Finite-dimensional Hilbert space and frame quantization

Finite dimensional Hilbert space and frame quantization Nicolae Cotfas1 , Jean Pierre Gazeau2 and Apostol Vourdas3 1 Faculty of Physics, University of Bucharest, PO Box 76 - 54, Post Office 76, Bucharest, Romania 2 Laboratoire APC, Université Paris Diderot, 10, rue A. Domon et L. Duquet, 75205 Paris Cedex13, France 3 Department of Computing, University of Bradford, Bradford BD7 1DP, United Kingdom E-mail: [email protected], [email protected], [email protected] Abstract. The quantum observables used in the case of quantum systems with finitedimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder-Berezin-Toeplitz type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables. Finite quantum systems 2 1. Introduction Weyl’s formulation of quantum mechanics [46] opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finite-dimensional systems. Based on Weyl’s approach, generalized by Schwinger [38], several authors (e.g., [35, 36, 39, 40, 10, 22, 47, 28, 26, 41, 42, 43, 44, 37]) investigated the analogue of the harmonic oscillator in finite-dimensional Hilbert spaces. This work combines concepts from quantum mechanics with discrete mathematics (number theory, finite sums [8], combinatorics, etc). The general formalism of coherent states [33, 34, 17, 48, 15] can also be used in the context of finite-dimensional quantum systems. This powerful method defines continuous systems of coherent states as well as finite systems of coherent states, usually called finite tight frames [3, 11, 12, 18]. In finite quantum systems mainly continuous systems of coherent states have been studied. Among recent explorations of finite systems of coherent states, see for instance [13, 15]. In this paper we study finite frames in the context of finite quantum systems. This is a finite frame quantization, and it is a finite version of a conventional Klauder-Berezin-Toeplitz type coherent state quantization. In section 2 we briefly present topics from the theory of finite quantum systems, which are needed later. In section 3 we present the general formalism for finitedimensional tight frames and the related quantization. We define covariant and contravariant symbols and we briefly recall and complete interesting probabilistic and semiclassical aspects of the coherent states/frame formalism which have been developed in previous works [2, 12]. We also define Wigner functions and Weyl functions in this context. In section 4 we present various examples of tight frames. The first example is a ‘lattice frame’ and consists of d2 vectors (where d is the dimension of the Hilbert space). It depends on a ‘fiducial’ vector, and we use an eigenvector of the discrete Fourier transformation which is related to the harmonic oscillator vacuum through a Zak [50] or Weil [45] transform. This example has been previously defined in [49] in terms of the Perelomov method by starting from a discrete version of the Heisenberg group. The second example is a ‘sparse frame’ and consists of d2 /n vectors where n is a divisor of d such that the (n, d/n) are coprime. This is a novel example and is discussed in section 4.2. In section 5 we show how the formalism can be used for practical calculations (matrix elements, spectra, etc) in the quantization context. We conclude in section 6 with a discussion of our results. 2. Finite quantum systems We consider a quantum system where position and momentum take values in the ring Zd = Z/dZ = {0, 1, ..., d−1} of the integers modulo d, where d is a fixed positive integer. The Hilbert space H of this system is d-dimensional, and we describe it by using the Finite quantum systems 3 orthonormal basis of ‘position states’ {|e0 i, |e1 i, ..., |ed−1 i}. The finite Fourier transform d−1 1 X 2πi kℓ e d |ek iheℓ | F =√ d k,ℓ=0 F : H −→ H, (1) allows us to consider the directly related orthonormal basis of ‘momentum states’ {|f0 i, |f1 i, ..., |fd−1 i} satisfying the relations d−1 1 X 2πi kℓ |fk i = F |ek i = √ e d |eℓ i , d ℓ=0 Each state |ψi ∈ H can be expanded as |ψi = d−1 X ℓ=0 ψℓ |eℓ i = d−1 X k=0 d−1 1 X − 2πi ℓk |eℓ i = F |fℓ i = √ e d |fk i . d k=0 + ψ̃k |fk i where the functions ψ : Zd −→ C : ℓ 7→ ψℓ and ψ̃ : Zd −→ C : k 7→ ψ̃k satisfying d−1 d−1 1 X 2πi ℓk e d ψ̃k , ψℓ = √ d k=0 1 X − 2πi kℓ ψ̃k = √ e d ψℓ . d ℓ=0 (2) are the corresponding ‘wavefunctions’ in the position and momentum representations. The parity operator with respect to the origin is defined as P (0, 0) = F 2 and P (0, 0)2 = F 4 = IH . The self-adjoint ‘position’ and ‘momentum’ operators x, p : H −→ H, x= d−1 X ℓ=0 ℓ |eℓ iheℓ | , p= d−1 X k=0 k |fk ihfk | , (3) are defined modulo d and satisfy the relations F xF + = p , x|eℓ i = ℓ|eℓ i , x|fk i = 1 d F pF + = −x , p|fk i = k|fk i , p|eℓ i = 1 d The function [42] Pd−1 ℓ,j=0 Pd−1 k,m=0 le 2πi ℓ(k−j) d ke |fj i , 2πi k(m−ℓ) d |em i . d−1 ∆0 (x) = and its derivatives 1 X 2πi ℓx ed , d ℓ=0 (4) d−1 ds 1X ∆s (x) = s ∆0 (x) = dx d ℓ=0  s 2πi 2πi ℓ e d ℓx d (5) are useful in the calculation of matrix elements. For x ∈ Zd , the ∆0 (x) = δ(x, 0) where δ(x, 0) is the Kronecker delta in Zd . We can now calculate the commutator d hen |[x, p]|em i = (n − m)∆1 (n − m) (6) 2πi We note here that matrix elements of ‘angular operators’ like x, p involve the summation over a ‘period’ from N to N +d−1, and the result does depend on N . Only exponentials of these operators (like in the displacement operators below) are single-valued. Finite quantum systems 4 Table 1. List of the eigenvalues ξn of the commutator [x, p] in the case d = 21. n ξn n 1 2 3 4 5 6 7 -133.9652067678112 i -27.11600086881775 i 0.5632328492846453 i 3.198831527436455 i 3.337619084687670 i 3.342159991389251 i 3.342252660619915 i 8 9 10 11 12 13 14 ξn n 3.342253797136738 3.342253804904009 3.342253804929771 3.342253804929802 3.342253804929803 3.342253804930850 3.342253805426666 i i i i i i i 15 16 17 18 19 20 21 ξn 3.342253907182515 3.342264884479544 3.342954064008703 3.369561581989102 4.015171698810642 13.73901531616308 92.75011344338981 i i i i i i i In table 1 we give the eigenvalues of this d × d matrix for the case d = 21. It is seen that the commutator is approximately a multiple of the unit matrix, as it should (for large d). The displacement operators [38, 39, 49] A , B : H −→ H , A=e 2πi p d , B =e 2πi x d , (7) are single-valued and 2πi k d A|eℓ i = |eℓ−1 i , A|fk i = e B|eℓ i = e B|fk i = |fk+1 i , 2πi ℓ d |eℓ i , |fk i , Ad = B d = IH , Aα B β = e 2πi αβ d B β Aα . General displacements operators with respect to (α, β) ∈ Zd × Zd are given by πi D(α, β) = Aα B β e− d αβ ; [D(α, β)]† = D(−α, −β) . (8) For an arbitrary operator Θ, it has been shown [44] that d−1 1 X D(α, β)Θ[D(α, β)]† = IH TrΘ . d α,β=0 (9) The parity operator with respect to (α, β) ∈ Zd × Zd is given by P (α, β) = D(α, β)P (0, 0)[D(α, β)]† , [P (α, β)]2 = IH (10) and it is related to the displacement operators through the Fourier transform d−1 1 X 2πi (αδ−βγ) P (γ, δ) . ed D(α, β) = d γ,δ=0 (11) 3. Tight finite frames and finite frame quantization Let H be a d-dimensional Hilbert space and {|e0 i, |e1 i, ..., |ed−1 i} an orthonormal basis of H. Let X = {a0 , a1 , ..., aM −1 } be a set of ‘parameters’ or ‘indices’, with M ≥ d. An example will be given later with X = Zd × Zd for which M = d2 . We consider an orthonormal system of d functions {φk : X −→ C}d−1 k=0 : M −1 X n=0 φj (an ) φk (an ) = δjk (12) Finite quantum systems 5 such that κn = d−1 X k=0 It follows that M −1 X n=0 |φk (an )|2 6= 0 for any n ∈ {0, 1, ..., M − 1} . (13) κn = d and κn ≤ 1 . The latter inequality is strict if d < M . We then consider the map X ∋ an 7→ |an i ∈ H defined by d−1 1 X |an i = √ φk (an ) |ek i , κn k=0 n = 0, ..., M − 1 . (14) The M vectors |an i overlap as: d−1 X 1 φk (an ) φk (am ) . han |am i = √ κn κm k=0 They solve the unity in H in the following way: M −1 X κn |an ihan | = IH . (15) (16) n=0 Therefore the vectors |an i form a tight finite frame in H [11, 12]. For later use we give the relation d−1 2πi 1 − πi αβ X d φk (an ) e d βk |ek−α i. D(α, β)|an i = √ e κn k=0 (17) M −1 The considered tight frame {|an i}n=0 defines two probability distributions which can be interpreted in terms of a Bayesian duality. The latter underlies practically all overcomplete families of states in finite- or infinite- dimensional resolving the unity. The intimate connection between certain of these families and statistical distributions has been explained in [19, 20, 21, 2]: (i) A prior distribution on the set of indices k ∈ {0, ..., d − 1}, with parameter an ∈ X , |φk (an )|2 k 7→ = |hek |an i|2 , (18) κn with d−1 X |hek |an i|2 = 1 for n ∈ {0, 1, ..., M −1} . k=0 Adopting a quantum mechanical language, this probability could be considered as concerning experiments performed on the system within some experimental protocol in order to measure the spectral values of a certain self-adjoint operator (a “quantum observable”) A acting in H (e.g. the position x in (3) and having P the spectral resolution A = k λk |ek ihek | (e.g. the measured positions “k” in (3)). Precisely, |hek |an i|2 is the probability to get the value λk from a measurement of A which is performed on the system when the latter is prepared in the “state” |an i. Finite quantum systems 6 (ii) A posterior distribution on the original set of parameters an ∈ X , equipped with uniform (discrete) measure, and with parameter k ∈ {0, ..., d − 1}, an 7→ |φk (an )|2 , (19) with M −1 X n=0 √ | κn |φk (an )|2 = 1 . The Bayesian duality stems in the two interpretations: the resolution of the unity verified by the states |an i, introduces a preferred prior measure on X , which is the |φk (an )|2 , with this distribution itself playing set of parameters of the distribution k 7→ κn the role of the likelihood function. The associated distributions an 7→ |φk (an )|2 on the original set X , indexed by k, become the related conditional posterior distributions. For a concrete example of such a duality in the discrete-continuous case, see Section 2 in [2]. 3.1. Covariant and contravariant symbols To each function f : X −→ C we associate the operator Af : H −→ H, Af = M −1 X n=0 κn f (an ) |an ihan | . (20) The matrix elements of Af with respect to the orthonormal basis {|ek i} are hek |Af |ej i = M −1 X n=0 κn f (an ) hek |an ihan |ej i = M −1 X f (an ) φk (an ) φj (an ) . (21) n=0 The operator Af corresponding to a real function f is self-adjoint, and this can be regarded as a Klauder-Berezin-Toeplitz type quantization [4, 5, 6, 7, 23, 24, 25] of f , the notion of quantization being considered here in a wide sense [15]. The eigenvalues of the matrix Af form the “quantum spectrum” of f (by opposition to its “classical spectrum” that is the set of its values f (an )). Given an operator A = Af , the function f , possibly not unique, is called upper [29] or contravariant symbol [6] (its analog in Quantum Optics is the so-called P -function) of A. We also introduce the function fˇ= Ǎf : X −→ R, such that fˇ(an ) = han |Af |an i = 1 = κn M −1 X m=0 M −1 X m=0 f (am ) κm f (am ) |han |am i|2 d−1 X 2 φk (an ) φk (am ) , (22) k=0 which is called lower [29] or covariant symbol [6] (its analog in Quantum Optics is the so-called Q-function) of Af . It can be viewed as finite version of the so-called Berezin transform of the original function f (see for instance [14]). Finite quantum systems 7 3.2. Stochastic aspect of a finite tight frame In [12] two of us have presented some stochastic properties displayed by finite tight frames. We give here a more complete account of these remarkable features. Let us introduce the real M × M matrix U with matrix elements Umn = |ham |an i|2 . (23) These elements obey Unn = 1 for 0 ≤ n ≤ M − 1 and 0 ≤ Umn = Unm ≤ 1 for any pair (m, n), with m 6= n. Now we suppose that there is no pair of orthogonal elements, i.e. 0 < Umn if m 6= n, and no pair of proportional elements, i.e. Umn < 1 if m 6= n, in the frame. Then from the Perron-Frobenius theorem for (strictly) positive matrices (e.g., [27]), the spectral radius r = r(U ) is > 0 and is dominant simple eigenvalue of U . It is proved (Collatz-Wielandt formula) that (U v)n def r = max min where N = {v | v ≥ 0 v 6= 0} . v∈N 0≤n≤M −1 ,vn 6=0 vn There exists a unique vector, vr , kvr k = 1, which is strictly positive (all components are > 0 and can be interpreted as probabilities) and U vr = rvr . All other eigenvalues α of U lie within the open disk of radius r : |α| < r. Since tr U = M , and that U has M eigenvalues, one should have r > 1. The value r = 1 represents precisely the limit case in which all eigenvalues are 1, i.e. U = I and the frame is just an orthonormal basis of CM . It is then natural to view the number def η = r−1 (24) as a kind of “distance” of the frame to the orthonormality. The question is to find the relation between the set {κ0 , κ1 , ..., κM −1 } of weights defining the frame and the distance η. By projecting on each vector |an i from both sides the frame resolution of the unity (16), we easily obtain the M equations 1 = ham |am i = def M −1 X n=0 κn |ham |an i|2 , def i.e. U vκ = vδ , (25) where t vκ = (κ0 κ1 ... κM −1 ) and t vδ = (1 1 ... 1) is the first diagonal vector in CM . Note that if U is not singular we have vκ = U −1 vδ . In the “uniform” case for which κn = d/M for all n, i.e. in the case of a finite equal norm Parseval frame, which means √ that vκ = (d/M ) vδ , then the radius r = M/d and vr = 1/ M vδ . In this case, the distance to orthonormality is just M −d , (26) η= d a relation which clearly exemplifies what we can expect at the limit d → M . def def Let us now examine the matrix P = U K, where K = diag(κ0 , κ1 , . . . , κM −1 ). def t Its (right) stochastic   nature is evident from (25). The row vector ̟ = vκ /d = κ0 κ1 κM − 1 ··· is a stationary probability vector: d d d ̟P =̟. (27) Finite quantum systems 8 As is well known, this vector obeys the ergodic property:
κn . lim P k mn = ̟n = k→∞ d (28) 3.3. Semi-classical aspects of finite frame quantization through lower symbols With the above stochastic matrix approach developed above, the relation between upper and lower symbols, fˇ(an ) = han |Af |an i = is rewritten as M −1 X m=0 κm f (am ) |han |am i|2 , f̌ = P f , (29)
def def with t f = (f (a0 ) f (a1 ) ... f (aM −1 )) and t f̌ = fˇ(a0 ) fˇ(a1 ) ... fˇ(aM −1 ) . This formula is interesting because it can be iterated: f̌ [k] = P k f , f̌ [k] = P f̌ [k−1] , f̌ [1] ≡ f̌ , (30) and so we find from the property (28) of P that the ergodic limit (or “long-term average”) of the iteration stabilizes to the “classical” average of the observable f defined as: def f̌ [∞] = hf icl vδ , where hf icl = M −1 X n=0 κn f (an ) . d (31) We can conclude from this fact that the chain of maps P : fˇ[k−1] 7→ fˇ[k] corresponds to a sort of increasing regularization of the original function f . We can evaluate the “distance” between the lower symbol f̌ and its classical counterpart f through the inequality: def kf̌ − f k∞ = max 0≤n≤M −1 |f (an )− fˇ(an )| ≤ kI − P k∞ kf k∞ , where the induced norm [31] on matrix A is kAk∞ = max0≤m≤M −1 present case, because of the stochastic nature of P , we have   kI − P k∞ = 2 1 − min κn . 0≤n≤M −1 (32) PM −1 n=0 |amn |. In the (33) In the uniform case, κn = d/M for all n, we thus have an estimate of how far the two functions f and fˇ are: kf̌ − f k∞ ≤ 2(M − d)/M kf k∞ . In the general case, we can view the parameter def ζ = 1− min 0≤n≤M −1 κn (34) as an alternative “distance” of the “quantum world” to the classical one, of noncommutativity to commutativity, or again of the frame to orthonormal basis, like the distance η = r − 1 introduced in (24). Note that in the uniform case κn = d/M these two parameters are simply related: η ζ = 1 − d/M = . 1+η Finite quantum systems 9 3.4. Wigner and Weyl functions in the context of frame quantization ff (α, β) of Af is The Weyl (or ambiguity) function W ff (α, β) = Tr[D(α, β)Af ] = W πi = e− d αβ d−1 X e 2πi rβ d r=0 n=0 M −1 X Wf (α, β) = Tr[P (α, β)Af ] = = e− 4πi αβ d r=0 κn f (an )han |D(α, β)|an i f (an )φr−α (an )φr (an ) . (35) n=0 The Wigner function Wf (α, β) of Af is d−1 X M −1 X e M −1 X κn f (an )han |P (α, β)|an i n=0 M −1 X rβ − 4πi d f (an )φ−r−2α (an )φr (an ) . (36) n=0 If f is real, then Af is a Hermitian operator and the Wigner function is real. Using Eq.(11) we prove that the Weyl and Wigner functions are related through the Fourier transform X 2πi ff (α, β) = 1 (37) e d (αδ−βγ) Wf (γ, δ). W d γ,δ The operator Af can be written as X X ff (−α, −β)D(α, β) Af = Wf (α, β)P (α, β) = W α,β (38) α,β The proof is based on the expression (21) for the matrix elements of Af with respect to the orthonormal basis |ej i. 4. Examples of uniform tight frames 4.1. A ‘lattice frame’ It is known that some of the formulas are different in odd-dimensional Hilbert spaces from their counterparts in even-dimensional Hilbert spaces. This is also true in the related area of finite sums [8]. In this example, and also in some cases below, we consider the odd-dimensional case. Let d = 2s + 1 be an odd number. In this case Zd = {−s, −s + 1, ..., s − 1, s}. We consider an example of the general tight frames discussed earlier, where X = Zd×Zd and 2πi 1 φk : Zd × Zd −→ C, φk (α, β) = √ e− d βk w̄α+k . (39) d P Here the wℓ ’s where ℓ ∈ Zd are d complex numbers normalized so that sℓ=−s |wℓ |2 = 1. The spectral radius of the matrix U is r = d and “distances” of such a frame to orthonormality are respectively η = d − 1 and ζ = 1 − 1/d. This frame is fairly appropriate to our purpose to explore quantum versions of the finite phase space X obtained through frame quantization. Finite quantum systems 10 It is easily proved that the functions φk form an orthonormal system for the inner product (12). Hence our general formalism leads to the d2 unit vectors |α, βi = s X e 2πi βk d k=−s wα+k |ek i, (α, β) ∈ Zd × Zd . (40) They form a finite tight frame in the Hilbert space H with uniform distribution καβ = d/d2 = 1/d (which means that no point in the phase space X = Zd × Zd is privileged): s 1 X |α, βihα, β| = IH . d α,β=−s (41) The overlap between two elements of the frame is s X 2πi hα, β|γ, δi = e d (δ−β)k wα+k wγ+k (42) k=−s (note that the Fourier transform of wα+k wγ+k appears here). In Ref.[49] a similar frame is obtained by starting from a discrete version of the Heisenberg group [39, 41, 43] defined in terms of the displacement operators. Then the resolution of the identity of Eq.(41) follows from the general Eq.(9). In this section we derive some of their properties through our approach, rather than through the properties of the displacement operators. wk ✻ q 0.5 q q wk ✻ 0.5 qqq q q q 0.25 q q q q q q q k q q q q ✲ q q q q q q q −10 q q 0.25 5 −5 10 q q qq qqqqqqqqqqqqqqq −20 k qqqqqqqqqqqqqq ✲ 10 −10 20 Figure 1. The wk of Eq.(43) in the cases d = 21 (left) and d = 41 (right). We choose as wk the following numbers (for examples see figure 1): ∞ 1 X − π (αd+k)2 wk = √ ; e d N0 α=−∞ N0 = ∞ X 2π 2 e− d γ . (43) γ=−∞ This is similar to a Zak [50] or Weil [45] transform of the Gaussian function, and wk = wk+d = w−k for any k. Using the Jacobi theta function ∞ X 2 θ3 (z; τ ) = eiπτ j +2ijz j=−∞ (44) (45) Finite quantum systems 11 we rewrite wk as
;i 1 θ3 πk d d wk = √ q
. d θ3 0; 2i (46) d The components wk have significant values only near 0 (modulo d). This is expected 2 because wk is related to the Gaussian function e−x /2 (which is the ground state wave function in the harmonic oscillator formalism). It is known [30] that wk are the coordinates of an eigenvector of the finite Fourier transform s 1 X 2πi kℓ √ e d wℓ = wk . (47) d ℓ=−s We can now prove the following relation F |α, βi = e− 2πi αβ d |β, −αi (48) 4.2. A ‘sparse frame’ We assume that n is a divisor of d and we consider the following partition of the set Zd : Sk = {k, n + k, 2n + k, ..., d − n + k}; n−1 [ Sk = Zd (49) k=0 Here k = 0, ..., n − 1. We also consider the subspaces Hk of the Hilbert space H, defined as: n−1 M Hk = span{|er i | r ∈ Sk }; H = Hk . (50) k=0 We call Πk the projector to the subspace Hk . We use the notation D = d/n and assume that n, D are coprime. We now give S another example of tight frames where X = (Sk × Sk ). Clearly this is a subset of Zd × Zd , with cardinality d2 /n. Let |wi be a fiducial vector such that 1 (51) hw|Πk |wi = n We consider the following sets of vectors Lk = {|α, β; ki = D(α, β)|wi | α, β ∈ Sk }; Each of these sets has (d/n)2 vectors, so their union L = appendix we show that n2 X |α, β; kihα, β; k| = Πk . d α,β∈S S k = 0, ..., n − 1. (52) Lk has d2 /n vectors. In the (53) k The Πk |α, β; ki can be viewed as a tight frame within Hk . From this follows that n−1 n2 X X |α, β; kihα, β; k| = IH . d k=1 α,β∈S k and therefore the set L of vectors, are a tight frame in H. (54) Finite quantum systems 12 5. Quantum mechanics for finite systems based on finite frame quantization Let d = 2s+1 be a positive odd number. The space X = Zd × Zd identified with the set Zd = {−s, −s+1, ..., s} × {−s, −s+1, ..., s} (55) is now considered as a finite version of a phase space, i.e. the mechanical phase space for the motion of a particle on the periodic set Zd . In accordance with the content of Section 3 we associate to each classical observable f : {−s, −s+1, ..., s}×{−s, −s+1, ..., s} −→ R the linear operator s 1 X Af : H −→ H , Af = f (α, β) |α, βihα, β| , (56) d α,β=−s and the lower symbol fˇ= Ǎf : {−s, −s+1, ..., s}×{−s, −s+1, ..., s} −→ R fˇ(a, b) = ha, b|Af |a, bi . by (57) We know from (32) that the deviation in the sense of the norm k · k∞ is bounded kf − f̌ k∞ ≤ 2ζ kf k∞ , with ζ = 1 − 1 . d (58) 5.1. Example The linear operators corresponding to the functions q ν : {−s, −s+1, ..., s}×{−s, −s+1, ..., s} −→ R , q ν (α, β) = αν , pν : {−s, −s+1, ..., s}×{−s, −s+1, ..., s} −→ R , where ν ∈ {1, 2, 3, ...}, are Aqν , Apν : H −→ H, Aqν s 1 X ν α |α, βihα, β| , = d α,β=−s and they satisfy the relation Apν pν (α, β) = β ν s 1 X ν β |α, βihα, β| , = d α,β=−s Aqν = F Apν F + = −F + Apν F. (59) (60) (61) The ‘position states’ {|eℓ i}sℓ=−s are eigenvectors of Aq and the ‘momentum states’ {|fk i}sk=−s are eigenvectors of Ap . More exactly, we have Aq |eℓ i = λℓ |eℓ i Therefore, where λj = Ap |fk i = λ−k |fk i = −λk |fk i Aq = s X ℓ=−s λℓ |eℓ iheℓ | , s X 2 α wα+j . (62) α=−s Ap = s X k=−s λ−k |fk ihfk | = − s X k=−s λk |fk ihfk | . (63) Finite quantum systems 13 Ǎq2 (a,b) ✻ Ǎq (a,b) ✻ q q 80 q q qqq 6 q q q q 60 q 3 q q q −10 −5 5 q q 10 q q a ✲ q q 40 q q q q −3 q q q q q qqq −6 20 q q q q qq qq q −10 −5 5 10 a ✲ Figure 2. The a-dependence of Ǎq (a, b) and Ǎq2 (a, b) in the case d = 21 i.e. M = 441. As expected the lower symbol Ǎqν (a, b) does not depend on b Ǎqν (a, b) = s X s X 2 wa+k 2 αν wα+k , (64) α=−s k=−s and the lower symbol Ǎpν (a, b) does not depend on a Ǎpν (a, b) = s X q̌(a, b) = s X 2 αν wα+k . 2 λk wa+k , p̌(a, b) = k=−s and 2 ha, b|(Aq ) |a, bi = (65) α=−s k=−s In particular, we have s X 2 wb+k s X 2 λk wb+k , (66) k=−s s X k=−s 2 λ2k wa+k , 2 ha, b|(Ap ) |a, bi = s X 2 λ2k wb+k . (67) k=−s The a-dependence of q̌(a, b) and Ǎq2 (a, b) in the case d = 21, i.e. M = 441, is presented in figure 2. Since kqk∞ = d − 1 = kpk∞ , the bound to deviation (58) gives for the position and momentum: (d − 1)2 (d − 1)2 , kp − p̌k∞ ≤ 2 , (68) d d which is clearly not an optimal bound in view of the figure 2 (actually we should think in terms of relative deviation). We can also show that ! s s X X 2 2 2 2 2 (∆Aq ) = ha, b|(Aq ) |a, bi − (ha, b|Aq |a, bi) = λ k λk − . (69) λℓ wa+ℓ wa+k kq − q̌k∞ ≤ 2 k=−s ℓ=−s Finite quantum systems 14 The operators X, P : H −→ H, s X X= ℓ |eℓ iheℓ | , P = ℓ=−s satisfy the relation s X k=−s k |fk ihfk | , (70) X = F + P F = −F P F + (71) and the corresponding lower symbols are X̌(a, b) = ha, b|X|a, bi = λa (72) P̌ (a, b) = ha, b|P |a, bi = λ−b = −λb . The distribution of the eigenvalues of X, Aq and the distribution of the eigenvalues of 1 (P 2 +X 2 ), 21 (A2p +A2q ), A 1 (p2+q2 ) obtained by using the matrix elements 2 2 Ps 1 1 2 1 (k−ℓ)j 2 2πi 2 2 d hek | 2 (P +X )|eℓ i = 2 k δkℓ + 2d j=−s j e Ps 1 (k−ℓ)j 2 2πi d hek | 21 (A2p +A2q )|eℓ i = 21 λ2k δkℓ + 2d (73) j=−s λj e P P 2πi s 1 2 2 d β(k−ℓ) + 2d wα+k wα+ℓ hek |A 1 (p2+q2 ) |eℓ i = 12 δkℓ sα=−s α2 wα+k α,β=−s β e 2 are presented in figure 3. One can remark a tendency, as M = d2 becomes larger and larger w.r.t. d, to have λk = k, and a tendency of the levels of A 1 (p2+q2 ) to become 2 equidistant levels. ✻ ✻ 10 90 80 5 70 60 50 0 40 30 −5 20 10 −10 0 X Aq 1 (P 2 +X 2 ) 2 1 (A2p +A2q ) 2 A 1 (p2+q2 ) 2 Figure 3. Spectra of X, Aq and 21 (P 2 +X 2 ), 21 (A2p +A2q ), A 12 (p2+q2 ) in the case d = 21. 6. About the non-uniqueness of the upper symbol The function f : Zd −→ R is an upper symbol of our position operator X, s s X 1 X f (α, β) |α, βihα, β| = j |ej ihej | d α,β=−s j=−s Finite quantum systems 15 if and only if the values f (α, β) satisfy the system of d2 equations s 2πi 1 X f (α, β) e d β(k−ℓ) wα+k wα+ℓ = k δkℓ . d α,β=−s Generally the upper symbol corresponding to a linear operator is not unique. Looking for a function f depending only on α, that is, a function of the form f (α, β) = ϕ(α) the previous sytem of equations becomes s 2πi 1 X ϕ(α) e d β(k−ℓ) wα+k wα+ℓ = k δkℓ . d α,β=−s and is equivalent to s X 2 ϕ(α) wα+k = k, k ∈ {−s, −s+1, ..., s−1, s}. α=−s that is,      2 w−s 2 w−s+1 ... ws2 2 w−s+1 2 w−s+2 ... 2 w−s 2 ws−1 ws2 ... 2 ws−2 ... ... ... ...  ws2 2 w−s ... 2 ws−1     ϕ(−s) ϕ(−s+1) .. . ϕ(s) (74)       =   −s −s+1 .. . s    .  ✻ ϕ(α) q q2000 q q q q q q q q q −10 5 −5 q 10 q α ✲ q q q q q q q q −2000 Figure 4. The upper symbol ϕ(α) of X obtained by solving (74) in the case d = 21. The matrix of the system is (up to a permutation of rows) a circulant matrix with the absolute value of the derminant ! s s Y X 2πi e d kℓ wk2 . (75) ℓ=−s k=−s A discussion of necessary and sufficient conditions for circulant matrices to be nonsingular is presented in [16]. The upper symbol f (α, β) = ϕ(α) of the position operator x, obtained by solving (74) in the case d = 21, is presented in the figure 4. Finite quantum systems 16 7. Concluding remarks The finite-dimensional quantum systems represent an essential ingredient in the development of several fields including spin systems, ensembles of two-level atoms, quantum dots, quantum optics and quantum information [1, 9, 32, 43]. In the present paper we have studied tight frames in this context and related phase space quantities (covariant and contravariant symbols and Wigner and Weyl functions). The general theory has been exemplified with novel tight frames (in section 4) which have been used for calculations of matrix elements, spectra, etc. If we use a different fiducial vector, or if we perform unitary transformations on |α, βi, we get other tight frames. But it is an interesting problem for further work to give examples of tight frame, that they be uniform or not. There are several other problems related to finite quantum systems: symplectic transformations and tomography, mutually unbiased bases, multipartite systems comprised of many finite quantum systems and their entanglement, etc. In future works we also plan to approach these questions from a finite tight frame point of view. Acknowledgment NC acknowledges the support provided by CNCSIS under the grant IDEI 992 - 31/2007. 8. Appendix We consider the matrix elements of both sides of Eq.(53) with her | and |er′ i, and we get n2 X X her |D(α, β)|em ihem |sihs|em′ ihem′ |[D(α, β)]† |er′ i d α,β∈S m,m′ ∈Z k d = δ(r, r′ ) (76) where r, r′ ∈ Sk . We then use the relation αβ 2πi he |D(α, β)|e i = e d (− 2 +βr) δ(r, m − α) r m (77) A comment here is that since r, α ∈ Sk , the right hand side can be nonzero only when m belongs to S2k . Now Eq.(76) simplifies to n2 X X 2πi (βr−βr′ ) δ(r, m − α) δ(r′ , m′ − α) ed d α,β∈S m,m′ ∈Z k d × hem |sihs|em′ i = δ(r, r′ ). (78) This simplifies further to n2 X 2πi (βr−βr′ ) ed her+α |sihs|er′ +α i = δ(r, r′ ). d α,β∈S (79) k We next consider the following subset of Zd : T = {0, D, 2D, ..., (n − 1)D}, ; D= d n (80) Finite quantum systems and prove that n X 2πi βℓ X 2πi kt e d δ(ℓ, t). ed = d β∈S t∈T 17 (81) k In the special case that ℓ ∈ S0 and n, D are coprime this simplifies to n X 2πi βℓ e d = δ(ℓ, 0). d β∈S (82) k This is because ℓ is an integer multiple of n and t is an integer multiple of D. Therefore ℓ can be equal to t only if ℓ = t = 0 (the case ℓ = t = Dn is the same because Dn = d = 0(mod d)). In Eq.(79) r − r′ ∈ S0 , and therefore it simplifies to X n δ(r − r′ , 0) her+α |sihs|er′ +α i = δ(r, r′ ). (83) α∈Sk But for fiducial vectors which obey the constraint of Eq.(51), this is true. This proves Eq.(53). 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