthermore, the properties F1 and F5b together imply that the invariant sets are composed by a single state, i.e., Fi = {ρi }, where ρi ∈ S (H), for i = 1, 2. We make use of property F5b in Lem. 22, shown in appendix D.2, which itself is used to prove some essential properties of the set of bank states, see Def. 6. This property is ultimately used to show that the exchange rate between resources is given by the relative entropy distance from the set of states describing the bank, see Cor. 11. It is important to stress that property F5b is not satisfied by every multi-resource theory. For example, this property is satisfied by the multi-resource theory of thermodynamics, but it is violated by other theories, like entanglement theory, where the set of free states is composed of separable states. We are currently working to weaken this property, following the ideas presented in Ref. [63], by requiring the invariant sets to be closed under permutations of copies. This less demanding property should allow us to use the approximate de Finetti’s theorems [68], and to obtain similar conditions to those obtained with F5b. To study the interconversion of entanglement with some other resource, however, one can think of restricting the state space of the theory in a way in which the resulting subset of separable states satisfies property F5b, 15 see the example in Sec. 5.2. Finally, it is worth noting that all the results we obtain in this section also apply if one of the monotones, or both, is of the form shown in Eq. (31). Indeed, these monotones satisfy the same properties of the relative entropy distances, with the difference that the corresponding invariant set can be composed by multiple states, and these states do not need to have full rank. Let us now consider an example of resource interconversion which will highlight the properties that we are searching for in a bank system. Suppose we have a certain amount of euros and pounds in our wallet, and we want to convert one into the other, for example, from pounds to euros. In order to convert these two currencies we need to go to the bank, that we would expect to satisfy the following properties. First of all, if we do not hand in some pounds, we cannot receive any euros (and vice versa). Secondly, the bank will convert the two currency at a certain exchange rate, and this exchange rate can be different depending on the bank we go to. Finally, we would like the bank not to change the exchange rate between pounds and euros as a consequence of our transaction (this last property is approximately satisfied by real banks, at least for the amount exchanged by average costumers). The previous example shows that, in order to achieve resource interconversion, we need to introduce in our framework an additional system, the bank, with some specific properties. Within our formalism, we consider the same multi-partite system introduced in Sec. 3.2, with the main system S, and two batteries B1 and B2 . The system S is now used as a bank, that has to satisfy the three essential properties listed before. First of all, we need the states describing the bank to be passive, meaning that we should not be able to extract from this system both resources at the same time, since we always need to pay one resource to gain another one. Thus, the set of bank states is defined as Definition 6. Consider a multi-resource theory Rmulti satisfying the asymptotic equivalence property with respect to the monotones EF1 and EF2 . The set of bank states of the theory is a subset of the state space S (H) defined as,  Fbank = ρ ∈ S (H) | ∀ σ ∈ S (H) , EF1 (σ) > EF1 (ρ) or EF2 (σ) > EF2 (ρ) or EF1 (σ) = EF1 (ρ) and EF2 (σ) = EF2 (ρ) . (34) Within the set Fbank we can find different subsets of bank states with a fixed value of EF1 and EF2 . We define each of these subsets as
 (35) Fbank ĒF1 , ĒF2 = ρ ∈ Fbank | EF1 (ρ) = ĒF1 and EF2 (ρ) = ĒF2 . Notice that Eq. (34) implies that no state can be found with smaller values of both monotones EFi ’s. In this way, the agent is not able to transform the state of the bank in a way in which both resources are extracted from it and stored in the batteries. Instead, they always need to trade resources. The set of bank states Fbank can be visualised in the resource diagram of the theory, see Fig. 4. This set is represented by a curve on the boundary of the state space, connecting the points associated with F1 to those associated with F2 . In appendix B we show that, under the current assumptions, this curve is always convex, and in the following we focus our attention to those segments where the curve is strictly convex.
The subsets Fbank ĒF1 , ĒF2 ’s represent individual points in the resource diagram describing the multi-resource theory, and they obey many of the properties satisfied by the invariant sets Fi ’s. Indeed, one can show that • For all n ∈ N, we have that each subset of bank states is such that

(n) ⊗n Fbank ĒF1 , ĒF2 = Fbank ĒF1 , ĒF2 , (36) that is, these subsets satisfy property F5b. This equality is proved in Prop. 23 of appendix D.2.
• Every subset Fbank ĒF1 , ĒF2 is convex, property F2, as shown in Prop. 24 in appendix D.2.
• Every subset Fbank ĒF1 , ĒF2 , and its extensions to the many-copy case, is invariant under the class of allowed operations Cmulti of the multi-resource theory, as shown in Lem. 25 in appendix D.2. The second essential property for a bank is that the exchange rate needs only to depend on which state of the bank
we choose to use. In our framework, it is the choice of the values ĒF1 and ĒF2 , defining the subset Fbank ĒF1 , ĒF2 , that determines the exchange rate at which the resources are converted. In order to obtain this exchange rate we introduce the following function, which quantifies how much the properties of the bank change during a transformation,
and generalises the Helmholtz free energy used in thermodynamics. Given the subset of bank states Fbank ĒF1 , ĒF2 , this function is defined as ĒF1 ,ĒF2 (ρ) := α EF1 (ρ) + β EF2 (ρ) − γ, fbank (37) 16 Figure 4: The set of bank states introduced in Eq. (34) is represented in the EF1 –EF2 diagram. Only part of the state-space S (H) is shown, in blue, together with the invariant sets of the theory F1 and F2 , the two yellow segments. The black curve connecting these
segments is the set of all the bank states of the theory Fbank . A specific subset of bank states, labelled by Fbank ĒF1 , ĒF2 , is shown on the curve, see Eq. (35). Notice that, graphically, a bank state is one for which there exists no other state in the region immediately below and left. The red line, which is tangent to the set of bank states and passes through the ĒF ,ĒF2
point Fbank ĒF1 , ĒF2 , is parametrised by fbank1 = 0, see Eq. (37). where α, β, and γ are non-negative constant factors, which depend on the subset of bank states we have chosen. In order to define the linear coefficients, we impose the following two properties for this function,
ĒF1 ,ĒF2 is equal to zero over the subset Fbank ĒF1 , ĒF2 . B1 The function fbank
B2 The value of this function on the states contained in the subset Fbank ĒF1 , ĒF2 is minimum. Notice that property B1 is there to set the zero of the function, and implies that γ = α ĒF1 + β ĒF2 . (38) Property B2, instead, fixes the ratio between the constants α and β. This condition can be visualised in the resource diagram, and is equivalent to the request that, in such diagram, the bank monotone is tangent to the state space, so that
ĒF1 ,ĒF2 ĒF1 ,ĒF2 (39) (σ), ∀ ρ ∈ S (H) , ∀ σ ∈ Fbank ĒF1 , ĒF2 . (ρ) ≥ fbank fbank The above property is always satisfied under our working assumptions, since the curve of bank states is convex, see Fig. 4. We refer to this function as the bank monotone. The bank monotone can be easily extended to the state space of n copies of the system. The main difference is that, when we consider
states in S (H⊗n ), the coefficient γ is proportional to the number of copies n, and
we write γ = n α ĒF1 + β ĒF2 . This follows from property B1, together with the fact that the subset Fbank ĒF1 , ĒF2 satisfies property F5b, see Eq. (36). Since the function in Eq. (37) is a linear combination of the monotones EF1 and EF2 , it is easy to show (see also appendix D.2) that it satisfies the properties listed in the following proposition Proposition 7. Consider a resource theory Rmulti with allowed operations Cmulti , satisfying asymptotic equivalence with respect to the monotones EF1 and EF2 , i.e. the relative entropy distances from the invariant sets of the theory. Suppose Ē F1 that these sets satisfy the properties F1, F2, F3, and F5b. Then, the function fbank the following properties. Ē ,ĒF2 Ē ,ĒF2 F1 B3 The function fbank F1 B4 The function fbank is additive. is monotonic under partial tracing. 17 ,ĒF2 introduced in Eq. (37) satisfies Ē ,ĒF2 Ē ,ĒF2 Ē ,ĒF2 F1 B5 The function fbank F1 B6 The function fbank F1 B7 The function fbank Ē F1 scales extensively. For any sequence {ρn ∈ S (H⊗n )}, we have fbank ,ĒF2 (ρn ) = O(n). is asymptotic continuous. is monotonic under the set of allowed operations Cmulti , since α and β are non-negative. The third and last property we demand from a bank concerns the back-reaction it experiences during interconversion of resources. We want that, after the transformation, the state of the bank only changes infinitesimally with respect to the bank monotone associated with it. If this is the case, we can show that the exchange rate only changes infinitesimally, and therefore we can keep using the bank to inter-convert between resources at the same exchange rate. More concretely, we now consider a tripartite system composed by a bank S and and two batteries, B1 and B2 . Each of these subsystems is composed by many copies of the same fundamental system described by H, for which we defined the notion of bank states. Thus, the bank S is described by HS = H⊗n , with n ∈ N, and its initial state is given by n copies of the
bank state ρ ∈ Fbank ĒF1 , ĒF2 . The batteries are described by HBi = H⊗mi , mi ∈ N, where i = 1, 2. The states describing the batteries are ω1 ∈ S (HB1 ), and ω2 ∈ S (HB2 ), respectively. A resource interconversion is an asymptotically reversible transformation asympt ρ⊗n ⊗ ω1 ⊗ ω2 ←−−→ ρ̃⊗n ⊗ ω1′ ⊗ ω2′ , (40) where ρ̃ ∈ S (H), ω1′ ∈ S (HB1 ), and ω2′ ∈ S (HB2 ), satisfying the following property, see also Fig. 5, X1 The state of the bank changes infinitesimally during the resource interconversion.
If ρ ∈ Fbank ĒF1 , ĒF2 ⊂ S (H), then the state ρ̃ ∈ S (H) is such that Ē F1 fbank ,ĒF2 Ē F1 (ρ̃⊗n ) = fbank ,ĒF2 (ρ⊗n ) + δn , (41) where δn > 0 is such that δn → 0 as n → ∞. We are now ready to introduce the interconversion relation which links the different amounts of resources exchanged, weighted by the exchange rate given by the bank. The theorem is proved in appendix D.1. Theorem 8. Consider a resource theory Rmulti with two resources, equipped with the batteries B1 and B2 . Suppose the theory satisfies asymptotic equivalence with respect to the monotones EF1 and EF2 , i.e. the relative entropy distances from the invariant sets of the theory, and that these sets satisfy the properties F1, F2, F3, and F5b. Then, the resource interconversion of Eq. (40), where the bank has to transform in accord to condition X1, is solely regulated by the following relation, α ∆W1 = −β ∆W2 + δn . (42) Furthermore, when the number of copies of the bank system n is sent to infinity, we have that the above equation reduces to the following one, which we refer to as the interconversion relation, ∆W1 = − β ∆W2 , α (43) where the amount of resources exchanged ∆Wi is non-zero. Let us analyse this interconversion relation. Since both parameters α and β are non-negative we find that, whenever we exchange between resources, we increase the amount contained in one of the batteries (for example, ∆W1 > 0) while decreasing the amount contained in the other (∆W2 < 0). However, the change in these two resources also depends on the transformation of the bank state, see Eq. (24). Therefore, one has to consider the bank state used for interconversion, and the amount of resources contained in it. If the bank state ρ is such that EF1 (ρ) > 0 and EF2 (ρ) > 0, then interconversion can be achieved (in both directions) between ∆W1 and ∆W2 , at the rate specified by Eq. (43). Moreover, as far as the amount of resources in the bank is non-zero, we can exchange any amount of one resource for the other (since we can take the number of copies of the bank to be infinite). This is the case of thermodynamics, where thermal states indeed contain a positive amount of both energy and entropy, the two resources of the theory, and Eq. (43) gives the conversion rate for Landauer’s erasure. When the bank state is such that EF1 (ρ) > 0 and EF2 (ρ) = 0 (or vice versa), we can only exchange in one direction, since we can gain the first resource while paying the second resource (or vice versa). Finally, if the bank state does not 18 Figure 5: The state-space of the theory Rmulti is represented in the EF1 –EF2 diagram. The invariant sets of the theory, F1 and F2 , are represented by the two yellow segments. The set of bank states Fbank lies on the boundary of the state-space,
and is represented by the curve connecting the two invariant sets, see appendix B. The subset of bank states Fbank ĒF1 , ĒF2 , where ρ is contained, is represented by a point in the diagram. The red line which is tangent to the state-space and passes by the point ĒF ,ĒF 2 equal to 0. The other line is given by all those associated to ρ represents the set of states with value of the monotone fbank1 states with a value δ > 0 of this monotone. We see that, by mapping ρ into ρ̃, we can extract an amount ∆W1 of the first resource, while paying an amount ∆W2 of the second resource. Furthermore, one can show that when δ → 0, these two quantities 1 tend to 0 as δ 2 , i.e., with a slower rate. It is then possible to keep the ∆Wi ’s finite if we take n ∝ δ −1 copies of the bank states, see the proof of Thm. 8, in appendix D.1. Thus, in the limit n → ∞, the overall back-action on bank states associated with the conversion of resources can be made arbitrarily small. contain any amount of resources, EF1 (ρ) = 0 and EF2 (ρ) = 0, then we cannot perform interconversion, because we would have to reduce the amount of one of them within the bank. However, this is not possible since the amount of resource stored in a (bank) state cannot be negative. As a result, the multi-resource theories in which an interesting interconversion relation can be found are the ones in which the invariant sets of the theory do not intercept, see the right panel of Fig. 2. 4.2 Bank monotones and the relative entropy distance We start this section with an example concerning different models to describe thermodynamics, and the connection between these models. In the last part of Sec. 2.2, we have introduced a multi-resource theory whose resources are energy and entropy (or, information). For this theory, the bank states are thermal states at a given temperature T . We can move from this description of thermodynamics to a different one, based on a single-resource theory, by enlarging the class of operations in such a way that the agent can freely add ancillary systems in a thermal state with temperature T . This corresponds to the physical situation in which the system is put in contact with an infinite thermal bath. The single-resource theory we obtain is analogous to the one of Thermal Operations [10, 11], and its resource quantifier is unique. In fact, we can show that the bank monotone of the multi-resource theory and the resource quantifier of the single resource theory both coincides (modulo a multiplicative factor) with F − Fβ , where F is the Helmholtz free energy of the state whose resource we are quantifying, and Fβ is the Helmholtz free energy of the thermal state with temperature T = β −1 . In the following we study the connection between a general multi-resource theory and the single-resource theory obtained by enlarging the allowed operations with the possibility of adding ancillary systems described by bank states
ĒF1 ,ĒF2 , coincide with the unique measure of in Fbank ĒF1 , ĒF2 . We find that the bank monotone of Eq. (37), fbank resource for the obtained single-resource theory. As a result, we find that property X1, which regulates the exchange of resources in the multi-resource theory, can be understood as the condition that the resource characterising the bank does
not increase during the transformation. Furthermore, we show that, when the subset of bank states Fbank ĒF1 , ĒF2 Ē F1 contains a full-rank state, the monotone fbank ,ĒF2 is proportional to the relative entropy distance from this subset. Let us 19 now introduce the single-resource theory which can be derived from Rmulti by allowing the possibility of adding ancillary systems described by specific bank states. Definition 9. Consider the two-resource theory Rmulti with allowed operations Cmulti
and invariant sets F1 and F2 which satisfy the properties F1, F2, F3, and F5b. Consider the bank set Fbank ĒF1 , ĒF2 introduced in Eq. (35). We define the single-resource theory Rsingle as that theory whose class of allowed operations Csingle is composed by the following three fundamental operations,
1. Add an ancillary system described by n ∈ N copies of a bank state ρP ∈ Fbank ĒF1 , ĒF2 . 2. Apply any operation E ∈ Cmulti to system and ancilla. 3. Trace out the ancillary systems. The most general operation in Csingle which does not change the number of systems between its input and output is 
 E (s) (ρ) = TrP (n) E ρ ⊗ ρ⊗n , P (44) where we are partial tracing over the degrees of freedom P (n) , that is, over the ancillary system initially in ρ⊗n P .
The bank monotone associated with the bank set Fbank ĒF1 , ĒF2 , see Eq. (37), is the unique quantifier for the single-resource theory Rsingle . In order to show the uniqueness of this monotone, we first have to show that the singleresource theory satisfies asymptotic equivalence. Theorem 10. Consider the two-resource theory Rmulti with allowed operations Cmulti , and invariant sets F1 and F2 which property with respect satisfy the properties F1, F2, F3, and F5b. Suppose the theory satisfies the asymptotic equivalence
to the monotones EF1 and EF2 . Then, given the subset of bank states Fbank ĒF1 , ĒF2 , the single-resource theory Ē F1 Rsingle with allowed operations Csingle satisfies the asymptotic equivalence property with respect to fbank ,ĒF2 . The proof of this theorem can be found in appendix D.1, and we provide a geometric sketch of it in Fig. 6. As a Csingle . This side remark, notice that the functions EF1 and EF2 are not monotonic under the set of allowed operations
follows from the fact that we can now replace any state of the system with a state in Fbank ĒF1 , ĒF2 , since we are free to add an ancillary system in such state, and to perform a swap between main system and ancilla (since this operation belongs to Cmulti ). Then, if the bank state contains a non-zero amount of resources, meaning that ĒFi > 0 for i = 1, 2, we can always find a state in S (H) with lower value of either EF1 or EF2 (but not both at the same time), and therefore the above transformation would increase the value of this monotone. ĒF1 ,ĒF2 , defined in Eq. (37), From the above theorem it follows an interesting link between the bank
monotone fbank and the relative entropy distance from the set of bank states Fbank ĒF1 , ĒF2 . Indeed, when this set of states contains at least one full-rank state, we can prove that these two functions have to coincide, modulo a multiplicative factor. This is a consequence of the fact that Rsingle satisfies asymptotic equivalence, which implies the uniqueness of the resource measure, and of the fact that both the bank monotone and the relative entropy distance from the bank set satisfy the same properties, in particular monotonicity under the operations in Csingle and asymptotic continuity. We can express this fact in the following corollary, whose proof can be found in appendix D.1. Corollary 11. Consider the two-resource theory Rmulti with allowed operations Cmulti , and invariant sets F1 and F2 which equivalence property with respect satisfy the properties F1, F2, F3, and F5b. Suppose the theory satisfies the asymptotic
to the monotones EF1 and EF2 . If the subset of bank states Fbank ĒF1 , ĒF2 contains a full-rank state, then the Ē F1 bank monotone fbank constant. ,ĒF2 coincides with the relative entropy distance from this subset of states, modulo a multiplicative We close the section with the remark that, in the currently known scenarios, the bank subsets always contain at least a full-rank state, and in fact we find that, for these theories, the above correspondence between the bank monotone of Eq. (37) and the relative entropy distance is satisfied. An example is the multi-resource theory of thermodynamics, in which the relative entropy distance from a thermal state at a given temperature is indeed equal to the linear combination of the average energy and the entropy of a system. Other examples can be found in Sec. 5. 20 Figure 6: We sketch a geometric proof of Thm. 10 using the EF1 –EF2 diagram. The blue region is the state-space S (H), the yellow segments are the invariant sets F1 and F2 , and the red lines highlight the states with same value of monotone ĒF ,ĒF 2 . Notice that in this figure we are using the fact that, when Fi satisfies property F5b, the monotone EFi is such that fbank1 EFi (ρ ⊗ σ) = EFi (ρ) + EFi (σ) for any two states ρ and σ in S (H), see Lem. 22. To represent the state ρ ⊗ σ in the diagram, we renormalise its values of the EFi ’s by dividing them by the number of copies considered, in this case by two. Left. We first ĒF ,ĒF 2 is monotonic under the set of allowed operations Csingle . Consider
a system initially described sketch why the function fbank1 by the state ρ, and add to it an ancillary system described by the bank state ρP ∈ Fbank ĒF1 , ĒF2 . The global system is then represented by a point in the middle of the segment connecting ρ and ρP . We can transform the global state with the help of a sub-linear ancilla and of the operation E ∈ Cmulti , mapping it into the state σ ⊗ ρ̃P with same value of EF1 and EF2 . If we take ρ̃P to be on the boundary of the state-space, we can easily see that σ ≡ TrP [E(ρ ⊗ ρP )] always lies below the red line passing ĒF ,ĒF2 through ρ, i.e., its value of fbank1 is smaller than the one for ρ. Right. We now sketch how to map between states with the ĒF ,ĒF2 , fbank1 using the set of operations Csingle . In this case, we compose the main system, initially same value of the monotone described by ρ, with an additional one described by n copies of ρP . We then use an operation E ∈ Cmulti , together with a sub-linear ancilla, and we ask the final state of the system, σn = TrP (n) E(ρ ⊗ ρ⊗n P ) to have the same value of EF1 of the target state σ. It is then easy to show that, as n → ∞, the state σn tends to σ, while the n copies of the final state of the ancilla, ρ̃n , tends to the bank state ρP . 4.3 First law for multi-resource theories We can now introduce a general first law for multi-resource theories with disjoint invariant sets, see the right panel of Fig. 2. In order for this law to be valid, we need access to the batteries, the bank, and the main system. Within this setting, the first law consists of a single relation which links the different amount of resources exchanged with the batteries, the ∆Wi ’s, with the change in bank monotone over the state of the main system. The idea is that, contrary to what seen in Sec. 3.2, a state transformation over the main system is possible, when a bank is present, if this single relation is satisfied. Indeed, we do not need to use a fixed amount of each resource, since they are inter-convertible using the bank system. In more detail, we consider a theory Rmulti that, for simplicity, has just two resources. The invariant sets are F1 and F2 , they satisfy the properties F1, F2, F3 and F5b, and the theory satisfies the asymptotic equivalence property with respect to the monotones EF1 and EF2 . The global system is divided into four partitions, the main system S, the
bank P , and the batteries B1 and B2 . We assume the bank to be initially described by a state ρP ∈ Fbank ĒF1 , ĒF2 , where this subset contains at least one full-rank state. The relevant
monotone for the interconversion of resources is then the relative entropy distance from the subset Fbank ĒF1 , ĒF2 , as shown in Cor. 11. Suppose that the main system is initially described by the state ρ ∈ S (HS ), and we want to map it into the state σ ∈ S (HS ), with possibly a different value of EF1 and EF2 . If we do not have access to the bank, then the amount of resources we need to exchange is given by the difference of the monotones EFi ’s between the initial and final state of the main system, see Eq. (28) in Sec. 3.3. But since we have access to the battery, we can exchange between the resources, and we are not obliged any more to provide a fixed amount for each resource. In order to show this, consider the global initial state ρ ⊗ ρP ⊗ ω1 ⊗ ω2 , describing the main system, the bank, and the two batteries B1 and B2 . Then, we (asymptotically) map this global state, using the allowed operations Cmulti , into the final state σ ⊗ ρ̃P ⊗ ω1′ ⊗ ω2′ , 21 where the final state of the bank is ρ̃P , and the batteries B1 and B2 have final state ω1′ and ω2′ , respectively. Due to asymptotic equivalence, this state transformation is possible only if the monotones EFi ’s are preserved. However, the final state of the bank only has to satisfy property X1, and we have shown in Sec. 4.1 that such constraint still allows us to exchange an arbitrary amount of resources, see Thm. 8. As a result, there is a single relation that regulates the state transformation over the main system, Corollary 12. Consider the two-resource theory Rmulti with allowed operations Cmulti , and invariant sets F1 and F2 which satisfy the properties F1, F2, F3, and F5b. Suppose the theory satisfies the asymptotic equivalence property with respect to the monotones EF1 and E
F2 , and that the global system is divided into a main system S, a bank described by the set of states Fbank ĒF1 , ĒF2 (which contains at least one full-rank state), and two batteries B1 and B2 . Then, a transformation which maps the state of the main system from ρ into σ, where these states are completely general, only has to satisfy the following relation α ∆W1 + β ∆W2 = EFbank (ĒF 1 ,ĒF2 ) (ρ) − EFbank (ĒF1 ,ĒF2 ) (σ), (45) where each ∆Wi is defined as the difference in the monotone EFi over the final and initial state of the battery Bi , see Eq. (23), and EFbank (ĒF ,ĒF ) is the relative entropy distance from the set of states describing the bank. 1 2 We refer to Eq. (45) as the first law of multi-resource theories. Indeed, for the resource theory of thermodynamics, where energy and entropy are the two resources, and the bank is given by an infinite thermal reservoir with a given temperature T , this equation corresponds to the First Law of Thermodynamics. In fact, in the thermodynamic scenario we have that ∆W1 = −∆U , where U is the internal energy of the system, while ∆W2 = ∆S is the change in entropy in the system. The change in relative entropy distance on the main system is proportional to the change in Helmholtz free-energy, which in turn is equal to the work extracted from the system, W . The linear coefficients in the equation can be computed from Eq. (37), knowing that the bank monotone is equal to the relative entropy distance from the thermal state with temperature T . It is easy to show that α = T −1 and β = 1 . If we re-arrange the equation, and we define Q = T ∆S as the amount of heat absorbed by the system, we obtain ∆U = Q − W , that is, the First Law of Thermodynamics. 5 Examples In this section we present two examples of multi-resource theories where an interconversion relation can be derived. The first one is thermodynamics for multiple conserved quantities (even non-commuting ones), while the second one concerns local control under energetic restrictions. In both examples we describe the state-space (and we represent it with a resource diagram), we find the bank states of the theory, and we derive an interconversion relation for the different resources. Furthermore, in both cases we find that the bank monotone is proportional to the relative entropy distance from the given set of bank states, as expected from Cor. 11. Before we introduce the examples, we provide a flowchart that should help the reader in building a multi-resource theory. In particular, the flowchart clarifies in which situations each of the results we obtain hold for a specific theory. This tool should be used as follow. Once decided which resources are the fundamental ones for the theory of interest, m and therefore once defined the class of allowed operations Cmulti and the invariant sets of the theory {Fi }i=1 , one has to first check whether asymptotic equivalence holds for the theory (by finding a protocol which maps between states with same values of the monotones). If this is the case, then the properties of the monotones and of the invariant sets have to be considered, and depending on these properties, one obtains a theory with different features. The following flowchart is used in the first example to clarify how to characterise a multi-resource theory. 5.1 Thermodynamics of multiple-conserved quantities In this example we consider the resource theory of thermodynamics in the presence of multiple conserved quantities (even in the case in which these quantities do not commute) [45, 46, 48]. Our system is a d-level quantum system, and for simplicity, we only consider two conserved quantities A and B. The allowed operations are Thermal Operations [10, 11], composed by unitary operators which commute with both A and B. This set of maps can be obtained as a proper subset of the intersection between the allowed operations of the following single-resource theories, 22 Does the theory satisfy the asymptotic equivalence property of Def. 1? NO YES Is the theory asymptotically equivalent with respect to the relative entropy distances from the invariant sets, or equivalently to the monotones introduced in Sec. 3.4? YES NO The resource theory does not allow for the use of batteries, see Sec. 3.2. a For NO YES NO Modify the class of allowed operations Cmulti , if possible. YES Which additional properties do the invariant sets satisfy, F4 and F5, or F5b? Do the monotones satisfy the properties M4, M5, M6, and particularly asymptotic continuity, property M7? NO Can we find candidate battery subsystems such that the monotones satisfy properties M1, M2, and M3? Do the invariant sets satisfy the properties F1, F2, and F3a ? Can we find batteries satisfying property M1 with respect to these monotones? F4 ∧ F5 YES The state-space of the theory can be represented in a resource diagram, see Fig. 3, but the representation is not unique. Resources cannot be exchanged, as the bank does not contain one or more of them. NO It is possible to exchange one resource for another, see Thm. 8, and the theory admits a first law, see Cor. 12 YES F5b The state-space of the theory can be represented in a resource diagram, and the representation is unique, see Thm. 4. The state-space of the theory can be represented in a resource diagram, and the representation is unique. Are the invariant sets disjoint, see right panel of Fig 2? the monotones introduced in Sec. 3.4, property F3 is not relevant. Figure 7: Flowchart: how to apply the results of this paper to an arbitrary resource theory. • The resource theory of the quantity A. The allowed operations are all the average-A-non-increasing maps, whose invariant set is composed by a single state, FA = {|a0 i ha0 |}, the eigenstate of A associated with its minimum eigenvalue a0 (for simplicity, we here assume it to be non-degenerate). From Sec. 3.4 it follows that this theory has a monotone of the form MA (ρ) = Tr [Aρ] − a0 . • The resource theory of the quantity B. The allowed operations are all the average-B-non-increasing maps, whose invariant set is composed by a single state, FB = {|b0 i hb0 |}, the eigenstate of B associated with its minimum eigenvalue b0 (for simplicity, we here assume it to be non-degenerate). From Sec. 3.4 it follows that this theory has a monotone of the form MB (ρ) = Tr [Bρ] − b0 . • The resource theory of purity, where the allowed operations are all the maps whose fix point is the maximally-mixed state FS = dI (unital maps). One monotone of the theory is the relative entropy distance from dI , that is, EFS (ρ) = log d − S(ρ) where S(·) is the von Neumann entropy. The first box in the flowchart asks whether or not the considered multi-resource theory satisfies asymptotic equivalence. In Refs. [36, 37] it has been shown that, indeed, a resource theory of this kind does satisfy the asymptotic equivalence property of Def. 1 with respect to the monotones MA , MB and EFS . Furthermore, it is easy to see that these monotones 23 are either relative entropy distances from the set of invariant states, or that they are of form of Eq. (31). This implies that we can answer positively to the second box we have reached in the flowchart. We now need to consider the properties of the invariant sets of the theory, which in turn determine the properties of the monotones. It is easy to show that these sets are closed (property F1) and convex (property F2). Furthermore, FS contains a full-rank state (property F3), that implies asymptotic continuity of the associated monotone, see Refs. [63, 64]. The fact that the other sets do not contain a full-rank state is not problematic since we are considering monotones of the form of Eq. (31), that are nevertheless asymptotic continuous, see Prop. 21. Additionally, all invariant sets satisfy (n) property F5b, that is, Fi = Fi⊗n , for i = A, B, S. In order to answer the next box of the flowchart, we need to find batteries that only store one kind of resource each. For example, we can search for two pure states with different average values of A, and same average values of B. Then, the battery BA , storing the first kind of resource, is composed by a certain number of copies of these two states, where the number varies when we extract/store the resource. A similar construction can be done for the other battery BB . For the purity battery, we can take a system with degenerate A and B, and take states with a certain number of copies of a pure state and mixed state. If this is possible, then we can answer positively the box of the flowchart we have reached. Let us now consider a reversible transformation, described by the following equation asympt ′ ′ ρ⊗n ⊗ ωA ⊗ ωB ⊗ ωS ←−−→ σ ⊗n ⊗ ωA ⊗ ωB ⊗ ωS′ , (46) where the n copies of ρ and ρ′ describe the main system at the beginning and the end of the transformation, and the states ωi and ωi′ are the initial and final states of the battery Bi , for i = A, B, S. According to asymptotic equivalence, the transformation is possible if ∆WA = MA∞ (ρ) − MA∞ (σ) = Tr [A (ρ − σ)] , ∆WB = ∆WS = (47) MB∞ (ρ) − MB∞ (σ) = Tr [B (ρ − σ)] , ∞ ∞ EF (ρ) − EF (σ) = S(σ) − S(ρ). S S (48) (49) The last box in the flowchart we need to consider concerns the bank states. In order to get an interconversion relation and a first law, we need the bank states to contain a non-zero amount of each resource. This has to be the case for the current resource theory, since the invariant sets do not intercept each other. Therefore, this theory admits a first law, as we are going to show. Indeed, it can be easily shown, using Jaynes principle [67], that the bank states are of the following form e−β1 A−β2 B , (50) τβ1 ,β2 = Z  −β A−β B  2 where the parameters β1 , β2 ∈ [0, ∞), and Z = Tr e 1 is the partition function of the system. These states are known in thermodynamics are the grand-canonical ensemble. Each τβ1 ,β2 is a bank state with a different value of resource A, resource B, and purity. The value of these three resources only depends on the parameters β1 and β2 . In order to find the interconversion relation we need to construct the bank monotone β̄1 ,β̄2 fbank (ρ) = αβ̄1 ,β̄2 MA (ρ) + γβ̄1 ,β̄2 MB (ρ) + δβ̄1 ,β̄2 EFS (ρ) − ξβ̄1 ,β̄2 (51) which is equal to zero over the bank state τβ̄1 ,β̄2 . Properties B1 and B2 provide a geometrical way of building the monotone. If we represent the state space in a three-dimensional diagram (where the axes are given by MA , MB , and β̄1 ,β̄2 EFS ), then the hyperplane defined by the equation fbank = 0 is tangent to the state space and only intercepts it in τβ̄1 ,β̄2 , see Fig. 8 for an example. β̄1 ,β̄2 The hyperplane defined by fbank = 0 is identified by the normal vector n̂ = r̂1 × r̂2 , where r̂i =  ∂MA (τβ̄1 ,β̄2 ) ∂MB (τβ̄1 ,β̄2 ) ∂EFS (τβ̄1 ,β̄2 ) ; ; ∂βi ∂βi ∂βi T for i = 1, 2. (52) The parametric equation of the hyperplane then gives us the expression of the monotone,


β̄1 ,β̄2 (ρ) = n1 MA (ρ) − MA (τβ̄1 ,β̄2 ) + n2 MB (ρ) − MB (τβ̄1 ,β̄2 ) + n3 EFS (ρ) − EFS (τβ̄1 ,β̄2 ) , fbank 24 (53) Figure 8: The state space of the multi-resource theory of thermodynamics and conserved angular momenta (along the x and z axes). On the surface we find the states τβ1 ,β2 defined in Eq. (50), where β1 and β2 take values in R. The red surface is the set of bank states, with β1 and β2 are both non-negative, and the green plane is tangent to the state space in the point associated β̄1 ,β̄2 with τβ̄1 ,β̄2 . The equation of the plane gives the monotone fbank . where ni is the i-th component of the normal vector n̂. By evaluating the monotones MA , MB , EFS , and their derivatives β̄1 ,β̄2 we find that fbank is equal (modulo a positive multiplicative factor depending on the parameters β̄1 and β̄2 ) to the relative entropy distance from τβ̄1 ,β̄2 , β̄1 ,β̄2 fbank (ρ) ∝ Eτβ̄1 ,β̄2 (ρ) = β̄1 Tr [ρA] + β̄2 Tr [ρB] − S(ρ) + log Z. (54) Thus, the bank state τβ̄1 ,β̄2 allows us to obtain the following interconversion relation between the three resources, β̄1 ∆WA + β̄2 ∆WB = ∆WS , (55) while the state of the bank only changes by an infinitesimal amount in terms of Eτβ̄1 ,β̄2 . 5.2 Local control theory under energetic restrictions We now introduce a multi-resource theory describing local control under energetic restrictions. Specifically, we consider the situation in which a quantum system is divided into two well-defined partitions A and B, and we can only act on the individual partitions with non-entangling operations, which furthermore need to not increase the energy of the overall system. This kind of simultaneous restrictions on locality and thermodynamics has also been considered in other previous works, see for example Refs. [69–73]. The multi-resource theory is obtained by considering two single-resource theories, the one of entanglement and the one of energy. While this is a well define multi-resource theory, it is not straightforward to prove that it is also a reversible theory. Therefore, to provide a first law in this setting, we have to restrict the state-space to a subset of all bipartite density operators. 5.2.1 Set-up Let us consider a bipartite system, whose partitions are labelled as A and B, with a non-local Hamiltonian HAB (that is, the two partitions interact with each other, and the ground state of the system is an entangled state). The set of allowed operations of this multi-resource theory is obtained from the intersection of the allowed operations of the following single-resource theories, • The resource theory of energy. The allowed operations are all the average-energy-non-increasing maps, defined in Sec. 3.4. When the Hamiltonian has non-degenerate ground state |gi, the fix state of the maps is FH = |gi hg|. The monotone of this resource theory is MH (ρ) = Tr [Hρ] − Eg , where Eg is the eigenvalue associated with the ground state |gi. 25 • The resource theory of entanglement. The allowed operations are the asymptotically non-entangling maps [53]. These maps are relevant to us for two reasons. Firstly, all our results hold in the asymptotic limit, and therefore it is reasonable to consider the set of maps which do not create entanglement in this limit. Secondly, this is the only set of operations which provides a reversible theory for entanglement. The monotone is EFsep (·), where Fsep is the set of separable states, invariant under the class of operations. While the current multi-resource theory is well-defined and meaningful, it is not straightforward to prove whether it is reversible in the sense given in Def. 1. Furthermore, it is known that the relative entropy of entanglement, EFsep , is not additive (or even extensive) for all bipartite density operator. Therefore, if we want to study interconversion of resources in this setting, we need to consider a subset of the state-space (as well as of the invariant set Fsep ). In the following we will focus on the simplest example of a multi-resource theory of this kind. The bipartite system is composed by two qubits, so that its Hilbert space is HAB = C2 ⊗ C2 . The Hamiltonian of the system is HAB = E0 |Ψsinglet i hΨsinglet | + E1 Πtriplet , (56) where E0 < E1 , the ground state is the singlet state, 1 |Ψsinglet i = √ (|01i − |10i) , 2 and Πtriplet = P3 i=1 (i) (57) (i) |Ψtriplet i hΨtriplet | is the projector on the triplet subspace, where 1 (1) |Ψtriplet i = √ (|01i + |10i) , 2 1 (2) |Ψtriplet i = √ (|00i − |11i) , 2 1 (3) |Ψtriplet i = √ (|00i + |11i) . 2 (58) (59) (60) In order to get a reversible multi-resource theory, and therefore to be able to define the interconversion relations, we consider a restricted state-space, given by the following subset of bipartite density operators, ( ) 3 X 1 (i) (i) pi |Ψtriplet i hΨtriplet | , with p0 ≥ S1 = ρ ∈ S(HAB ) | ρ = p0 |Ψsinglet i hΨsinglet | + . (61) 2 i=1 There are two additional reasons why we are interested in this set of states. First of all, because the relative entropy of entanglement EFsep has an analytical expression for states which are diagonal in the Bell basis [74–76] (that here coincides with the energy eigenbasis). Secondly, because it is easy to show, see Eq. (34), that S1 contains the bank states of the theory, that are the interesting ones when it comes to study interconversion. Finally, it is worth noting that the state-space S1 contains all the Gibbs states of the non-local Hamiltonian HAB with positive temperatures. Within this restricted state-space we find the following subset of separable states, ) ( 3 X 1 (i) (i) pi |Ψtriplet i hΨtriplet | . (62) Fcss = ρ = |Ψsinglet i hΨsinglet | + 2 i=1 It is worth noticing that the above subset Fcss contains all the closest-separable states to the entangled states in our restricted state-space S1 (see Ref. [76]). As a result, for any state ρ ∈ S1 we have that EFsep (ρ) = EFcss (ρ) = 1 − h (hΨsinglet | ρ |Ψsinglet i) , (63) where h(·) is the binary entropy function. Since our focus is restricted to the sole states in the subset S1 , we will now re-define7 the set of allowed operations of the multi-resource theory as those energy-non-increasing maps which only preserve the subset of separable states Fcss = Fsep ∩ S1 . We can define this class of operation as Cmulti = {E : S(HAB ) → S(HAB ) | E(Fcss ) ⊆ Fcss and Tr [E(ρ)HAB ] ≤ Tr [ρ HAB ] ∀ ρ ∈ S(HAB )} , (64) 7 The modified set of allowed operations makes it easier for us to find a protocol for inter-converting resources. However, we do not exclude the possibility of being able to perform interconversion with the original set of allowed operations, that preserve all separable states. However, finding this protocol might be non-trivial, and could be material of future work. 26 Figure 9: We represent (in two different ways) the state-space of the multi-resource theory of local control under energy restrictions. We consider a bipartite system composed by two qubits, with a non-local Hamiltonian given in Eq. (56). Left. The state-space is represented by the blue region,while the green region is the set of all separable states, and the orange set on its boundary is the set of separable states Fcss , defined in Eq. (62). The black curve represents the states diagonal in the energy eigenbasis (Bell’s basis) whose probability of occupation of the ground state (the singlet) is p0 ≥ 41 . The extremal points of this curve are, respectively, the maximally-mixed state and the singlet. In particular, in our simplified example we restrict the state-space to the states on the curve’s section connecting the set Fcss to the singlet, that is, to S1 . Furthermore, the allowed operations will have to leave these two sets invariant. Right. The diagram represents the set of states diagonal in the energy eigenbasis (blue region). On the left part of the diagram we find the curve representing the set S1 , whose extreme points are the singlet and the invariant set Fcss . On the right hand side, we have all those diagonal states with p0 < 12 . The red line is the set of all separable states Fsep . It is easy to see that, although the set of diagonal states in the Bell’s basis is convex, its representation in the diagram has not to be convex, see comments after Lem. 18, in the appendix. where each E ∈ Cmulti is a completely positive and trace preserving map. The two batteries we use in the theory store, respectively, energy and entanglement. One can imagine different kinds of energy batteries. For example, we could have that only Alice (or Bob) has access to the battery, which would imply that only one of them can change the energy of the non-local system. However, we prefer to consider a symmetric situation in which both Alice and Bob can interact with the battery. Moreover, we chose the battery to be non-local, so that they are effectively using the same battery, and not two local batteries. Thus, the battery BW is composed by m copies of a two-qubit system with the same Hamiltonian of the main system, that is, HW = E0 |Ψsinglet i hΨsinglet | + E1 Πtriplet . (65) The state of the battery is ⊗k ωW (k) = |Ψsinglet i hΨsinglet | (1) (1) ⊗m−k ⊗ |Ψtriplet i hΨtriplet | , (66) (1) where the excited state |Ψtriplet i could be replaced by any other triplet state. Notice that, in order to store/provide energy, we have to change the number of triplet and singlet states contained in the battery, and this can be done locally by both Alice and Bob. Moreover, even if we are changing the energy of the battery, we are not modifying its entanglement, in accord with property M1. The second battery BE is composed by ℓ copies of a two-qubit system with trivial Hamiltonian HE ∝ I (so as to be able to exchange entanglement while preserving the energy of the battery). We choose the state of the battery to be ⊗h ωE (h) = |Ψsinglet i hΨsinglet | ⊗ℓ−h ⊗ σmm , (67) where the state σmm ∈ Fcss , and we take it to be the maximally-mixed state on the subspace spanned by |Ψsinglet i and (1) |Ψtriplet i, that is 1 1 (1) (1) σmm = |Ψsinglet i hΨsinglet | + |Ψtriplet i hΨtriplet | . (68) 2 2 The change in entanglement is measured by the change in the number of singlet states h. 27 5.2.2 Reversibility and the interconversion relation In order for the present multi-resource theory to admit an interconversion relation, we first need to show that the asymptotic equivalence property of Def. 1 is satisfied. Let us consider the subset of states Sp0 ⊂ S1 , where p0 > 12 , defined as (69) Sp0 = {ρ ∈ S1 | hΨsinglet | ρ |Ψsinglet i = p0 } . It is easy to show that all the states in this subset have the same value of the energy and entanglement monotones, which we label M̄H and ĒFcss respectively. Furthermore, for any two states in this set, we can find an allowed operation in Cmulti , see Eq. (64), which maps one into the other. Indeed, consider an ancillary qutrit system described by the state P3 η = i=1 qi |θi i hθi |, and the global unitary operation U acting on main system and ancilla. The unitary operation maps (i) (j) |Ψtriplet i |θj i into |Ψtriplet i |θi i, for i, j ∈ {1, 2, 3}, and acts trivially on the remaining basis states. Then, the operation   Eη (·) = TrA U (· ⊗ ηA ) U † ∈ Cmulti maps any state ρ ∈ Sp0 into the state Eη (ρ) = p0 |Ψsinglet i hΨsinglet | + (1 − p0 ) 3 X i=1 (i) (i) qi |Ψtriplet i hΨtriplet | , (70) 3 where the probability distribution {qi }i=1 is defined by η. By choosing different ancillary states η, we can reach different states in Sp0 , proving in this way that the resource theory satisfies asymptotic equivalence8 . We can now consider the interconversion of energy and entanglement. Together with the two batteries BW and BE , one for energy and the other for entropy, we need to use a bank system. One can show that, when diagonal states in the energy eigenbasis are considered, bank states belongs to the set S1 introduced in the previous section. Thus, we describe the bank system using n ≫ 1 copies of a state ρin ∈ Sp0 , where p0 > 12 (the actual form of the state is not relevant, since we can use the allowed operation Eη to freely select any state in this set). In order to obtain an interconversion relation, we need to find an allowed operation in Cmulti , acting on the global state of bank and batteries, which modifies the state of the batteries (by exchanging resources) while leaving the state of the bank almost unchanged with respect to the relative entropy distance from Sp0 . In appendix C we provide a protocol which performs the following resource interconversion using an allowed operation Cmulti , asympt ⊗n ′ ′ (71) ρ⊗n in ⊗ ωW (k) ⊗ ωE (h) ←−−→ ρfin ⊗ ωW (k ) ⊗ ωE (h ). In the above transformation, the initial state of the bank ρin is mapped into a state ρfin ∈ Sp′0 , where p′0 = p0 + O(n−1 ). The energy battery BW is mapped from the initial state ωW (k), containing k copies of the ground state of HAB , into the final state ωW (k ′ ) with k ′ = k + ∆k copies of this ground state, where ∆k > 0 is arbitrary big. Likewise, the entanglement battery BE changes from the initial state ωE (h), containing h singlets, to the final state ωE (h′ ) containing p0 ∆k singlets. From the above transformation one is able to derive an interconversion relation between h′ = h − log 1−p 0 energy and entanglement, ∆E ∆WW = − (72) p0 ∆WE , log 1−p 0 where ∆WW = MH (ωW (k ′ ))−MH (ωW (k)) is the amount of energy exchanged, ∆WE = EFcss (ωE (h′ ))−EFcss (ωE (h)) is the amount of entanglement exchanged, and ∆E = E1 − E0 is the energy gap of the Hamiltonian HAB . Additionally, we find that the change in monotone ESp0 between the initial and final global state of the bank is negligible (for n → ∞), in accord with property X1. 6 Conclusions From multiple constraints to a resource theory. With the present work we set the mathematical ground for the development of resource theories with multiple resources able to describe new physical scenarios. Our construction of multi-resource theories is based on the definition of their class of allowed operations. First, we pinpoint the resources that compose the theory, and we introduce the corresponding single-resource theories. Then, we define the set of allowed operations for the multi-resource theory as the one composed by the maps in the intersection of the different classes of allowed operations of each single-resource theory, Eq. (11). This construction leaves the theory with multiple invariant 8 The operation Eη (·) we introduce is allowed since we restricted the invariant set Fsep to Fcss . Indeed, the above map would not leave invariant the set of separable states Fsep . 28 sets, some of which are the sets of free states of the relevant single-resource theories. It is worth remarking again that, in multi-constraint theories, there is a difference between the set of free states and the invariant sets (in contrast with the case of single-resource theories), and a multi-resource theory can have multiple invariant sets and no free states, Fig. 2. Reversibility. Together with the introduction of a general framework for multi-resource theories, we have studied the properties of these reversible theories. In particular, to analyse reversibility when multiple resources are present, we have first introduced the asymptotic equivalence property, see Def. 1. This property implies that a unique monotone can be used to quantify each resource. Furthermore, in the case of single-resource theories, it coincides with the usual notion of reversible rates of conversion. We know of multi-resource theories that satisfy this property, see the two examples provided in Sec. 5. However, it would be interesting to study which of the other, already existing, multi-resource theories satisfy the property of Def. 1. Ultimately, one would hope to find some general condition according to which a multi-resource theory is reversible, similarly to what has been found in Ref. [52]. The role of batteries. A crucial feature of our framework is the presence of batteries, used to store and quantify the resources exchanged during a state transformation over the main system. While batteries can be defined for singleresource theories as well, they do not seem to play the same fundamental role in that case, since one can quantify the amount of resource contained in a system using the conversion rate, see Def. 13 in appendix A. However, the conversion rate is linked to a change in the number of copies, for example ρ⊗n → σ ⊗k , where it is implicitly assumed that the remaining |n − k| copies of the system are in a free state. Since the framework allows us to model theories with no free states, we cannot change the number of systems with the allowed operations, and therefore we need to use batteries to quantify the amount of resources. We have seen in this paper what are the main properties for these batteries, primarily property M1, which requires each battery to store one and only one of the resource. It would be interesting to study these systems more carefully, possibly linking them to the kind of batteries used for fluctuation theorems [77–80], which are described by states in a big superposition, so as to always remain uncorrelated from the main system during a state transformation [81, 82]. Interconversion and further examples. We have studied the interconversion of resources and we have introduced a first law for multi-resource theories, Eq. (45), valid when the theories are reversible and the invariant sets are disjoint. We have provided two examples of theories with a first law, one related to thermodynamics, and the other concerning a theory of local control under energy restriction. In this latter example, we have studied an extremely simplified case, due to the fact that reversibility has not been proved in general for this theory. Due to the high importance of both non-locality and thermodynamics in the field of quantum technology and many-body physics, we believe that a complete analysis of this multi-resource theory would be useful. Furthermore, it would be interesting to know which other multi-resource theories allow for an interconversion relation, and whether it is possible to define interconversion for theories with a different structure of invariant sets, by for instance relaxing the assumptions made on the bank. For example, one could consider bank states from which both resources could in principle be extracted, and forbid such extraction by further constraining the class of allowed operations. Multiple ways to build a multiple-resource theory. In general, there could be different ways to intersect constraints in order to obtain the same final resource theory, and some of these constructions are a better fit for the analysis presented here than others. For example, the resource theory of thermodynamics equipped with Thermal Operations can be built as the intersection of either (1) the resource theories of information and energy, as we have done in Sec. 4.2, or (2) the resource theories of athermality and coherence [83–85]. However, the most convenient setting for the study of this latter construction is the single-copy regime, since in the many-copy scenario coherence is lost, as this quantity scales sub-linearly in the number of copies of the system considered. Beyond the asymptotic limit. The concrete results presented here for reversibility and interconversion of resources are only valid in the asymptotic limit where many independent and identically distributed copies of a system are considered. However, the general framework we introduced to describe resource theories with multiple resources and batteries can also be applied to scenarios with a single system. Understanding how resources can be exchanged in the single-copy regime, and studying the corrections to the first law in such a regime are worthwhile questions to pursue. Acknowledgement We thank the anonymous TQC referees for feedbacks, and Tobias Fritz for detailed comments on a previous version of this manuscript, CS is supported by the EPSRC (grant number EP/L015242/1). LdR acknowledges support from the Swiss National Science Foundation through SNSF project No. 200020 165843 and through the National Centre of Competence in Research Quantum Science and Technology (QSIT), and from the FQXi grant Physics of the observer. 29 CMS is supported by the Engineering and Physical Sciences Research Council (EPSRC) through the doctoral training grant 1652538, and by Oxford-Google DeepMind graduate scholarship. CMS would like to thank the Department of Physics and Astronomy at UCL for their hospitality. PhF acknowledges support from the Swiss National Science Foundation (SNSF) through the Early PostDoc.Mobility Fellowship No. P 2EZP 2 165239 hosted by the Institute for Quantum Information and Matter (IQIM) at Caltech, from the IQIM which is a National Science Foundation (NSF) Physics Frontiers Center (NSF Grant P HY − 1733907), and from the Department of Energy Award DE − SC0018407. JO is supported by the Royal Society, and by an EPSRC Established Career Fellowship. We thank the COST Network M P 1209 in Quantum Thermodynamics. Author contributions All authors contributed significantly to the ideas behind this work and to the development of the general framework (Sec. 2). CS, LdR and JO developed the results on batteries, bank states and the first law (Secs. 3, 4, 5). CS wrote the proofs and initial draft. 30 Appendix A Reversibility and asymptotic equivalence for single-resource theories In this section we show that, for a single-resource theory, the asymptotic equivalence property of Def. 1 is equivalent to the notion of reversibility given in terms of rates of conversion. Let us first introduce the concept of rate of conversion for a single-resource theory, see Ref. [30]. The definition of rate we use coincides with the one used in the literature, with the difference that we are making explicit use of the partial trace and of the addition of free states. In fact, we prefer not to include these operations within the set C, as we want the allowed operations to preserve the number of copies of the system they act over (with the exception of sub-linear ancillae). Definition 13. Consider a single-resource theory with allowed operations C and free states F, and two states ρ, σ ∈ S (H). We define the rate of conversion from ρ to σ as      kn ⊗n ⊗kn =0 R(ρ → σ) = sup | either lim min Trn−kn Ẽn (ρ ) − σ 1 n→∞ n Ẽn !  or lim n→∞ min Ẽkn (ρ⊗n ⊗ γkn −n ) − σ ⊗kn Ẽkn 1 = 0 , where γkn −n ∈ F (kn −n) . (73) h i (A) where the maps Ẽn have been defined in Eq. (13), and they are of the form Ẽn (·) = TrA En (· ⊗ ηn ) , with En ∈
(A) C (n+o(n)) and ηn ∈ S H⊗o(n) . Now that the notion of rate is defined, we introduce the concept of reversible single-resource theory, Definition 14. A single-resource theory with allowed operations C and free states F is reversible if, given any non-free states ρ, σ ∈ S (H), the rate of conversion from ρ to σ is such that R(ρ → σ) ∈ (0, ∞), and R(ρ → σ)R(σ → ρ) = 1. The above notion of reversibility is based on the rates of conversion between two resourceful states. However, it is not clear how to extend Def. 13 to the case of multiple resources, since the set of free states might be empty for multi-resource theories. For this reason, we have introduced the property of asymptotic equivalence in Sec. 3.1. This property also apply to the single-resource theory case, when m = 1. Now we want to show that Defs. 1 and 14, for a single-resource theory, coincide. First, let us introduce a function f : S (H⊗n ) → R (more formally, a family of functions) with the following properties, SM1 For each n ∈ N, the function f is monotonic under the set of allowed operations C (n) , that is
f (En (ρn )) ≤ f (ρn ) , ∀ ρn ∈ S H⊗n , ∀ En ∈ C (n) . SM2 For each n ∈ N, the function f is equal to 0 for all states γn ∈ F (n) , that is
f (γn ) = 0, ∀ γn ∈ S H⊗n . (74) (75) SM3 The function f is asymptotic continuous. SM4 The function f is monotonic under partial tracing, that is f (Trk [ρn ]) ≤ f (ρn ) ,
∀ n, k ∈ N , k < n , ∀ ρn ∈ S H⊗n . (76) SM5 For each n, k ∈ N, the function f is sub-additive, that is ∀ ρn ∈ S H⊗n f (ρn ⊗ ρk ) ≤ f (ρn ) + f (ρk ) ,

, ∀ ρk ∈ S H⊗k . SM6 For any given sequence of states {ρn ∈ S (H⊗n )}, the function f scales extensively, that is, f (ρn ) = O(n). 31 (77) Notice that property SM6 implies that the function f is regularisable. Furthermore, the value of f is preserved if we add free states, that is,
f (ρn ⊗ γk ) = f (ρn ), ∀ ρn ∈ S H⊗n , ∀ γk ∈ F (k) , (78) which follows from properties SM2, SM4, and SM5. The first lemma we introduce show that the rate of conversion of a reversible single-resource theory is linked to the function f satisfying the above properties. Notice that this proof is analogous to the one of Ref. [62], with the difference that we are allowing for the presence of a sub-linear ancilla in the definition of rate, following the notion of “seed regularisation” introduced in Ref. [23, Sec. 9]. Lemma 15. Consider a reversible resource theory with allowed operations C and free states F, and the function f satisfying SM1 – SM6. Then, for all non-free states ρ, σ ∈ S (H), we have that R(ρ → σ) = f ∞ (ρ) f ∞ (σ) (79) Proof. Let consider ρ and  σ such that R(ρ → σ) ≤ 1 (the proof of the other case is equivalent). Then, there exists a sequence of operations Ẽn of the form given in Eq. (13) such that   (80) lim Trn−kn Ẽn (ρ⊗n ) − σ ⊗kn 1 = 0 n→∞ where limn→∞ kn n = R(ρ → σ). If we use the asymptotic continuity of the function f , property SM3, we obtain
 
f Trn−kn Ẽn (ρ⊗n ) = f σ ⊗kn + o(kn ). (81) Let us now consider the lhs of the above equation. Using the properties of the monotone f , together with the definition of Ẽn in terms of sub-linear ancillae and allowed operations, we can prove the following chain of inequalities,     h i   

f Trn−kn Ẽn (ρ⊗n ) ≤ f Ẽn (ρ⊗n ) = f TrA En (ρ⊗n ⊗ ηn(A) ) ≤ f En (ρ⊗n ⊗ ηn(A) ) ≤ f ρ⊗n ⊗ ηn(A)  

(82) ≤ f ρ⊗n + f ηn(A) ≤ f ρ⊗n + o(n) where the first and second inequalities follow from property SM4, the equality follows from the definition of Ẽn , see Eq. (13), the third inequality follows from monotonicity under allowed operations, property SM1, the forth inequality from sub-additivity, property SM5, and the last one from the fact that the ancillary system is sub-linear in n together with property SM6. Thus, combining the last two equations, we get

f ρ⊗n ≥ f σ ⊗kn + o(n). (83) We can now divide the left and right hand side of the above equation by n, obtaining

kn 1 1 f σ ⊗kn + o(1). f ρ⊗n ≥ n n kn (84) By taking the limit of n → ∞, and using the fact that f is regularisable (which follows from property SM6) together with the definition of rate, we get f ∞ (ρ) ≥ R(ρ → σ) f ∞ (σ) . (85) We can also consider the reverse transformation, mapping n copies of the state σ into kn′ copies of ρ. Using the same steps used above, together with the fact that the monotone f is equal to zero over free states, property SM2, we can show that f ∞ (σ) ≥ R(σ → ρ) f ∞ (ρ) . (86) If we now use the reversibility property, which implies R(σ → ρ) = 1 R(ρ→σ) , f ∞ (ρ) f ∞ (ρ) ≥ R(ρ → σ) ≥ f ∞ (σ) f ∞ (σ) which proves the lemma. 32 we find that (87) Furthermore, we introduce a second small lemma, that can be found in Ref. [86, Prop. 13], Lemma 16. Given a regularisable function f : S (H⊗n ) → R, the regularised version is extensive, f ∞ (ρ⊗k ) = k f ∞ (ρ) , ∀ ρ ∈ S (H) , ∀ k ∈ N. (88) Proof. Consider a function h : R → R, such that limn→∞ h(n) = L < ∞. This is equivalent to say that ∀ ǫ > 0, ∃ c ∈ R : |h(n) − L| < ǫ, ∀ n > c. (89) Let us now consider an invertible function g : R → R, and consider m ∈ R such that n = g(m). Then, we can rewrite Eq. (89) as ∀ ǫ > 0, ∃ c ∈ R : |h(g(m)) − L| < ǫ, ∀ g(m) > c, (90) and by defining c̃ = g −1 (c), we get ∀ ǫ > 0, ∃ c̃ ∈ R : |h(g(m)) − L| < ǫ, ∀ m > c̃. (91) Therefore, we have limm→∞ h(g(m)) = L. If we choose h(n) = n1 f (ρ⊗n ), whose limit is L = f ∞ (ρ), and we use the reversible function g(m) = k · m where k ∈ N is fixed, we get f ∞ (ρ) = lim m→∞ 1 1 1 1 f (ρ⊗k·m ) = lim f ((ρ⊗k )⊗m ) = f ∞ (ρ⊗k ), m→∞ k·m k m k (92) which proves the lemma. We can now show that a single-resource theory which is reversible also satisfies the asymptotic equivalence property, and vice versa. Theorem 17. Consider the resource theory with allowed operations C and free states F. If the theory is reversible, then it satisfies the asymptotic equivalence property with respect to a function f satisfying the properties SM1 – SM6, and viceversa. Proof. (a) Let us first assume that the theory is reversible. Then, if we consider two non-free states ρ, σ ∈ S (H) such that f ∞ (ρ) = f ∞ (σ), and  we use Lem. 15, we find that the rate of conversion is R(ρ → σ) = 1. Then, there exists a sequence of operations Ẽn that approach this limit in one of two ways. In one case, we have   Trn−kn Ẽn (ρ⊗n ) − σ ⊗kn 1 → 0. (93) Notice that, since we have knn → 1, it follows that n − kn = o(n). Then, the above equation coincides with the second (A) ′(A) (A) part of Def. 1, where we are mapping ρ⊗kn into σ ⊗kn , and the sub-linear ancilla is ηn = ηn ⊗ ρn−kn , where ηn is completely arbitrary, and come from the definition of Ẽn . Alternatively, we can have that the sequence of maps is such that (94) Ẽkn (ρ⊗n ⊗ γkn −n ) − σ ⊗kn 1 → 0. We now use the monotonicity of the trace distance under discarding subsystems to obtain   Trkn −n Ẽkn (ρ⊗n ⊗ γkn −n ) − σ ⊗n 1 → 0. (95) Again, the above equation coincides with the second part of Def. 1, where we are mapping ρ⊗n into σ ⊗n , and the sub′(A) (A) linear ancilla is ηn = ηn ⊗ γkn −n . This proves the validity of one direction of the asymptotic equivalence property. To prove the other direction (existence of a sequence of maps  implies same value of the monotone on the two states), we can use the fact that, if there exists a sequence of maps Ẽn sending ρ⊗n into σ ⊗n , then the rate of conversion is R(ρ → σ) = 1. Then, with the help of Lem. 15, which is valid for reversible theories, we obtain that f ∞ (ρ) = f ∞ (σ). This proves the other direction of the asymptotic equivalence property. (b) Let now assume that the theory satisfies the asymptotic equivalence property. Consider any two non-free states ρ, σ ∈ S (H), and suppose that f ∞ (ρ) ≤ f ∞ (σ) (in the other case, the proof would follow analogously to the one we 33 are presenting). Take n, k ∈ N such that n f ∞ (ρ) = k f ∞ (σ), and let us use the extensivity of f ∞ , Lem. 16. Then, we have f ∞ (ρ⊗n ) = f ∞ (σ ⊗k ). Using the property of the function f shown in Eq. (78), we find that f ∞ (ρ⊗n ) = f ∞ (σ ⊗k ⊗ γn−k ), (96) where we add the free state γn−k ∈ F (n−k) to the right hand side since  n ≥ k. Then, we can use the asymptotic equivalence property, which implies the existence of a sequence of maps Ẽm·n m , see Eq. 13, such that lim m→∞ ⊗m Em·n (ρ⊗m·n ) − σ ⊗m·k ⊗ γn−k 1 = 0. If we use the monotonicity of the trace distance under partial tracing, we find that   lim Trm·(n−k) Em·n (ρ⊗m·n ) − σ ⊗m·k 1 = 0. m→∞ (97) (98) The existence of this sequence of maps implies that the rate of conversion R(ρ → σ) ≥ nk . At the same time, we can use  ′ asymptotic equivalence to find a sequence of maps Ẽm·n performing the reverse process. Using a similar argument m to the one presented above, we find that R(σ → ρ) ≥ nk . As a result, we find that the product of the forward and reverse rates of conversion is R(ρ → σ)R(σ → ρ) ≥ 1. However, this product cannot be higher than one, as otherwise we would be able to perform a cyclic transformation turning free states intro resourceful one, which is forbidden under allowed operations, see also Ref. [28]. Therefore, we find that R(ρ → σ)R(σ → ρ) = 1, which closes the proof. B Convex boundary and bank states In the following, we consider the case of a two-resource theory Rmulti defined on the Hilbert space H. The set of allowed operations is Cmulti = C1 ∩ C2 , where each Ci is a subset of the set of all CPTP maps that leave the set of states Fi invariant, i = 1, 2. We ask the resource theory Rmulti to satisfy the asymptotic equivalence property with respect to the monotones EF1 and EF2 . Furthermore, we assume that the two invariant sets satisfy the properties F1–F5. Thus, it ∞ ∞ and EF uniquely quantify the resources in this theory. follows from Thm. 4 and Prop. 5 that the two monotones EF 1 2 As a result, we can represent the state-space of Rmulti in a two-dimensional diagram, as shown in Fig. 3. We choose the two invariant sets of the theory to be disjoints, i.e., F1 ∩ F2 = ∅, and we focus on the set of bank states Fbank ⊂ S (H). Since in this section we are not making any assumption on the additivity (or extensivity) of the monotones EFi ’s, we have that the set of bank states is here defined as  ∞ ∞ Fbank = ρ ∈ S (H) | ∀ σ ∈ S (H) , EF (σ) > EF (ρ) or 1 1 ∞ ∞ EF2 (σ) > EF2 (ρ) or ∞ ∞ ∞ ∞ (σ) = EF (ρ) and EF (σ) = EF (ρ) . EF 1 2 2 1 (99) Notice that this set coincides with the one of Eq. (34) when property F5b is satisfied, and therefore the results we ∞ ∞ (F2 ) = 0 ∀ ρ ∈ F1 , and similarly (ρ) > EF obtain in this appendix apply to Sec. 4 as well. It is easy to show that EF 2 2 ∞ ∞ EF1 (ρ) > EF1 (F1 ) = 0 ∀ ρ ∈ F2 . Moreover, inside both invariant sets F1 and F2 we can find a subset of states with ∞ ∞ minimum value of the monotones EF and EF , respectively. We define these sets as 2 1   ∞ ∞ (ρ) ⊆ F1 , (100) E (σ) = min F1,min = σ ∈ F1 | EF F2 2 ρ∈F1   ∞ ∞ E F2,min = σ ∈ F2 | EF (ρ) ⊆ F2 . (101) (σ) = min F1 1 ρ∈F2 Given these two subsets, we can then define the following real intervals,  ∞  ∞ I1 = EF (F1 ) = 0 ; EF (F2,min ) , 1 1  ∞  ∞ I2 = EF (F2 ) = 0 ; EF (F1,min ) . 2 2 (102) (103) ∞ In what follows, we make use of the following two properties of the monotones EF ’s, i • Asymptotic continuity, which follows from the assumptions F1–F5 over the sets Fi ’s, as shown in Refs. [52, 87]. 34 ∞ ∞ –EF diagram. In the figure, the green segment is the invariant Figure 10: We represent part of the state-space S (H) in the EF 1 2 set F1 , the yellow one is F2 , and the black curve connecting these two segments is γbank , the curve of bank states of the theory, ∞ ∞ see Eq. (104). On the EF -axis we highlight the interval I1 defined in Eq. (102), and similarly for the interval I2 on the EF -axis. 1 2 Furthermore, the action of the function cbank : I1 → I2 , defined in Eq. (105), is shown for the input value ĒF1 . • Convexity, which follows from the assumptions F2 and F4 over the sets Fi ’s, as shown in Ref. [86], Prop. 13. ∞ We can now state the following lemma, concerning the value of the monotones EF ’s for bank states. i Lemma 18. Consider the multi-resource theory Rmulti with allowed operations Cmulti , and invariant sets F1 and F2 which satisfy properties F1–F5, and F1 ∩ F2 = ∅. If the theory satisfies the asymptotic equivalence property with respect to ∞ ∞ (ρ) ∈ I1 and EF (ρ) ∈ I2 . the monotones EF1 and EF2 , then for all bank states ρ ∈ Fbank we have that EF 1 2 ∞ Proof. Suppose, for example, that there exists a bank state ρ ∈ Fbank such that EF (ρ) ∈ / I1 , that is, ∃ σ ∈ F2,min 1 ∞ ∞ ∞ ∞ such that EF (σ) < E (ρ). By definition of F we also have that E (σ) ≤ E (ρ). These two inequalities, however, 2 F1 F2 F2 1 contradict the fact that ρ is a bank state, see Eq. (99), and conclude the proof. ∞ (ρ) = ĒF1 , and It is easy to show that for all ĒF1 ∈ I1 there exists (at least) one state ρ ∈ S (H) such that EF 1 ∞ the same holds for I2 . The proof that ∀ ĒF1 ∈ I1 , ∃ ρ ∈ S (H) : EF1 (ρ) = ĒF1 follows from two facts: (i) S (H) is a ∞ compact and path-connected set, and therefore its image under the (asymptotic) continuous function EF is a compact 1 and path-connected set in R, that is, a closed and bounded interval I1,S(H) , and (ii) I1 ⊆ I1,S(H) . ∞ ∞ diagram, the curve of bank states, which lies on part of the boundary of the –EF Let us now define, in the EF 2 1 state-space, as per definition in Eq. (99). The curve is defined as  ∞
∞ γbank = EF (ρ), EF (ρ) | ρ ∈ Fbank , (104) 1 2 where Fbank is the set of bank states of the theory. It is easy to see that this curve is completely contained within the subset of R2 given by I1 × I2 . Together with this curve, we can introduce the real-valued function cbank : I1 → I2 , defined as (105) cbank (EF1 ) = if (∃ P ∈ γbank such that P [0] = EF1 ) return P [1]. Essentially, this function checks the first element of the tuples in γbank , and returns the second element of the tuple whose first element is equal to EF1 . Since I1 is a closed interval in R, we have that for all EF1 ∈ I1 , the function cbank is well-defined. See Fig. 10 for the representation of the above curve of bank states in the resource diagram of the theory. ĒF1 ,ĒF2 of Eq. (37) satisfies the We will now prove the following two propositions, which assure that the monotone fbank property B2. This first proposition essentially tells us that the function cbank is monotonic decreasing.     (A) (A) (B) (B) Proposition 19. For all PA , PB ∈ γbank , where PA = EF1 , EF2 and PB = EF1 , EF2 , we have that (A) (B) (A) (B) EF1 < EF1 ⇔ EF2 > EF2 . 35 (106) (A) Proof. We prove the propositions in a single direction, as the other follows in analogue manner. Suppose that EF1 < (B) (A) (B) ∞ ∞ EF1 , and consider the states ρA , ρB ∈ Fbank such that EF (ρA ) = EF1 , and EF (ρB ) = EF1 . Since ρB belongs 1 1 to the set of bank states, we have that one of the following conditions, see Eq. (34), has to be satisfied for all states σ ∈ S (H), ∞ ∞ (ρB ). 1. EF (σ) > EF 1 1 ∞ ∞ 2. EF (σ) > EF (ρB ). 2 2 ∞ ∞ ∞ ∞ (ρB ). (σ) = EF (ρB ) and EF (σ) = EF 3. EF 2 2 1 1 Let us then take σ = ρA . In this case, options 1 and 3 are not possible, since they contradict the hypothesis. Therefore, (A) (B) ∞ ∞ (ρB ). In a similar manner, if EF1 = EF1 , the only possible (ρA ) > EF option 2 has to be valid, which implies that EF 2 2 ∞ ∞ option for ρB would have been EF2 (ρA ) = EF2 (ρB ), which concludes the proof. The second propositions tells us, instead, that the function cbank is convex.     (A) (A) (B) (B) Proposition 20. For all PA , PB ∈ γbank , where PA = EF1 , EF2 and PB = EF1 , EF2 , and for all λ ∈ [0, 1],   (C) (C) there exists a PC ∈ γbank , where PC = EF1 , EF2 , such that (C) (A) (B) (107) (C) EF2 (A) λ EF 2 (B) λ) EF2 (108) EF1 = λ EF1 + (1 − λ) EF1 , ≤ + (1 − (A) (B) ∞ (ρC ) = Proof. Let us consider, without losing in generality, that EF1 < EF1 , and take ρC ∈ Fbank such that EF 1 (A) (B) λ EF1 + (1 − λ) EF1 . This state always exists since I1 is a closed interval (and therefore is path-connected). Let us (A) (B) ∞ ∞ now define ρA , ρB ∈ Fbank such that EF (ρA ) = EF1 , and EF (ρB ) = EF1 . By convexity of the regularised relative 1 1 ∞ entropy distance EF1 , it follows that (B) (A) ∞ ∞ (λ ρA + (1 − λ) ρB ) . EF (ρC ) = λ EF1 + (1 − λ) EF1 ≥ EF 1 1 (109) Then, it is easy to show that (A) (B) ∞ ∞ EF (ρC ) ≤ EF (λ ρA + (1 − λ) ρB ) ≤ λ EF2 + (1 − λ) EF2 , 2 2 (110) ∞ . Since ρC ∈ Fbank , the where the first inequality follows from Prop. 19, and the second one from the convexity of EF 2
∞ ∞ point PC = EF1 (ρC ), EF2 (ρC ) is a point on the curve γbank . It is easy to see that the above propositions imply that cbank is (strictly) monotonic decreasing, and convex. Since this function is defined on the closed interval I1 ∈ R, we have that cbank is continuous (except, maybe, at its endpoints). Ē ,Ē F2 F1 of Eq. (37), and it always satisfies condition B2. Finally, it is Therefore, we can always define the monotone fbank worth noticing that all the results apply if one (or both) the monotones are of the form of Eq. (31), since they satisfy all the necessary properties, in particular they are linear in both the tensor product and the admixture of states. C Energy-entanglement interconversion protocol In this section we provide a protocol, based on the compression theorems [88] known in quantum information theory, to perform interconversion of energy and entanglement using two batteries and a bank, see Sec. 5.2.1 for revising the set-up we use. In our protocol, we assume that the bank is initially described by n ≫ 1 copies of a generic state ρ ∈ Sp0 , where p0 > 21 , see Eq. (69), while the batteries BW and BE are initially in the states ωW (k) and ωE (h), respectively. Our first step consists in using the allowed operation Eη ∈ Cmulti , see Eq. (70), with η = |θ1 i hθ1 |, to map the generic bank state ρ into (1) (1) ρin = p0 |Ψsinglet i hΨsinglet | + (1 − p0 ) |Ψtriplet i hΨtriplet | . (111) 36 Thus, the bank system is now described by n copies of the state ρin . Due to the central limit theorem, we can well approximate the state of the bank with an ensemble of its typical states, and in the following we will focus on the strongly typical ensemble,   dst ⊗n (1−p0 ) 1 X (1) (1) ⊗n p0 πi |Ψsinglet i hΨsinglet | ⊗ |Ψtriplet i hΨtriplet | , (112) Πst = dst i=1 where dst ≈ 2nh(p0 ) is the number of states contained in the strongly typical set, the πi ’s are the elements of the symmetric group acting on n copies of the two-qubit system, and h(·) is the binary entropy. Then, we can use a unitary operation to re-order the states in Πst so as to obtain ⊗n(1−h(p0 )) ⊗nh(p0 ) Π′st = σmm ⊗ |Ψsinglet i hΨsinglet | , (113) where σmm is the separable state introduced in Eq. (68). It is easy to see that this transformation, while leaving the ′ amount of entanglement in the bank constant, EFcss (ρ⊗n in ) = EFcss (Πst ), might not preserve the average energy. For this reason, while transforming the bank we also transform the energy battery, mapping ωW (k) into ωW (k + ∆k) to keep the energy fixed. We can now exchange some singlets with the entanglement battery. For example, we can perform a swap between the bank and the battery, moving in this way an integer number r of singlets from the bank into the battery. This transformation maps the state of the bank into ⊗n(1−h(p0 ))−r ⊗nh(p0 )+r ⊗ |Ψsinglet i hΨsinglet | Π′′st = σmm , (114) and transforms the state of the entanglement battery from ωE (h) into ωE (h + r). Furthermore, the transformation also modify the energy of the bank, so that we need to map the state of the energy battery from ωW (k +∆k) to ωW (k +∆k ′ ). It is then possible to map the state Π′′st into ′ Π′′′ st   dst ⊗n (1−p′0 ) 1 X (1) (1) ⊗n p′0 = ′ πi |Ψsinglet i hΨsinglet | , ⊗ |Ψtriplet i hΨtriplet | dst i=1 (115) where p′0 is chosen in order to satisfy the equality nh(p0 ) + r = nh(p′0 ), nh(p′0 ) and d′st = 2 (116) . The state Π′′′ st is the strongly typical ensemble associated with n copies of the state (1) (1) ρfin = p′0 |Ψsinglet i hΨsinglet | + (1 − p′0 ) |Ψtriplet i hΨtriplet | , where it is easy to show that the probability of occupation of the singlet is p′0 ≈ p0 − Π′′st Π′′′ st (117) r 1 p0 n log 1−p 0 for n ≫ 1. The transformation mapping into preserves the entanglement of the bank, while changing its energy. Therefore, while acting on the bank we have to modify the state of the energy battery as well, from ωW (k + ∆k ′ ) to ωW (k + ∆k ′′ ). In this way, we have modified the bank system by mapping n copies of ρin into n copies of ρfin , and we kept entanglement and energy fixed on the global system by modifying the states of the batteries. Notice that the protocol can be extended to the typical ensembles by using √ a sub-linear ancillary system, and by considering corrections to the exchanged energy and entanglement of order O( n). During the protocol, the bank has exchanged r singlets with the battery BE , so that the gain in entanglement for this battery is (118) ∆WE = EFcss (ωE (h + r)) − EFcss (ωE (h)) = r. In order to compute the amount of energy exchanged between the bank and the battery BW , we consider the difference ⊗n in average energy between ρ⊗n in and ρfin . In this way, we find that the amount of energy exchanged is ∆WW = MH (ωW (k + ∆k ′′ )) − MH (ωW (k)) = − ∆E p0 r, log 1−p 0 (119) that is, energy has been paid in order to gain entanglement during the process. The interconversion relation between the two resources is given by ∆E (120) ∆WW = − p0 ∆WE , log 1−p 0 37 and we only need to show that the bank state has changed in a negligible way with respect to the related bank monotone. It is worth noting that, since the current theory satisfies all the properties we have considered in the main text, the bank monotone coincides, modulo a multiplicative constant, with the relative entropy distance from the set of states Sp0 initially describing the bank. Indeed, it is easy to show that the relative entropy distance from this set is given by a linear combination of the monotones EFcss and MH . For ρ ∈ S1 we find that p0

log 1−p 0 MH (ρ) − M̄H , ESp0 (ρ) = inf D(ρ k σ) = EFcss (ρ) − ĒFcss + σ∈Sp0 ∆E (121) where we recall that ĒFcss = EFcss (σ) and M̄H = MH (σ), for any state σ ∈ Sp0 . The linear coefficient in the rhs of Eq. (121) is the (inverse) exchange rate that we find in the interconversion relation, Eq. (120). If we now consider the initial and final state of the bank, and we study how much the state is changed by the above protocol with respect to ESp0 , we find that ⊗n −1 ), (122) ESp0 (ρ⊗n fin ) − ESp0 (ρin ) = O(n so that, when n → ∞, we obtain that the state of the bank is only infinitesimally changed, and can be used again to perform another resource interconversion with the same initial exchange rate. D Proofs D.1 Main results In the first part of this appendix we provide the proofs of the results presented in the main text. We start with the proof of the following theorem, where it is shown that a multi-resource theory which satisfies the asymptotic equivalence property of Def. 1 has a unique quantifier for each of the resources present in the theory. This theorem is introduced in Sec. 3.3. Theorem 4. Consider the resource theory Rmulti with m resources, equipped with the batteries Bi ’s, where i = 1, . . . , m. m Suppose the theory satisfies the asymptotic equivalence property with respect to the set of monotones {fi }i=1 . If these monotones satisfy the properties M1 – M7, then the amount of i-th resource contained in the main system S is uniquely quantified by the regularisation of the monotone fi (modulo a multiplicative constant). Proof. Let us prove that f1∞ uniquely quantifies the amount of 1-st resource contained in the main system (the proof for the other fi6=1 ’s is analogous). We prove the theorem by contradiction. Suppose that there exists two monotones f1 and g1 satisfying the properties M1 – M7, such that 1. ∃ ρ ∈ S (HS ), where ρ 6∈ F1 , for which f1∞ (ρ) = g1∞ (ρ) (this is always possible by rescaling the monotone g). 2. ∃ σ ∈ S (HS ), where σ 6∈ F1 , for which f1∞ (σ) 6= g1∞ (σ) (that is, f1 is not unique). Consider now the values of f1∞ (ρ) and f1∞ (σ). If these are equal, it is easy to see, using the asymptotic equivalence property, that f1 is unique. Suppose instead that they are not equal. Then, there exists n, k ∈ N9 such that n f1∞ (ρ) = k f1∞ (σ). (123) Let us consider the system together with the batteries Bi ’s, initially in the state ρ⊗n ⊗ ω1 ⊗ . . . ⊗ ωm . Then, we take the states ωi′ ∈ S (HBi ), where i = 1, . . . , m, such that ′ fi∞ (ρ⊗n ⊗ ω1 ⊗ . . . ⊗ ωm ) = fi∞ (γn ⊗ ω1′ ⊗ . . . ⊗ ωm ) , ∀ i ∈ {1, . . . , m} , ∞ ∞ ′ fj (ωi ) = fj (ωi ) , ∀ i, j ∈ {1, . . . , m} , i 6= j, (n) (124) (125) where γn ∈ F1 .  Due to the asymptotic equivalence property, the conditions in Eq. (124) imply that there exists a sequence of maps ẼN N of the form of Eq. (13) such that 
⊗N  ′ ⊗N lim ẼN ρ⊗n ⊗ ω1 ⊗ . . . ⊗ ωm − (γn ⊗ ω1′ ⊗ . . . ⊗ ωm ) = 0, (126) N →∞ 9 Where we assume that all physically meaningful values of the 1 fi∞ ’s 38 are in Q, which we recall is dense in R. as well as another sequence of maps performing the reverse transformation. From the asymptotic continuity of g1 , property M7, it then follows that    
⊗N  ′ ⊗N g1 ẼN ρ⊗n ⊗ ω1 ⊗ . . . ⊗ ωm = g1 (γn ⊗ ω1′ ⊗ . . . ⊗ ωm + o(N ). (127) ) Let us consider the lhs of the above equation, and recall that the map ẼN is obtained by applying an allowed operation to N (A) copies of the system together with a sub-linear ancilla ηN , see Eq. (13). For simplicity, let us refer to ρ⊗n ⊗ω1 ⊗. . .⊗ωm as Ω in the following chain of inequalities,  h  i     

(A) (A) (A) g1 ẼN Ω⊗N = g1 TrA EN Ω⊗N ⊗ ηN ≤ g1 EN Ω⊗N ⊗ ηN ≤ g1 Ω⊗N ⊗ ηN  

(A) ≤ g1 Ω⊗N + g1 ηN ≤ g1 Ω⊗N + o(N ) (128) where the first inequality follows from property M4, the second one from the monotonicity of g1 under allowed operations, the third one from the sub-additivity of g1 , property M5, and the last inequality from property M6 and the fact that the ancilla is sub-linear in N . If we now combine this equation with the previous one, we divide both sides by N , and we send it to infinity, we obtain that the regularised version of g1 is such that,
′ g1∞ ρ⊗n ⊗ ω1 ⊗ . . . ⊗ ωm ≥ g1∞ (γn ⊗ ω1′ ⊗ . . . ⊗ ωm ). (129) By using the same argument for the sequence of maps performing the reverse transformation, we find that the above equation needs to hold as an equality, that is,
′ g1∞ ρ⊗n ⊗ ω1 ⊗ . . . ⊗ ωm = g1∞ (γn ⊗ ω1′ ⊗ . . . ⊗ ωm ). (130) We can now separate each contribution to g1 thanks to the property M2, use the fact that the batteries Bi6=1 ’s are not changing their value of g1 , property M1, and the fact that the final state of the system does not contain any resource associated with g1 , property M3. Then, we find that n g1∞ (ρ) = g1∞ (ω1′ ) − g1∞ (ω1 ) , (131) where we have also used Lem. 16. The same result follows for f1 , so that we find that n f1∞ (ρ) = f1∞ (ω1′ ) − f1∞ (ω1 ) . (132) If we now consider Eqs. (123) and (132), we find that k f1∞ (σ) = f1∞ (ω1′ ) − f1∞ (ω1 ) . (133) (k) We can add to the above equation the term f1∞ (γk ), where γk ∈ F1 , since this term is equal to zero due to property M3. Then, we find k f1∞ (σ) + f1∞ (ω1 ) = f1∞ (γk ) + f1∞ (ω1′ ) . (134) Now, we want to introduce the initial and final states of the batteries Bi6=1 ’s, so as to be sure that the transformation from σ ⊗k into γk does not violate the conservation of the other resources. Specifically, we introduce ωi , ωi′′ ∈ S (HBi ) for i 6= 1, such that
′′ fi∞ σ ⊗k ⊗ ω1 ⊗ ω2 ⊗ . . . ⊗ ωm = fi∞ (γk ⊗ ω1′ ⊗ ω2′′ ⊗ . . . ⊗ ωm ) , ∀ i ∈ {2, . . . , m} , (135) f1∞ (ωi ) = f1∞ (ωi′′ ) , ∀ i ∈ {2, . . . , m} , fj∞ (ωi ) = fj∞ (ωi′′ ) , ∀ i, j ∈ {2, . . . , m} , i 6= j. (136) (137) Then, using the constraints of Eq. (136) over the states of the Bi6=1 ’s batteries, we can re-write Eq. (134) as ′′ k f1∞ (σ) + f1∞ (ω1 ) + f1∞ (ω2 ) + . . . + f1∞ (ωm ) = f1∞ (γk ) + f1∞ (ω1′ ) + f1∞ (ω2′′ ) + . . . + f1∞ (ωm ). (138) If we now use Lem. 16 and property M1, we find that
′′ f1∞ σ ⊗k ⊗ ω1 ⊗ ω2 ⊗ . . . ⊗ ωm = f1∞ (γk ⊗ ω1′ ⊗ ω2′′ ⊗ . . . ⊗ ωm ) 39 (139) From  ′ Eqs. (135) and (139) it follows, using the asymptotic equivalence property, that there exists a sequence of maps ẼN N such that lim N →∞ ′ ẼN  σ ⊗k ⊗ ω1 ⊗ ω2 ⊗ . . . ⊗ ωm
⊗N  ⊗N ′′ − (γk ⊗ ω1′ ⊗ ω2′′ ⊗ . . . ⊗ ωm ) 1 = 0, (140) as well as a related sequence of maps performing the reverse transformation. Using the properties of g1 , as we did before, we find that k g1∞ (σ) = g1∞ (ω1′ ) − g1∞ (ω1 ) . (141) Then, combining Eqs. (131) and (141), we obtain that n g1∞ (ρ) = k g1∞ (σ) . (142) Finally, using Eq. (123) and the initial assumption on the state ρ, we find that f1∞ (σ) = g1∞ (σ) , (143) which contradicts our initial assumption. Therefore, f1∞ uniquely quantify the amount of 1-st resource contained in the main system. In the next theorem, first stated in Sec. 4.1, we show that in the presence of a bank two resources can always be exchanged one for the other, while the state of the bank is only infinitesimally modified by the resource interconversion. Theorem 8. Consider a resource theory Rmulti with two resources, equipped with the batteries B1 and B2 . Suppose the theory satisfies asymptotic equivalence with respect to the monotones EF1 and EF2 , i.e. the relative entropy distances from the invariant sets of the theory, and that these sets satisfy the properties F1, F2, F3, and F5b. Then, the resource interconversion of Eq. (40), where the bank has to transform in accord to condition X1, is solely regulated by the following relation, α ∆W1 = −β ∆W2 + δn . (42) Furthermore, when the number of copies of the bank system n is sent to infinity, we have that the above equation reduces to the following one, which we refer to as the interconversion relation, ∆W1 = − β ∆W2 , α (43) where the amount of resources exchanged ∆Wi is non-zero. Proof. Let us consider the resource interconversion of Eq. (40), where a global operation is performed over bank and batteries, and the sole constraint over the bank system is given by condition X1. As we discussed in Sec. 3.2, in order for the transformation to happen, the conditions of Eq. (21) need to be satisfied for both monotones EF1 and EF2 , which in particular implies that the amount of resources exchanged with the batteries is ∆Wi = n (EFi (ρ) − EFi (ρ̃)) , Ē i = 1, 2, (144) ,Ē F2 F1 is monotonic under the set of allowed operations, where we have used property F5b. Furthermore, since fbank property B7, we find that ĒF1 ,ĒF2 ĒF1 ,ĒF2 (ρ̃⊗n ⊗ ω1′ ⊗ ω2′ ). (145) (ρ⊗n ⊗ ω1 ⊗ ω2 ) = fbank fbank Then, since the global system is given by many copies of H, and since the bank monotone is additive, property B3, we can separate the contribution given by bank and batteries. Furthermore, from the definition of bank monotone, Eq. (37), and the main property of the batteries, condition M1, it follows that



α EF1 (ρ⊗n ) + EF1 (ω1 ) + β EF2 (ρ⊗n ) + EF2 (ω2 ) = α EF1 (ρ̃⊗n ) + EF1 (ω1′ ) + β EF2 (ρ̃⊗n ) + EF2 (ω2′ ) . (146) Now, if we re-order the terms in the above equation, and we use Eq. (37) again, we obtain Ē F1 fbank ,ĒF2 Ē F1 (ρ⊗n ) − fbank ,ĒF2 (ρ̃⊗n ) = α (EF1 (ω1′ ) − EF1 (ω1 )) + β (EF2 (ω2′ ) − EF2 (ω2 )) . 40 (147) If we use property X1 together with the definitions of ∆W1 and ∆W2 given in Eq. (23), we get that α ∆W1 = −β ∆W2 + δn , (148) where δn → 0 as n tends to infinity. However, we are still left to show that, when n → ∞, the amount of resources exchanged by the batteries remains finite. ĒF1 ,ĒF2 Let us first recall that the way in which the monotone fbank is built implies that this monotone is tangent to the the curve of bank states, see Eq. (104) in appendix B, state-space, see property B2 and Fig. 5. As a result, we have that
can be approximate, in the neighbourhood of Fbank ĒF1 , ĒF2 , by a line. This implies that, if we take the state ρ̃ in the set of bank states Fbank , such that (149) EF1 (ρ̃) = EF1 (ρ) − ǫ,
where we recall ρ ∈ Fbank ĒF1 , ĒF2 , and ǫ ≪ 1, we find that the value of the monotone EF2 for this state is EF1 (ρ̃) = EF2 (ρ) + α ǫ + O(ǫ2 ). β (150) Then, it is easy to see that, if we map ρ into ρ̃ during the resource interconversion, we obtain the following ∆W1 = n ǫ , ∆W2 = −n α ǫ + O(n ǫ2 ) , β δn = O(n ǫ2 ), (151) where the first two equations follow from Eq. (144), while the last one is given by Eq. (41). Thus, if we take ǫ ∝ n1 , and we send n to infinity, we get that the amount of resources ∆Wi exchanged during the transformations are finite and their value is arbitrary, while the change in the bank monotone over the bank system δn is infinitesimal. The next theorem can be found in Sec. 4.2. The theorem states that, given a multi-resource theory with a non-empty set of bank states, we can always build a single-resource theory out of it, by extending the class of allowed operations with the possibility of adding ancillary systems described by the bank states, see Def. 9. In particular, we show that if the multi-resource theory satisfies the asymptotic equivalence property, so does the single-resource theory with respect to the bank monotone of Eq. (37). Theorem 10. Consider the two-resource theory Rmulti with allowed operations Cmulti , and invariant sets F1 and F2 which satisfy the properties F1, F2, F3, and F5b. Suppose the theory satisfies the asymptotic equivalence property with respect
to the monotones EF1 and EF2 . Then, given the subset of bank states Fbank ĒF1 , ĒF2 , the single-resource theory Ē F1 Rsingle with allowed operations Csingle satisfies the asymptotic equivalence property with respect to fbank ,ĒF2 . Proof. (a) We start the proof by showing that, for the single resource theory Rsingle , the second statement in Def. 1 implies the first one. In other words, we want to show that for any two states ρ, σ ∈ S (H) which can be asymptotically mapped into one another with the allowed operations Cnsingle ,othe value of the bank monotone on the two states is the (s) (s) = 0, where such that limN →∞ ẼN (ρ⊗N ) − σ ⊗N same. Suppose there exists a sequence of operations ẼN 1 N these maps are of the form h i (s) (s) (A) ẼN (·) = TrA EN (· ⊗ ηN ) , (152)
(A) (s) with ηN ∈ S H⊗o(N ) an arbitrary state of a sub-linear ancilla, and EN an allowed operation for Rsingle . Likewise, suppose there is a sequence of maps that perform the reverse transformation. If we use the asymptotic continuity of the bank monotone, property B6, it follows that  
ĒF1 ,ĒF2 ĒF1 ,ĒF2 (s) ẼN (ρ⊗N ) = fbank σ ⊗N + o(N ). (153) fbank Then, by using the properties B1 – B7 of the bank monotone, we can prove the following chain of inequalities for the lhs of the above equation    h i   ĒF1 ,ĒF2 ĒF1 ,ĒF2 ĒF1 ,ĒF2 (s) (s) (A) (s) (A) TrA EN (ρ⊗N ⊗ ηN ) ≤ fbank ẼN (ρ⊗N ) = fbank fbank EN (ρ⊗N ⊗ ηN )    
ĒF1 ,ĒF2 ĒF1 ,ĒF2 ĒF1 ,ĒF2 (A) (A) ρ⊗N ⊗ ηN ηN ≤ fbank = fbank ρ⊗N + fbank
ĒF1 ,ĒF2 ≤ fbank ρ⊗N + o(N ) (154) 41 where the first inequality follows from monotonicity under partial trace, property B4, the second one from monotonicity under the allowed operations Csingle (that we still need to show), the equality follows from additivity, property B3, and the last inequality from the extensivity of the monotone, property B5. If we use the same argument for the sequence of maps performing the reverse transformation, and we regularise the monotones by dividing the equations by the number of copies N , and sending N to infinity, we find that Ē ,ĒF2 F1 fbank Ē ,ĒF2 F1 (ρ) = fbank (σ) , (155) which proves the asymptotic equivalence property in one direction. We still need to show that the bank monotone is monotonic under the allowed operations Csingle of the single-resource theory. Recall that the most general of these operations, Eq. (44), is given by   E (s) (ρ) = TrP (n) E(ρ ⊗ ρ⊗n (156) P ) ,
where E ∈ Cmulti , and we add n ∈ N copies of the bank state ρP ∈ Fbank ĒF1 , ĒF2 . Then, using the properties of the bank monotone, we can show that    

ĒF1 ,ĒF2 ĒF1 ,ĒF2 ĒF1 ,ĒF2 E (s) (ρ) = fbank TrP (n) E(ρ ⊗ ρ⊗n fbank ≤ fbank E(ρ ⊗ ρ⊗n P ) P )

ĒF1 ,ĒF2 ĒF1 ,ĒF2 ĒF1 ,ĒF2 = fbank ≤ fbank ρ ⊗ ρ⊗n (ρ) + fbank ρ⊗n P P Ē ,ĒF2 F1 = fbank (ρ) , (157) where the first inequality follows from property B4, the second one from the monotonicity under the allowed operations Cmulti , property B7, and the last two equalities from additivity, property B3, and the fact that the bank monotone is equal to zero over the bank states, property B1, respectively. (b) We now want to prove the other direction of the asymptotic equivalence property for the resource theory Rsingle , i.e., that the first statement in Def. 1 implies the second one. In other words, we want tonshowothat for all states ρ, ĒF1 ,ĒF2 ĒF1 ,ĒF2 (s) of the form given (σ), there exists a sequence of operations ẼN (ρ) = fbank σ ∈ S (H) such that fbank N in Eq. (152), mapping N copies of ρ into N copies of σ, where
N → ∞. Before proving this part of the theorem, we recall that, given the bank state ρP ∈ Fbank ĒF1 , ĒF2 , all other bank states ρ̃P ∈ Fbank are such that, if EF1 (ρ̃P ) = EF1 (ρP ) + δ with δ ≪ 1, then α (158) EF2 (ρ̃P ) = EF2 (ρP ) − δ + O(δ 2 ), β Ē ,Ē F1 F2 = 0 parametrises the line which is tangent to the state space and passes which follows from the fact
that fbank through the point ĒF1 , ĒF2 , see appendix B. Ē ,Ē F2 F1 , let us introduce the sequences of Given the two states ρ, σ ∈ S (H) with same value of the monotone fbank states {σn ∈ S (H)}n and {ρ̃P,n ∈ Fbank }n such that, for n ∈ N big enough, we have EF1 (σn ) = EF1 (σ) EF1 (ρ ⊗ EF2 (ρ ⊗ ρ⊗n P ) ⊗n ρP ) (159) ⊗n ), (160) ⊗n ), (161) = EF1 (σn ⊗ (ρ̃P,n ) = EF2 (σn ⊗ (ρ̃P,n )
where ρP ∈ Fbank ĒF1 , ĒF2 . From the above equations, and from the additivity of EF1 , which follows from property F5b, we obtain that 1 EF1 (ρ̃P,n ) = EF1 (ρP ) + (EF1 (ρ) − EF1 (σ)) . (162) n Notice that, for n → ∞, we have that n1 (EF1 (ρ) − EF1 (σ)) → 0, and therefore, for n sufficiently big, it follows from Eq. (158) that α 1 EF2 (ρ̃P,n ) = EF2 (ρP ) − (163) (EF1 (ρ) − EF1 (σ)) + O(n−2 ). β n If we now combine Eq. (161) and (163) together, we use the additivity of EF2 , and we use the fact that ρ and σ have the same value of the bank monotone, we obtain the following EF2 (σn ) = EF2 (σ) + O(n−1 ). 42 (164) Let us now focus on the operations mapping ρ into σ. We do this in two steps. First, we use the fact that the theory Rmulti satisfies asymptotic equivalence, and we consider the Eqs. (160) and (161). These equations imply that, for all n ∈ N, there exists of a sequence of maps ẼN,n N such that lim N →∞ ẼN,n  ρ ⊗ ρ⊗n P
⊗N  − σn ⊗ (ρ̃P,n )⊗n
⊗N 1 = 0.

As per definition of asymptotic equivalence, the maps ẼN,n : S H⊗N (n+1) → S H⊗N (n+1) are of the form h  i (A) ẼN,n (·) = TrA EN,n · ⊗ ηN (165) (166) where the map EN,n is an allowed operation of Rmulti acting on system and ancilla, and the state of the ancilla is
⊗f (N ) (A) , where f (N ) = o(N ). Notice that, in particular, we can take n to be a monotonic function ηN ∈ S H⊗n+1 of N , n = g(N ), such that limN →∞ g(N ) = ∞ and f (N )g(N ) = o(N ). For example, if f (N ) ∝ N 1/2 , we can chose g(N ) ∝ N 1/4 , so that their product is N 3/4 =n o(No).
(s) acting on S H⊗N . These maps are defined as We can now define the sequence of maps ẼN N h  i (s) ⊗N g(N ) ẼN (ρ⊗N ) = TrP ẼN,g(N ) ρ⊗N ⊗ ρP , (167) ⊗N g(N ) where we are tracing out the part of the system which was initially in the state ρP . It is interesting to notice that this system is super-linear in the number of copies N of ρ, a condition that seems to be necessary to achieve the conversion, see Ref. [10] for an example in thermodynamics. We can re-write these maps as h  i (s) (s) (A) ẼN (ρ⊗N ) = TrA EN ρ⊗N ⊗ ηN , (168) where we recall that the ancillary system still lives on a sub-linear number of copies of H, due to our choice of the (s) function g(N ), and the operation EN is an allowed operations for the theory Rsingle – compare it with Eq. (44) – defined as h  i (s) ⊗N g(N ) EN (·) = TrP EN,g(N ) · ⊗ ρP . (169) If we now use Eq. (165) together with the monotonicity of the trace distance under partial tracing, we find that lim (s) N →∞ ẼN (ρ⊗N ) − σg(N )
⊗N 1 = 0. (170)  To conclude the proof, we notice that the sequence of states σg(N ) N does not need to converge to σ with respect to the trace distance. However, if we consider the regularisation of the EFi ’s on these states, we find that lim N →∞ 1 ⊗N EFi (σg(N ) ) = EFi (σ), N i = 1, 2, (171) which follows from Eqs. (159) and (164). Then, we can use the asymptotic equivalence of Rmulti , which tells us that there exists a second sequence of allowed operations, and a sub-linear ancilla, such that we can asymptotically transform the state of the system into σ. This concludes the proof. The following corollary is stated in Sec. 4.2, and it shows that the bank monotone introduced in Eq. (37) coincides
with the relative entropy distance from the set of bank states Fbank ĒF1 , ĒF2 . Corollary 11. Consider the two-resource theory Rmulti with allowed operations Cmulti , and invariant sets F1 and F2 which equivalence property with respect satisfy the properties F1, F2, F3, and F5b. Suppose the theory satisfies the asymptotic
to the monotones EF1 and EF2 . If the subset of bank states Fbank ĒF1 , ĒF2 contains a full-rank state, then the Ē F1 bank monotone fbank constant. ,ĒF2 coincides with the relative entropy distance from this subset of states, modulo a multiplicative 43 Proof. We first notice that Thm. 10 promises us that, under the current assumptions over the theory Rmulti , we can construct a single-resource theory Rsingle with allowed operations Csingle as in Def. 9, which satisfies asymptotic equivalence Ē ,Ē F2 F1 . Furthermore, since this monotone satisfies the properties SM1 – SM6 with respect to the bank monotone fbank listed in appendix A, we can use Thm. 17 in the same appendix to prove that this single resource theory is reversible. If we then use the results of Ref. [30], we obtain that this monotone is the unique measure of resource for the theory Rsingle . What we need to show in this proof is that, actually, both
the bank monotone defined in Eq. (37) and the relative entropy distance from the set of bank states Fbank ĒF1 , ĒF2 satisfy the properties from SM1 to SM6, and therefore by uniqueness these two functions need to coincide (modulo a multiplicative constant). That the bank monotone satisfies these properties is easy to show. Indeed, its monotonicity under the class of operations Csingle , property SM1, is proved in part (a) of Thm. 10. Furthermore, all other properties directly follow from property B1
and the ones listed in Prop. 7. Showing that the relative entropy distance from the set of states Fbank ĒF1 , ĒF2 satisfies the same properties is not difficult either. First, we recall that the invariant sets of the theory, F1 and F2 , satisfy the properties F1, F2, F3 and F5b by hypothesis. This in turn implies that the subset of bank states under consideration satisfies
properties F1, F2 and F5b, as it follows from the Props. 24 and 23 in appendix D.2. That the subset Fbank ĒF1 , ĒF2 contains a full-rank state, property F3, is an hypothesis of this corollary.
With the help of the above properties we can show that the relative entropy distance from Fbank ĒF1 , ĒF2 satisfies the same properties of the bank monotone. That this relative entropy is monotonic under the set of operations Csingle , property SM1, is shown in Prop. 26. Furthermore, property SM2 follows from
the definition of relative entropy distance, see Eq. (9), while property SM3 follows from the fact that Fbank ĒF1 , ĒF2 satisfies the properties F1, F2, and F3. The properties SM4 and SM5 follow from the additivity of the set of bank states, property F5b. Finally, the fact that the monotone scales extensively is a consequences of the additivity of the set of bank states, as well as of the fact that a full-rank state is contained in this set, properties F5b and F3, respectively. D.2 Technical results In this section we provide some minor results that are used to prove some of the main theorems in the paper. In particular, the next proposition is used in Sec. 3.3, together with Thm. 4, to show that a multi-resource theory satisfying asymptotic equivalence with respect to the relative entropy distances from its invariant sets has unique resource quantifiers. This proposition is already known in the literature, see the references inside the proof. Proposition 5. Consider a resource theory Rmulti with m resources, equipped with the batteries Bi ’s, where i = 1, . . . , m. m Suppose the class of allowed operations is Cmulti and the invariant sets are {Fi }i=1 . If the invariant set Fi is of the form of Eq. (27), and it satisfies the properties F1 – F5, then the relative entropy distances from this set, EFi , is a regularisable monotone under the class of allowed operations, and it obeys the properties M1 – M7. Proof. Let us first show that the relative entropy distance EFi is a monotone for the multi-resource theory Rmulti , and that its regularisation is well-defined. These are necessary assumptions we have made in Def. 1. The fact that EFi is monotonic under the class of allowed operations Cmulti , and that in particular it is monotonic under the allowed operations in Ci , follows from the argument provided in the last paragraph of Sec. 2.1, and from the fact that Cmulti is obtained from the intersection of all the other classes of allowed operations, see Eq. (11). Furthermore, that the regularisation of EFi exists follows from the properties F3 and F4. In fact, for all ρ ∈ S (H), we have that 1 1 1 EFi (ρ⊗n ) = inf D(ρ⊗n k γn ) ≤ inf D(ρ⊗n k γ ⊗n ) = inf D(ρ k γ) ≤ D(ρ k γfull-rank ) (n) γ∈F n n γn ∈F n γ∈F (172) where the first inequality follows from the fact that the invariant sets are closed under tensor product, property F4, and the second inequality from the fact that they contain at least one full-rank state γfull-rank , property F3. Since the rhs of Eq. (172) is finite, and independent of n, we have that the regularisation of the EFi ’s is well-defined. In order for the monotone to satisfy the property M1, we can simply choose the states of the battery Bi to have a fixed value of the monotones EFj6=i , for all j ∈ {1, . . . , m}. Property M2, instead, follows from the fact that we want the batteries to be independent from each other, so as to address them individually. As a result, we choose the global invariant sets to be of the form Fi = Fi,S ⊗ Fi,B1 ⊗ . . . ⊗ Fi,Bm , where i = 1, . . . , m, the main system is S, and the Bi ’s refer to the batteries. This implies that the relative entropy distances from these sets are additive over (n) system and batteries. However, it is still possible for Fi⊗n to be a proper subset of Fi , since on the main systems or batteries we do not ask any additivity property. The validity of property M3 for EFi follows straightforwardly from the 44 definition of relative entropy distance, see Eq. (9). That EFi satisfies property M4 follows from property F5, since for all ρn ∈ S (H⊗n ) we have that EFi (Trk [ρn ]) = inf (n−k) γn−k ∈Fi D(Trk [ρn ] k γn−k ) ≤ inf (n) γn ∈Fi D(Trk [ρn ] k Trk [γn ]) ≤ inf (n) γn ∈Fi D(ρn k γn ) = EFi (ρn ), (173) where the first inequality follows from property F5, and the second one from the monotonicity of the relative entropy ⊗n under CPTP maps.
The monotones EFi ’s are also sub-additive, property M5, since for any two states ρn ∈ S (H ) ⊗k and ρk ∈ S H we have that EFi (ρn ⊗ ρk ) = = inf (n+k) γn+k ∈Fi inf (n) γn ∈Fi D(ρn ⊗ ρk k γn+k ) ≤ D(ρn k γn ) + inf (k) γk ∈Fi inf (n) γn ∈Fi (k) ,γk ∈Fi D(ρn ⊗ ρk k γn ⊗ γk ) D(ρk k γk ) = EFi (ρn ) + EFi (ρk ), (174) where the inequality follows from property F4 of the set Fi . Property M6 for the relative entropy distance EFi follows from similar considerations to the one presented in Eq. (172). In fact, we have that for all ρn ∈ S (H⊗n ),   ⊗n ⊗n ) = −S(ρn ) − Tr ρn log γfull-rank EFi (ρn ) = inf D(ρn k γn ) ≤ D(ρn k γfull-rank (n) γn ∈Fi   ⊗n ≤ −Tr ρn log γfull-rank ≤ n log λ−1 min , (175) where the first inequality follows from the fact that Fi contains a full-rank state, property F3, the second one from the fact that the von Neumann entropy is non-negative, and the last one from the fact that the optimal case is obtained when ρn is given by n copies of the pure state associated with the minimum eigenvalue λmin of the full-rank state γfull-rank . Finally, in Ref. [64], Lem. 1, it was shown that the relative entropy distance from a set F satisfying properties F1, F2, and F3 is asymptotic continuous. In the proof, it was required the set F to contain the maximally-mixed state. However, as it was noticed in Ref. [63], Lem. C.3, one simply needs F to contain a full-rank state. Thus, under the above properties on the free set, we have that EFi satisfies the property M7. The next proposition collects the properties of the bank monotone defined in Eq. (37). Proposition 7. Consider a resource theory Rmulti with allowed operations Cmulti , satisfying asymptotic equivalence with respect to the monotones EF1 and EF2 , i.e. the relative entropy distances from the invariant sets of the theory. Suppose Ē F1 that these sets satisfy the properties F1, F2, F3, and F5b. Then, the function fbank the following properties. Ē ,ĒF2 Ē ,ĒF2 Ē ,ĒF2 Ē ,ĒF2 Ē ,ĒF2 F1 B3 The function fbank F1 B4 The function fbank F1 B5 The function fbank F1 B6 The function fbank F1 B7 The function fbank ,ĒF2 introduced in Eq. (37) satisfies is additive. is monotonic under partial tracing. Ē F1 scales extensively. For any sequence {ρn ∈ S (H⊗n )}, we have fbank ,ĒF2 (ρn ) = O(n). is asymptotic continuous. is monotonic under the set of allowed operations Cmulti , since α and β are non-negative. Proof. Most of the properties listed above follows straightforwardly from the ones of the invariant sets Fi ’s. Here, we only focus on property B4, stating that Ē F1 fbank ,ĒF2 Ē F1 (Trk [ρn ]) ≤ fbank ,ĒF2
∀ n, k ∈ N , k < n , ∀ ρn ∈ S H⊗n . (ρn ), (176) In order to prove the above property, we make use of Lem. 22 and of the definition of bank monotone, see Eq. (37). First, let us divide the n copies of the system into two sets, so that H⊗n = H⊗k ⊗ H⊗n−k , and in the following equation we refer to S1 as the system described by the first k copies, and to S2 as the system described by the last n − k copies. 45 

In particular, ρS1 = Trn−k [ρn ] ∈ S H⊗k , and ρS2 = Trk [ρn ] ∈ S H⊗n−k . Then, we have the following chain of inequalities Ē F1 fbank ,ĒF2

(ρn ) = α EF1 (ρn ) − n ĒF1 + β EF2 (ρn ) − n ĒF2

≥ α EF1 (ρS1 ) + EF1 (ρS2 ) − n ĒF1 + β EF2 (ρS1 ) + EF2 (ρS2 ) − n ĒF2

= α EF1 (ρS1 ) − k ĒF1 + β EF2 (ρS1 ) − k ĒF2

+ α EF1 (ρS2 ) − (n − k)ĒF1 + β EF2 (ρS2 ) − (n − k)ĒF2 Ē F1 = fbank ,ĒF2 Ē F1 (ρS1 ) + fbank ,ĒF2 Ē F1 (ρS2 ) ≥ fbank ,ĒF2 Ē F1 (ρS2 ) = fbank ,ĒF2 (Trk [ρn ]), (177) where the first inequality follows from Lem. 22, and the second one from the fact that the bank monotone is non-negative, which itself follows from properties B1 and B2. The following proposition is used in Sec. 3.4 to show that single-resource theories whose class of allowed operations does not increase the average value of a given observable admit a monotone that is asymptotic continuous, see property M7. Proposition 21. Consider an Hilbert space H with dimension d, an Hermitian operator A ∈ B (H), and the function MA : S (H) → R defined as MA (ρ) = Tr [Aρ] − a0 , (178) where ρ ∈ S (H) is an element of the state-space, and a0 is the minimum eigenvalue Pn of A. When n copies of the Hilbert space are considered, Hn = ⊗ni=1 H(i) , the above operator is extended as An = i=1 A(i) , where A(i) ∈ B (H) acts on the i-th copy of the Hilbert space. Then, the function MA is asymptotic continuous. Proof. Consider two states ρn , σn ∈ S(H⊗n ), such that kρn − σn k1 → 0 for n → ∞. We are interested in the difference between the value of the function MA evaluated on ρn and σn . By definition, |MA (ρn ) − MA (σn )| = |Tr [(ρn − σn ) An ]| . P Now, we can diagonalise the operator ρn − σn = λ λ |ψλ i hψλ |. Then, we find |Tr [(ρn − σn ) An ]| = X λ λ hλ| An |λi ≤ X λ |λ| |hλ| An |λi| ≤ (179) X λ |λ| kAn k∞ , (180) where we are using the operator norm kOk∞ = sup|ψi∈H kO|ψik k|ψik , and the last inequality straightforwardly follows from the definition of operator norm. Then, due to the way in which we have defined An , it is easy to show that kAn k∞ = n kAk∞ , and therefore X X |λ| kAn k∞ = n kAk∞ |λ| = n kAk∞ kρn − σn k1 . (181) λ λ n Finally, notice that dim Hn = d , where d is fixed by the initial choice of H. Then, we have, |MA (ρn ) − MA (σn )| ≤ n log d kAk∞ kρn − σn k1 . (182) If we now divide by n both side of the inequality, we get that |MA (ρn ) − MA (σn )| ≤ log d kAk∞ kρn − σn k1 , n and if we send n → ∞, we obtain that 1 n (183) |MA (ρn ) − MA (σn )| → 0, which proves the theorem. (n) The following lemma states that, when the sets Fi ’s are such that Fi = Fi⊗n for all n ∈ N, property F5b, the relative entropy distances from these sets are super-additive. This lemma is used in Prop. 23 and Thm. 10. 46 
Lemma 22. Consider a state ρS1 ,S2 ∈ S H⊗2 , and suppose that the sets F1 and F2 satisfy property F5b, that is, (n) Fi = Fi⊗n for all n ∈ N, i = 1, 2. Then, the relative entropy distances from these sets, EF1 and EF2 , are such that EFi (ρS1 ,S2 ) ≥ EFi (ρS1 ) + EFi (ρS2 ) , i = 1, 2, (184) where ρS1 = TrS2 [ρS1 ,S2 ], and similarly ρS2 = TrS1 [ρS1 ,S2 ]. Furthermore, the above inequality is saturated if and only if ρS1 ,S2 = ρS1 ⊗ ρS2 . The result extends trivially to the case in which n > 2 copies of the system are considered. Proof. Let us consider the monotone EF1 , as the following argument can be equally applied to EF2 . By definition of relative entropy distance, we have that EF1 (ρS1 ,S2 ) = inf (2) σS1 ,S2 ∈F1 D(ρS1 ,S2 kσS1 ,S2 ) = −S(ρS1 ,S2 ) + inf (2) (−Tr [ρS1 ,S2 log σS1 ,S2 ]) , (185) σS1 ,S2 ∈F1 where S(ρS1 ,S2 ) = −Tr [ρS1 ,S2 log ρS1 ,S2 ] is the Von Neumann entropy of the state ρS1 ,S2 . From the sub-additivity of the Von Neumann entropy, we have that − S(ρS1 ,S2 ) ≥ −S(ρS1 ) − S(ρS2 ), (186) while from the property F5b it follows that inf (2) (−Tr [ρS1 ,S2 log σS1 ,S2 ]) = σS1 ,S2 ∈F1 = = inf (−Tr [ρS1 ,S2 log σS1 ⊗ σS2 ]) inf (−Tr [ρS1 log σS1 ] − Tr [ρS2 log σS2 ]) σS1 ,σS2 ∈F1 σS1 ,σS2 ∈F1 inf σS1 ∈F1 (−Tr [ρS1 log σS1 ]) + inf σS2 ∈F1 (−Tr [ρS2 log σS2 ]) . (187) From Eqs. (185), (186), and (187) it follows that EF1 (ρS1 ,S2 ) ≥ inf σS1 ∈F1 (−S(ρS1 ) − Tr [ρS1 log σS1 ]) + inf σS2 ∈F1 (−S(ρS2 ) − Tr [ρS2 log σS2 ]) = EF1 (ρS1 ) + EF1 (ρS2 ). (188) The following proposition is used in Sec. 4.1, in Prop. 26, and in Cor. 11. The proposition states that, when the (n) curve of bank states is strictly convex, and we consider n copies of a bank system, the set of bank states Fbank is given by the tensor product of n copies of states that are in the set Fbank , each of them with the same value of monotones EF1 and EF2 . (n) Proposition 23. Suppose the sets F1 and F2 satisfy property F5b, that is, Fi = Fi⊗n for all n ∈ N, i = 1, 2, and the set of bank states
Fbank is represented by a strictly convex curve in the resource diagram. Consider the set of bank states Fbank ĒF1 , ĒF2 defined in Eq. (35), where EF1 and EF2 are the relative entropy distances from the sets F1 and F2 , respectively. Then, when n ∈ N copies of the bank system are considered, we find that the set of bank states coincides with

(n) (189) Fbank = ρ1 ⊗ . . . ⊗ ρn ∈ S H⊗n | ∃ ĒF1 , ĒF2 such that ρ1 , . . . , ρn ∈ Fbank ĒF1 , ĒF2 .
Furthermore, we have that for all subset of bank state Fbank ĒF1 , ĒF2 ⊂ S (H), the corresponding bank subset in S (H⊗n ) is such that

(n) ⊗n Fbank ĒF1 , ĒF2 = Fbank ĒF1 , ĒF2 . (190)
Proof. We prove the theorem for n = 2, as the argument extends trivially for n > 2. Consider a state σS1 ,S2 ∈ S H⊗2 . From Lem. 22, it follows that (191) EFi (σS1 ,S2 ) ≥ EFi (σS1 ) + EFi (σS2 ) , i = 1, 2, where σS1 = TrS2 [σS1 ,S2 ], σS2 = TrS1 [σS1 ,S2 ], and the inequality is saturated iff σS1 ,S2 = σS1 ⊗ σS2 . Now, for both the states σS1 , σS2 ∈ S (H), select the bank states ρP1 , ρP2 ∈ Fbank such that EFi (σSj ) ≥ EFi (ρPj ) , i, j = 1, 2. 47 (192) Recall now that, in the EF1 –EF2 diagram, the curve of bank state is convex (see Prop. 20), and therefore given ρP1 , ρP2 ∈ Fbank , we can find another ρP3 ∈ Fbank such that 1 1 EF (ρP1 ) + EFi (ρP2 ) ≥ EFi (ρP3 ) , i = 1, 2, 2 i 2 (193) where the inequality
(when the curve is strictly convex) is saturated iff ρP1 , ρP2 , and ρP3 all belong to the same subset Fbank ĒF1 , ĒF2 . By combining Eqs. (191), (192), and (193), together with property F5b of the sets F1 and
F2 (that implies the additivity of the corresponding relative entropy distances), we find that for all σS1 ,S2 ∈ S H⊗2 , it exists a ρP3 ∈ Fbank such that EFi (σS1 ,S2 ) ≥ EFi (ρ⊗2 (194) P3 ) , i = 1, 2
where the inequality is saturated iff σS1 ,S2 = σS1 ⊗σS2 , and both σS1 and σS2 belong to the same subset Fbank ĒF1 , ĒF2 . Due to the definition of bank states given in Eq. (34), the thesis of this proposition follows. The next proposition shows that, when the invariant sets F1 and F2 are convex sets, the set of bank states
Fbank ĒF1 , ĒF2 , defined in Eq. (35), is convex as well. This proposition is used in Sec. 4.1, as well as in Thm. 11. the relative entropy distances from Proposition 24. Suppose that F1 and F2 are convex sets, property F2, and consider
these two sets, EF1 and EF2 . Then, the set of bank states Fbank ĒF1 , ĒF2 is convex, as well as its extension to the
(n) n-copy case, Fbank ĒF1 , ĒF2 , defined in Eq. (190).
Proof. Let us consider two states ρ1 , ρ2 ∈ Fbank ĒF1 , ĒF2 . For these two states, there exists σ1 , σ2 ∈ F1 such that EF1 (ρ1 ) = D(ρ1 k σ1 ) = ĒF1 , EF1 (ρ2 ) = D(ρ2 k σ2 ) = ĒF1 . (195a) (195b) Then, for all λ ∈ [0, 1], we have
EF1 λ ρ1 + (1 − λ) ρ2 = inf D(λ ρ1 + (1 − λ) ρ2 k σ) σ∈F1 ≤ D(λ ρ1 + (1 − λ) ρ2 k λ σ1 + (1 − λ) σ2 ) ≤ λ D(ρ1 k σ1 ) + (1 − λ) D(ρ2 k σ2 ) = ĒF1 , (196) where the first inequality follows from the fact that F1 is convex, property F2, and the second inequality from the joint convexity of the relative entropy. In the same way, it follows that
EF2 λ ρ1 + (1 − λ) ρ2 ≤ ĒF2 . (197)
Since ρ1 , ρ2 ∈ Fbank ĒF1 , ĒF2 , they satisfy the properties of Eq. (34), and therefore it has to be that, for all λ ∈ [0, 1],

EF1 λ ρ1 + (1 − λ) ρ2 = ĒF1 and EF2 λ ρ1 + (1 − λ) ρ2 = ĒF2 . (198)
Thus, we have that λ ρ1 + (1 − λ) ρ2 ∈ Fbank ĒF1 , ĒF2 . This result can be extended to the case of n ∈ N copies of the
(n) system, where the bank set Fbank ĒF1 , ĒF2 is defined as in Eq. (190). In this case, the proof is analogous to the one considered above, with the exception that in the rhs of Eqs. (195), and of the following ones, we add the multiplicative factor n. The next lemma is used in Prop. 26. The lemma states that, given the class of operations Cmulti for which F1 and F2 are invariant sets, the set of bank states Fbank , defined in Eq. (34), is invariant as well. Lemma 25. Consider a resource theory R
multi with allowed operations Cmulti , and two invariant sets F1 and F2 . Consider
the subset of bank states Fbank ĒF1 , ĒF2 as defined in Eq. (35). Then, for all E ∈ Cmulti , we have that Fbank ĒF1 , ĒF2 is an invariant set, that is

E (ρ) ∈ Fbank ĒF1 , ĒF2 , ∀ ρ ∈ Fbank ĒF1 , ĒF2 (199) Analogously, the set of bank states describing n copies of the bank system is invariant under the class of allowed operations (n) Cmulti . 48 
Proof. Let us consider ρ ∈ Fbank ĒF1 , ĒF2 , as well as the state E(ρ) obtained by applying the map E ∈ Cmulti to the bank state. Due to the monotonicity of EF1 and EF2 , we have that EF1 (E(ρ)) ≤ EF1 (ρ), and EF2 (E(ρ)) ≤ EF2 (ρ). Recall now that ρ is a bank state, which implies that ∀ σ ∈ S (H), one (or more) of the following options holds 1. EF1 (σ) > EF1 (ρ). 2. EF2 (σ) > EF2 (ρ). 3. EF1 (σ) = EF1 (ρ) and EF2 (σ) = EF2 (ρ). However, the monotonicity conditions given by EF1 and EF2 implies that E(ρ) violates options 1 and 2, so that option 3 is the only possible one. But this implies that EF1 (E(ρ)) = EF1 (ρ) and EF2 (E(ρ)) = EF2 (ρ), meaning that E(ρ) ∈
(n) Fbank ĒF1 , ĒF2 . The same argument applies to the set Fbank , when n copies of the system are considered. Indeed, (n) this case is analogous to the one considered above, with the sole difference that now the state ρ ∈ Fbank , the state (n) σ ∈ S (H⊗n ), and the operations we use are in the class Cmulti defined in Sec. 2.2.
The last proposition of this section shows that the relative entropy distance from the set Fbank ĒF1 , ĒF2 is monotonic under the class of operations Csingle , introduced in Def. 9. This proposition is used in Cor. 11. Proposition 26. Consider a multi-resource theory Rmulti with two resources, whose allowed operations Cmulti leave the sets F1 and F2 invariant. Suppose these invariant sets satisfy the
properties F1, F2, F3, and F5b. Then, the relative entropy distance from the subset of bank states Fbank ĒF1 , ĒF2 is monotonic under both the class of operations Cmulti and the class Csingle introduced in Def. 9. Proof. 1. Here we show monotonicity of the relative entropy distance with respect to the addition of an ancillary system
described by n ∈ N copies of a bank states. Consider the state ρ ∈ S (H), and the bank state ρP ∈ Fbank ĒF1 , ĒF2 . Then, we have EFbank (ĒF 1 ,ĒF2 ⊗n ) (ρ ⊗ ρP ) = σ,σ = = inf P1 ,...,σPn ∈Fbank (ĒF1 ,ĒF2 ) D(ρ ⊗ ρ⊗n P k σ ⊗ σ P1 ⊗ . . . ⊗ σ P n ) n X D(ρP k σPi ) inf D(ρ k σ) + inf σ∈Fbank (ĒF1 ,ĒF2 ) i=1 σPi ∈Fbank (ĒF1 ,ĒF2 ) inf σ∈Fbank (ĒF1 ,ĒF2 ) D(ρ k σ) = EFbank (ĒF 1 ,ĒF2 ) (ρ), (200)
where the first equality follows from Prop. 23, and the last one from the fact that ρP ∈ Fbank ĒF1 , ĒF2 . 2. Now we show monotonicity of the relative entropy distance with respect to the allowed operations Cmulit . Let us consider a state ρ ∈ S (H), together with an operation E ∈ Cmulti . Then, we have that
D(E(ρ) k E(σ)) D(E(ρ) k σ) ≤ inf inf EFbank (ĒF ,ĒF ) E(ρ) = 2 1 σ∈Fbank (ĒF1 ,ĒF2 ) σ∈Fbank (ĒF1 ,ĒF2 ) ≤ inf σ∈Fbank (ĒF1 ,ĒF2 ) D(ρ k σ) = EFbank (ĒF 1 ,ĒF2 ) (ρ), (201) where the first inequality follows from Lem. 25, and the second one from the monotonicity of the relative entropy under CPTP maps. This result trivially extends to the case in which we have multiple copies of the system, since in Lem. 25 (n) (n) we have shown that Fbank is invariant under the allowed operations Cmulti for all n ∈ N. 3. We show the monotonicity of the relative entropy with respect to partial tracing when the ancillary system is composed by just one copy. However, the result straightforwardly extends to the case in which the ancillary system is
composed by n ∈ N copies. Let us consider the state ρS1 ,S2 ∈ S H⊗2 . Then, we have that EFbank (ĒF 1 ,ĒF2 ) (TrS2 [ρS1 ,S2 ]) = σ D(TrS2 [ρS1 ,S2 ] k σS1 ) inf (ĒF1 ,ĒF2 ) S1 ∈Fbank = ≤ inf D(TrS2 [ρS1 ,S2 ] k TrS2 [σS1 ⊗ σS2 ]) inf D(ρS1 ,S2 k σS1 ⊗ σS2 ) = EFbank (ĒF σS1 ,σS2 ∈Fbank (ĒF1 ,ĒF2 ) σS1 ,σS2 ∈Fbank (ĒF1 ,ĒF2 ) 1 ,ĒF2 ) (ρS1 ,S2 ), (202) where the second equality follows from Prop. 23, while the inequality follows from the monotonicity of the relative entropy distance under CPTP maps. 49 References [1] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Physical Review A, 54(5):3824, 1996. [2] Eric M. Rains. Entanglement purification via separable superoperators. arXiv:quant-ph/9707002, 1997. [3] Vlatko Vedral and Martin B. Plenio. Entanglement measures and purification procedures. Physical Review A, 57(3):1619–1633, 1998. [4] Eric M. Rains. Bound on distillable entanglement. 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