Papers by Bibhas Adhikari
Quantum Information and Computation
Recently, dimensionality testing of a quantum state has received extensive attention (Ac{\'i}... more Recently, dimensionality testing of a quantum state has received extensive attention (Ac{\'i}n et al. Phys. Rev. Letts. 2006, Scarani et al. Phys. Rev. Letts. 2006). Security proofs of existing quantum information processing protocols rely on the assumption about the dimension of quantum states in which logical bits are encoded. However, removing such assumption may cause security loophole. In the present paper, we show that this is indeed the case. We choose two players' quantum private query protocol by Yang et al. (Quant. Inf. Process. 2014) as an example and show how one player can gain an unfair advantage by changing the dimension of subsystem of a shared quantum system. To resist such attack we propose dimensionality testing in a different way. Our proposal is based on CHSH like game. As we exploit CHSH like game, it can be used to test if the states are product states for which the protocol becomes completely vulnerable.
Linear Algebra and its Applications
Advances in Complex Systems
In this paper, we propose a self-organization mechanism for newly appeared nodes during the forma... more In this paper, we propose a self-organization mechanism for newly appeared nodes during the formation of corona graphs that define a hierarchical pattern in the resulting corona graphs and we call it self-organized corona graphs (SoCG). We show that the degree distribution of SoCG follows power-law in its tail with power-law exponent approximately 2. We also show that the diameter is less equal to 4 for SoCG defined by any seed graph and for certain seed graphs, the diameter remains constant during its formation. We derive lower bounds of clustering coefficients of SoCG defined by certain seed graphs. Thus, the proposed SoCG can be considered as a growing network generative model which is defined by using the corona graphs and a self-organization process such that the resulting graphs are scale-free small-world highly clustered growing networks. The SoCG defined by a seed graph can also be considered as a network with a desired motif which is the seed graph itself.
Journal of Algebraic Combinatorics
Construction of non-isomorphic cospectral graphs is a nontrivial problem in spectral graph theory... more Construction of non-isomorphic cospectral graphs is a nontrivial problem in spectral graph theory specially for large graphs. In this paper, we establish that graph theoretical partial transpose of a graph is a potential tool to create non-isomorphic cospectral graphs by considering a graph as a clustered graph.
IEEE Transactions on Network Science and Engineering
International Journal of Quantum Information
This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomi... more This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some ...
Journal of Mathematical Chemistry
Journal of Statistical Mechanics: Theory and Experiment
Quantum Information Processing
Quantum discord refers to an important aspect of quantum correlations for bipartite quantum syste... more Quantum discord refers to an important aspect of quantum correlations for bipartite quantum systems. In our earlier works we have shown that corresponding to every graph (combinatorial) there are quantum states whose properties are reflected in the structure of the corresponding graph. Here, we attempt to develop a graph theoretic study of quantum discord that corresponds to a necessary and sufficient condition of zero quantum discord states which says that the blocks of density matrix corresponding to a zero quantum discord state are normal and commute with each other. These blocks have a one to one correspondence with some specific subgraphs of the graph which represents the quantum state. We obtain a number of graph theoretic properties representing normality and commutativity of a set of matrices which are indeed arising from the given graph. Utilizing these properties we define graph theoretic measures for normality and commutativity that results a formulation of graph theoretic quantum discord. We identify classes of quantum states with zero discord using the said formulation.
Expert Systems with Applications
IEEE Transactions on Knowledge and Data Engineering
Quantum Information Processing, 2017
Graph representation of quantum states is becoming an increasingly important area of research to ... more Graph representation of quantum states is becoming an increasingly important area of research to investigate combinatorial properties of quantum states which are nontrivial to comprehend in standard linear algebraic density matrix based approach of quantum states. In this paper, we propose a general weighted directed graph framework for investigating properties of a large class of quantum states which are defined by three types of Laplacian matrices associated with such graphs. We generalize the standard framework of defining density matrices from simple connected graphs to density matrices using both combinatorial and signless Laplacian matrices associated with weighted directed graphs with complex edge weights and with/without loops. We also introduce a new notion of Laplacian matrix which we call signed Laplacian matrix associated with such graphs. We produce necessary and/or sufficient conditions for such graphs to represent pure and mixed quantum states. Using these criteria we finally determine the graphs whose corresponding density matrices represent entangled pure states which are well-known and important for quantum computation applications. It is important to observe that all these entangled pure states share a common combinatorial structure.
Discrete Applied Mathematics, 2017
Product graphs have been gainfully used in literature to generate mathematical models of complex ... more Product graphs have been gainfully used in literature to generate mathematical models of complex networks which inherit properties of real networks. Realizing the duplication phenomena imbibed in the definition of corona product of two graphs, we define Corona graphs. Given a small simple connected graph which we call seed graph, Corona graphs are defined by taking corona product of a seed graph iteratively. We show that the cumulative degree distribution of Corona graphs decay exponentially when the seed graph is regular and cumulative betweenness distribution follows power law when seed graph is a clique. We determine explicit formulae of eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues of Corona graphs when the seed graph is regular. Computable expressions of eigenvalues and signless Laplacian eigenvalues of Corona graphs are also obtained when the seed graph is a star graph.
Physical Review A, 2016
In this paper, we consider the separability problem for bipartite quantum states arising from gra... more In this paper, we consider the separability problem for bipartite quantum states arising from graphs. Earlier it was proved that the degree criterion is the graph theoretical counterpart of the familiar PPT criterion for separability, although there are entangled states with positive partial transpose for which degree criterion fails. Here, we introduce the concept of partially symmetric graphs and degree symmetric graphs by using the well-known concept of partial transposition of a graph and degree criteria, respectively. Thus, we provide classes of bipartite separable states of dimension m × n arising from partially symmetric graphs. We identify partially asymmetric graphs which lack the property of partial symmetry. Finally we develop a combinatorial procedure to create a partially asymmetric graph from a given partially symmetric graph. We show that this combinatorial operation can act as an entanglement generator for mixed states arising from partially symmetric graphs.
In this paper, we propose a connection model for formation of networks having both positive and n... more In this paper, we propose a connection model for formation of networks having both positive and negative links. We define utility functions of a node and hence of a network by using balanced and unbalanced triads in the network. Further, using those utility functions we define strongly efficient and conditionally efficient networks. We also find sufficient conditions for which a network will be efficient. Further, we introduce pairwise stable networks having both positive and negative links. Finally, we provide certain interesting results regarding pairwise stable networks.
Siam Journal on Matrix Analysis and Applications, 2009
Structured backward perturbation analysis plays an important role in the accuracy assessment of c... more Structured backward perturbation analysis plays an important role in the accuracy assessment of computed eigenelements of structured eigenvalue problems. We undertake a detailed structured backward perturbation analysis of approximate eigenelements of linearly structured matrix pencils. The structures we consider include, for example, symmetric, skew-symmetric, Hermitian, skew-Hermitian, even, odd, palindromic, and Hamiltonian matrix pencils. We also analyze structured backward errors of approximate eigenvalues and structured pseudospectra of structured matrix pencils.
ABSTRACT In this paper, we propose a distorted information diffusion protocol to detect diversity... more ABSTRACT In this paper, we propose a distorted information diffusion protocol to detect diversity of different network models. The protocol is inspired by the fact that, in real social networks, the information diffusion get influenced by the property of nodes and conduct of the links. Thus, information get de-formed/distorted during diffusion process and true amount of information from a spreader never reach to all the nodes in the network. We consider a single spreader which has maximum de-gree in the network. We divide the entire network in to different layers where the nodes in a layer are defined by the nodes having a fixed distance from the spreader. We observe that the amount of information available at every node in a layer after the diffusion process reaches to saturation, is not equal. Thus, we define density of information profile of a layer that measures the density of information available in a layer. Finally, we define a vector, which we call information diversity vector whose components are density of information of the layers. The dimension of the diversity vector is the number of layers in the entire network. We implement the protocol in standard network models which include Albert-Barabasi preferential attachment Model (ABM), Hierarchial network generation Model (HM), and Watts-Strogatz Model (WSM). We also simulate the protocol in real world networks which include ego-Facebook Network, Collaboration network of ArXiv General Relativity, and Collaboration network of ArXiv High Energy Physics Theory. The simulated results show that the information diversity vectors of ABM and HM is far from reflecting the same in real world networks. However, diversity vector of WSM is similar to that of real world networks which we consider in this paper.
Quantum Information Processing, 2016
Building upon our previous work, on graphical representation of a quantum state by signless Lapla... more Building upon our previous work, on graphical representation of a quantum state by signless Laplacian matrix, we pose the following question. If a local unitary operation is applied to a quantum state, represented by a signless Laplacian matrix, what would be the corresponding graph and how does one implement local unitary transformations graphically? We answer this question by developing the notion of local unitary equivalent graphs. We illustrate our method by a few, well known, local unitary transformations implemented by single qubit Pauli and Hadamard gates. We also show how graph switching can be used to implement the action of the C N OT gate, resulting in a graphical description of Bell state generation.
Linear Algebra and its Applications, 2016
In this work we propose graphical representation of quantum states. Pure states require weighted ... more In this work we propose graphical representation of quantum states. Pure states require weighted digraphs with complex weights, while mixed states need, in general, edge weighted digraphs with loops; constructions which, to the best of our knowledge, are new in the theory of graphs. Both the combinatorial as well as the signless Laplacian are used for graph representation of quantum states. We also provide some interesting analogies between physical processes and graph representations. Entanglement between two qubits is approached by the development of graph operations that simulate quantum operations, resulting in the generation of Bell and Werner states. As a biproduct, the study also leads to separability criteria using graph operations. This paves the way for a study of genuine multipartite correlations using graph operations.
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Papers by Bibhas Adhikari