Quantum states defined by using the finite frame quantization

arXiv:2206.10602v2 [quant-ph] 15 Jul 2022 Quantum states defined by using the finite frame quantization Nicolae Cotfas Faculty of Physics, University of Bucharest, Romania E-mail: [email protected] , https://unibuc.ro/user/nicolae.cotfas/ 18 July 2022 Abstract. Finite frame quantization is a discrete version of the coherent state quantization. In the case of a quantum system with finite-dimensional Hilbert space, the finite frame quantization allows us to associate a linear operator to each function defined on the discrete phase space of the system. We investigate the properties of the density operators which can be defined by using this method. 1. Introduction The quantum particle moving along a straight line is described by using the Hilbert space L2 (R). For the corresponding classical system, R is the configuration space and R2 = R×R the phase space. The position operator q̂ψ(q) = q ψ(q) (1) and the momentum operator d p̂ = −i~ dq satisfy the commutation relation (2) [q̂, p̂] = i~ (3) and the relation p̂ = F̂ † q̂ F̂ (4) where F̂ [ψ](p) = √1 2π~ Z∞ − ~i pq e ψ(q) dq = −∞ √1 h Z∞ e− 2πi pq h ψ(q) dq (5) −∞ is the Fourier transform. In the odd-dimensional case, d = 2s+1, a discrete version can be obtained by using R = {−s, −s+1, ..., s−1, s} (6) Density operators obtained through frame quantization 2 as a configuration space, the Hilbert space (several representations are presented)   ψ(n+d) = ψ(n), d 2 H ≡ C ≡ ℓ (R) = { ψ : R → C } ≡ ψ : Z → C for all n ∈ Z with hψ, ϕi = s X (7) (8) ψ(n)ϕ(n) n=−s and the discrete Fourier transform F̂ : H → H, s X 2πi 1 F̂[ψ](k) = √d e− d kn ψ(n). (9) n=−s The standard basis {δ−s , δ−s+1 , ..., δs−1 , δs }, where  1 if n = m modulo d δm (n) = 0 if n 6= m modulo d (10) is an orthonormal basis. By using Dirac’s notation |mi instead of δm , we have s X hm|ki = δmk , |mihm| = I, (11) m=−s where I : H → H, Iψ = ψ, is the identity operator. In the discrete case, the position operator q̂ : H → H : ψ 7→ q̂ψ is q̂ψ(n) = n ψ(n). (12) For the momentum operator p̂ : H → H, the definition p̂ = F̂† q̂F̂ (13) is more adequate than the use of a finite-difference operator instead of In the discrete case, the set d . dq R2 = R×R = { (n, k) | n, k ∈ {−s, −s+1, ..., s−1, s} } (14) plays the role of phase space. The Gaussian function of continuous variable (κ > 0 is a parameter) κ 2 κπ 2 gκ (q) = e− 2~ q = e− h q gκ : R → R, (15) satisfies the relation F̂ [gκ ] = √1κ g 1 . (16) κ The corresponding Gaussian function of discrete variable, defined as [11, 15] ∞ X κπ 2 gκ : R → R, gκ (n) = e− d (n+αd) (17) α=−∞ satisfies the similar relation F̂[gκ ] = √1κ g 1 . κ (18) In this article, we restrict us to the odd-dimensional case, but most of the definitions and results can be extended in order to include the even-dimensional case d = 2s also. Density operators obtained through frame quantization 3 2. Coherent state quantization In the continuous case, the quantum state |0, 0i = √ 1 |g1 i hg1 ,g1 i (19) represents the vacuum state. The coherent states [13] |q, pi = D̂(q, p)|0, 0i, (20) defined by using the displacement operators [13, 16] i i i πi D̂(q, p) = e− 2~ pq e ~ pq̂ e− ~ qp̂ = e− h pq e 2πi pq̂ h e− 2πi q p̂ h (21) satisfy the resolution of the identity Z 1 I = 2π~ |q, pihq, p| dqdp. (22) By using the coherent state quantization, we associate the linear operator [9] Z 1 Âf = 2π~ f (q, p) |q, pihq, p| dqdp (23) R2 R2 to each function f : R× R → C, (24) defined on the phase space R2 , and such that the integral is convergent. For example, in the case f (q, p) = q, we get [9] Z 1 Âf = 2π~ q |q, pihq, p| dqdp = q̂, (25) R2 in the case f (q, p) = p, we get [9] Z 1 Âf = 2π~ p |q, pihq, p| dqdp = p̂, (26) R2 2 2 and, in the case f (q, p) = p +q , we get [9] 2 Z 2 2 ~2 d2 1 2 1 p +q 1 |q, pihq, p| dqdp = − + q + . Âf = 2π~ 2 2 dq 2 2 2 (27) R2 In the last case, the operator ~2 d2 1 2 1 + q Âf − = − 2 2 dq 2 2 is the Hamiltonian of the quantum harmonic oscillator. (28) Density operators obtained through frame quantization 4 3. Finite frame quantization The quantum state [11, 15] |0;0i = √ 1 |g1 i hg1 ,g1 i (29) can be regarded as a discrete counterpart of the vacuum state and D̂(n, k) : H → H, πi D̂(n, k) = e− d nk e 2πi kq̂ d e− 2πi np̂ d (30) as displacement operators [16, 17]. Theorem 1. The discrete coherent states [8, 17] |n;ki = D̂(n, k)|0;0i, (31) satisfy the resolution of the identity s X I = d1 |n;kihn;k|. (32) n,k=−s Proof. Since e− 2πi np̂ d g1 (m) = e− = √1d = √1d = √1d = = 2πi nF̂† q̂F̂ d s P a=−s s P a=−s s P g1 (m) = F̂† e− e 2πi ma d e e 2πi ma d e− 2πi na d e 2πi ma d e− 2πi na √1 d − 2πi nq̂ d a=−s s P b=−s s P 1 d s P e 2πi nq̂ d F̂g1 (m) F̂g1 (a) F̂g1 (a) d 2πi a(m−n−b) d s P e− 2πi ab d g1 (b) b=−s g1 (b) a=−s δb (m−n)g1 (b) = g1 (m−n) b=−s and hm|n;ki = hm|D̂(n, k)|0;0i 2πi 2πi πi = √ 1 e− d nk e d kq̂ e− d np̂ g1 (m) we get * m 1 d s P n,k=−s hg1 ,g1 i πi 1 e− d nk hg1 ,g1 i πi √ 1 e− d nk hg1 ,g1 i =√ e 2πi km d e− = e 2πi km d g1 (m−n) |n;kihn;k| ℓ + = d1 hg11,g1 i = = s P e 2πi np̂ d 2πi km d g1 (m) g1 (m−n)e− 2πi kℓ d g1 (ℓ−n) n,k=−s s s P P 2πi 1 1 e d k(m−ℓ) g1 (m−n) g1 (ℓ−n) d hg1 ,g1 i n=−s k=−s s P 1 δmℓ g1 (m−n) g1 (ℓ−n) = δmℓ .  hg1 ,g1 i n=−s Density operators obtained through frame quantization 5 By using the finite frame quantization, we associate the linear operator [7, 8, 9] Λ̂f = 1 d s X n,k=−s to each function f (n, k) |n;kihn;k| f : R×R → C (33) (34) defined on the discrete phase space R2 = R×R. Theorem 2. Let f, g : R×R → C and α, β ∈ C. We have: a) f (n, k) = 1 ⇒ Λ̂f = I. b) Λ̂αf +βg = αΛ̂f +β Λ̂g . (36) c) f (n, k) ∈ R for any n, k ⇒ Λ̂†f = Λ̂f . (37) d) f (n, k) ≥ 0 for any n, k ⇒ Λ̂f ≥ 0. (38) e) tr Λ̂f = d1 s X f (n, k). n,k=−s Proof. a) Direct consequence of (32). b) Direct consequence of the definition (33). s P c) Λ̂†f = d1 f (n, k) (|n;kihn;k|)† = 1 d n,k=−s s P n,k=−s f (n, k) |n;kihn;k| = Λ̂f . d) For any ψ ∈ H, we have s P hψ, Λ̂f ψi = d1 f (n, k) hψ|n;kihn;k|ψi = e) tr Λ̂f = = = = = (35) s P 1 d n,k=−s s P n,k=−s f (n, k) |hn;k|ψi|2 ≥ 0. hm|Λ̂f |mi m=−s s s P P 1 f (n, k) hm|n;kihn;k|mi d m=−s n,k=−s s s P P 1 f (n, k) hn;k|mihm|n;ki d m=−s n,k=−s s P 1 f (n, k)hn;k|I|n;ki d n,k=−s s P 1 f (n, k).  d n,k=−s (39) Density operators obtained through frame quantization 2 6 2 In the case f (n, k) = n +k , the operator Λ̂f − 12 can be regarded as a discrete version 2 of the Hamiltonian of the quantum harmonic oscillator. The eigenfunctions ψn of Λ̂f , considered in the increasing order of the number of sign alternations, can be regarded as a finite counterpart of the Hermite-Gauss functions Ψn (q). In the cases analyzed in [6], the eigenfunctions ψn of Λ̂f approximate Ψn (q) better than the Harper functions hn , and approximately satisfy the relation F̂[ψn ] = (−i)n ψn . (40) Instead of the standard definition of the discrete fractional Fourier transform [1, ?, 12] d−1 X α F̂ = (−i)nα |hn ihhn |. (41) n=0 one can use [6] α F̂ = d−1 X n=0 (−i)nα |ψn ihψn |. (42) as an alternative definition. The Harper functions (available only numerically) are defined as the eigenfunctions of a discrete version of the Hamiltonian of the quantum harmonic oscillator obtained by using finite-differences [1, ?, 12]. The finite frame quantization [6, 7, 8] seems to behave better than the method based on finite-differences when we have to obtain discrete versions of certain operators. 4. Density operators obtained through finite frame quantization The finite frame quantization allows us to define a remarkable class of quantum states. Theorem 3. If the function f : R×R → [0, d] is such that s X f (n, k) = d, (43) n,k=−s then the corresponding linear operator ̺ˆf : H → H, s X 1 f (n, k) |n;kihn;k| ̺ˆf = d (44) n,k=−s is a density operator. Proof. Direct consequence of theorem 2.  For example, the state corresponding to f (n, k) = d1 is the mixed state ̺ˆf = d1 I, and the state corresponding to ( d for (n, k) = (m, ℓ) f (n, k) = 0 for (n, k) 6= (m, ℓ) (45) is the pure state ̺ˆf = |m;ℓihm;ℓ|, that is, the discrete coherent state |m;ℓi. Theorem 4. The set Sfr of all the density operators of the form (44) is a convex set. Proof. If λ ∈ [0, 1] and ̺ˆf , ̺ˆg ∈ Sfr , then (1−λ)ˆ ̺f +λˆ ̺g = ̺ˆh , where h(n, k) = (1−λ)f (n, k)+λg(n, k).  Density operators obtained through frame quantization 7 Theorem 5. Sfr is the convex hull of the set of pure states { |n;kihn;k| | n, k ∈ R }. Proof. The purity of a state ̺ˆf is s s X X 2 1 f (n, k) f (m, ℓ) |hn;k|m;ℓi|2. (46) tr ̺ˆf = d2 n,k=−s m,ℓ=−s Since |hn;k|m;ℓi|2 ≤ hn;k|n;ki hm;ℓ|m;ℓi = 1, (47) ̺ˆf is a pure state if and only if f is a function of the form (45), that is, ̺ˆf is one of the discrete coherent states |m;ℓihm;ℓ|.  Theorem 6. If the function f : R×R → [0, d] is such that s X f (n, k) = d, (48) n,k=−s then the mean value D E  = tr(A ̺ˆf ) (49) ̺ˆf of an observable  : H → H in the state ̺ˆf is s D E X  = 1d f (n, k) hn;k|Â|n;ki. ̺ˆf Proof. We have (50) n,k=−s D E  = ̺ˆf = = = 1 d 1 d 1 d 1 d s P n,k=−s s P n,k=−s s P n,k=−s s P n,k=−s f (n, k) tr( |n;kihn;k|) f (n, k) f (n, k) s P hm|Â|n;kihn;k|mi m=−s s P hn;k|mihm|Â|n;ki m=−s f (n, k) hn;k|Â|n;ki. Theorem 7. If the function f : R×R → [0, d] is such that s X f (n, k) = d,  (51) n,k=−s then, under the Fourier transform, ̺ˆf maps as ̺ˆf 7→ F̂ˆ ̺f F̂† = ̺ˆg , where g(n, k) = f (−k, n). Proof. Since (52) Density operators obtained through frame quantization 8 F̂D̂(n, k)g1 (m) hm|F̂|n;ki = hm|F̂D̂(n, k)|0;0i = √ 1 hg1 ,g1 i s P mℓ − 2πi √1 d = √ 1 D̂(n, k)g1 (ℓ) e d hg1 ,g1 i = = = = = = n=−s s P 2πi πi 2πi 1 1 √ √ e− d mℓ e− d nk e d kℓ g1 (ℓ−n) hg1 ,g1 i d n=−s s P 2πi πi √ 1 e− d ℓ(m−k) g1 (ℓ−n) e− d nk √1d hg1 ,g1 i n=−s s P 2πi πi nk − 1 1 √ e− d (ℓ+n)(m−k) g1 (ℓ) e d √d hg1 ,g1 i n=−s s P 2πi 2πi πi nk − − √ 1 e− d ℓ(m−k) e d e d n(m−k) √1d hg1 ,g1 i n=−s πi nk − 2πi nm √ 1 d d e e F̂[g1 ](m−k) hg1 ,g1 i 2πi πi √ 1 e d nk e− d nm g1 (m−k) = hm|k;−ni, hg1 ,g1 i g1 (ℓ) we have F̂|n;ki = |k;−ni, and consequently s P F̂ˆ ̺f F̂† = d1 f (n, k) F̂|n;kihn;k|F̂† = = 1 d 1 d n,k=−s s P n,k=−s s P n,k=−s f (n, k) |k;−nihk;−n| f (−k, n) |n;kihn;k|.  Theorem 8. If the function f : R×R → [0, d] is such that s X f (n, k) = d, (53) n,k=−s then, under the displacement D̂(m, ℓ), the operator ̺ˆf maps as ̺ˆf 7→ D̂(m, ℓ)ˆ ̺f D̂† (m, ℓ) = ̺ˆg , (54) where g(n, k) = f (n−m (mod d), k−ℓ (mod d)). Proof. We have (see [3]) s P f (n, k) D̂(m, ℓ)|n;kihn;k|D̂† (m, ℓ) D̂(m, ℓ)ˆ ̺f D̂† (m, ℓ) = d1 n,k=−s = = 1 d 1 d s P n,k=−s s P n,k=−s f (n, k) |n+m (mod d);k+ℓ (mod d)ihn+m (mod d);k+ℓ (mod d)| f (n−m (mod d), k−ℓ (mod d)) |n;kihn;k|. Theorem 9. If the function f : R×R → [0, d] is such that s X f (n, k) = d,  (55) n,k=−s then, under the transposition map |jihℓ| 7→ |ℓihj|, the operator ̺ˆf transforms as ̺ˆf 7→ ̺ˆTf = ̺ˆg , where g(n, k) = f (n, −k). Proof. Since (56) Density operators obtained through frame quantization |n; ki = s X j=−s |jihj|n; ki = s X j=−s |ji √ 1 hg1 ,g1 i 9 πi e− d nk e 2πi kj d g1 (j −n), under the transposition map |jihℓ| 7→ |ℓihj|, the operator |n; kihn; k| = transforms to s X j,ℓ=−s s X j,ℓ=−s |ℓihj| hg11,g1 i e |jihℓ| hg11,g1 i e 2πi kj d e 2πi kℓ d 2πi kj d e− 2πi kℓ d g1 (j −n) g1 (ℓ−n) g1 (j −n) g1 (ℓ−n) = |n; −kihn; −k|.  Theorem 10. If the function f : R×R → [0, d] is such that s X f (n, k) = d, (57) n,k=−s then, under the parity transform |ji 7→ Π|ji = | −ji, the operator ̺ˆf maps as ̺ˆf 7→ Πˆ ̺f Π = ̺ˆg , (58) where g(n, k) = f (−n, −k). Proof. Since g1 (−n) = g1 (n), under the transform |ji 7→ Π|ji = | −ji, |n; ki = maps to s X j=−s s X j=−s 1 hg1 ,g1 i |ji √ 1 hg1 ,g1 i |−ji √ πi e− d nk e πi e− d nk e 2πi kj d 2πi kj d g1 (j −n), g1 (j −n) = |−n; −ki.  In the odd-dimensional case, the discrete Wigner function [10, 17, 19] of a density operator ̺ : H → H, is usually defined as W̺ : R×R → R, s 1 X − 4πi km hn+m|̺|n−mi. e d W̺ (n, k) = d m=−s (59) The discrete Wigner function of a pure state ̺ = |ψihψ| is [2, 3, 4, 5] Wψ (n,k) = s 1 X d e− 4πi km d (60) ψ(n+m) ψ(n−m). m=−s Theorem 11. If the function f : R×R → [0, d] is such that s X f (n, k) = d, (61) n,k=−s then the discrete Wigner function of ̺ˆf is W̺ˆf (m, ℓ) = C s X n,k=−s f (n, k) ∞ X 2π d 2 2π d 2 (−1)αβ e− d (m−n+α 2 ) e− d (ℓ−k+β 2 ) , α,β=−∞ (62) Density operators obtained through frame quantization 10 where C is a normalizing constant. Proof. We have (see [3, 4, 5]) s P f (n, k) W|n;ki(m,ℓ) W̺ˆf (m, ℓ) = 1d = 1 d =C n,k=−s s P n,k=−s s P f (n, k) W|0;0i(m−n,ℓ−k) f (n, k) ∞ P d 2 2π 2π d 2 (−1)αβ e− d (m−n+α 2 ) e− d (ℓ−k+β 2 ) .  α,β=−∞ n,k=−s 5. Composite quantum systems Let sA , sB ∈ {1, 2, 3, ...}, dA = 2sA +1, dB = 2sB +1, d = dA dB , RA = {−sA , −sA +1, ..., sA −1, sA }, HA = CdA ≡ { ψ : RA → C }, RB = {−sB , −sB +1, ..., sB −1, sB }, HB = CdB ≡ { ϕ : RB → C }, R = RA ×RB , H = HA ⊗HB ≡ { Ψ : RA ×RB → C }. The tensor product of two tight frames is a tight frame. Particularly, { |n,m;k,ℓi ≡ |n,m;k,ℓiAB = |n;kiA ⊗|m;ℓiB | n, k ∈ RA , m, ℓ ∈ RB } is a tight frame in H = HA ⊗HB , namely sA sA sB P P P 1 1 |n,m;k,ℓihn,m;k,ℓ| = d d n,k=−sA m,ℓ=−sB = = sB P |n;kiA ⊗|m;ℓiB Ahn;k| ⊗ Bhm;ℓ| n,k=−sA m,ℓ=−sB sA sB P P 1 |n;kiAhn;k|⊗|m;ℓiBhm;ℓ| dA dB n,k=−sA m,ℓ=−sB sA sB P P 1 1 |n;ki hn;k|⊗ |m;ℓiBhm;ℓ| A dA dB n,k=−sA m,ℓ=−sB = IHA ⊗ IHB = IH . By using the finite frame quantization, we associate the linear operator sB sA X X 1 f (n, m; k, ℓ) |n,m;k,ℓihn,m;k,ℓ| ̺ˆf = d (63) n,k=−sA m,ℓ=−sB to each function f : (RA ×RB )×(RA ×RB ) → [0, d], (64) defined on the discrete phase space R2 , and satisfying the relation sB sA X X f (n, m; k, ℓ) = d. (65) n,k=−sA m,ℓ=−sB If f : RA ×RA → [0, dA ] and g : RB ×RB → [0, dB ] are such that sA X n,k=−sA f (n, k) = dA , sB X m,ℓ=−sB g(m, ℓ) = dB (66) Density operators obtained through frame quantization 11 then ̺ˆf ⊗ ̺ˆg = = = = 1 dA 1 d 1 d 1 d sA P n,k=−sA sA P f (n, k) |n;kiAhn;k| ⊗ sB P n,k=−sA m,ℓ=−sB sA sB P P n,k=−sA m,ℓ=−sB sA sB P P n,k=−sA m,ℓ=−sB 1 dB sB P m,ℓ=−sB g(m, ℓ) |m;ℓiBhm;ℓ| f (n, k) g(m, ℓ) |n;kiAhn;k| ⊗ |m;ℓiBhm;ℓ| f (n, k) g(m, ℓ) |n;kiA ⊗|m;ℓiB Ahn;k|⊗ Bhm;ℓ| f (n, k) g(m, ℓ) |n,m;k,ℓihn,m;k,ℓ| = ̺ˆh , where h : (RA ×RB )×(RA ×RB ) → [0, d], h(n, m; k, ℓ) = f (n, k) g(m, ℓ). Theorem 12. If f : (RA ×RB )×(RA ×RB ) → [0, d] is such that sA X sB X f (n, m; k, ℓ) = d, (67) n,k=−sA m,ℓ=−sB then: trA ̺ˆf = ̺ˆfB , a) (68) where fB : RB ×RB → [0, dB ], trB ̺ˆf = ̺ˆfA , b) where fA : RA ×RA → [0, dA ], Proof. a) We have sA P trA ̺ˆf = = = = = = a=−sA sA P fB (m,ℓ) = d1A fA (n,k) = d1B sA P f (n,m;k,ℓ). n,k=−sA sB P (69) f (n,m;k,ℓ). m,ℓ=−sB ha|ˆ ̺f |aiA A sB sA P P 1 f (n, m; k, ℓ) Aha|n,m;k,ℓihn,m;k,ℓ|aiA d a=−sA n,k=−sA m,ℓ=−sB sA sB sA P P P 1 f (n, m; k, ℓ) ha|n;kiAhn;k|aiA |m;ℓiBhm;ℓ| A d a=−sA n,k=−sA m,ℓ=−sB sA sA sB P P P 1 f (n, m; k, ℓ) hn;k|aiAha|n;kiA |m;ℓiBhm;ℓ| A d a=−sA m,ℓ=−sB n,k=−sA sB sA P P 1 1 f (n, m; k, ℓ) |m;ℓiBhm;ℓ| dB dA m,ℓ=−sB n,k=−sA sB P 1 fB (m, ℓ) |m;ℓiBhm;ℓ|. dB m,ℓ=−sB b) Similar to the proof of a).  Theorem 13. In the case dA = dB , if f : (RA ×RB )×(RA ×RB ) → [0, d] is such that sA X sB X n,k=−sA m,ℓ=−sB f (n, m; k, ℓ) = d, (70) Density operators obtained through frame quantization 12 then, under the SWAP transform HA ⊗HB → HA ⊗HB : |ϕiA ⊗|ψiB 7→ |ψiA ⊗|ϕiB , (71) the density operator ̺ˆf maps as ̺ˆf 7→ SW AP (ˆ ̺f ) = ̺ˆg , (72) where g : (RA ×RB )×(RA ×RB ) → [0, d], g(n, m; k, ℓ) = f (m, n; ℓ, k). Proof. We have sB sA X X SW AP (ˆ ̺f ) = 1d f (n, m; k, ℓ) |m;ℓiA ⊗|n;kiB Ahm; ℓ|⊗ Bhn; k| = ̺ˆg .  n,k=−sA m,ℓ=−sB 6. Quantum channels obtained through finite frame quantization We continue to use the notations from the previous section and choose an auxiliary system HA′ such that dim HA′ = dim HA = 2sA+1, and consequently HA′ = { ψ : RA → C }. The pure quantum state sA sA X X |iiA′ ⊗|iiA = √1d |iii (73) |Φi = |ΦiA′A = √1d A A i=−sA i=−sA is the most entangled state in HA′ ⊗HA . In view of the channel-state duality (also called Choi-Jamiolkowski isomorphism), a quantum channel E : L(HA ) → L(HB ) satisfying the relation (I ⊗E)(|ΦihΦ|) = ̺ˆ corresponds to each state ̺ˆ : HA′ ⊗HB → HA′ ⊗HB , up to a normalization. Particularly, a quantum channel Ef : L(HA ) → L(HB ) corresponds to each state ̺ˆf : HA′ ⊗HB → HA′ ⊗HB with f : (RA ×RB )×(RA ×RB ) → [0, d] satisfying sA X sB X sA X sB X f (n, m; k, ℓ) = d. (74) n,k=−sA m,ℓ=−sB In the usual way, we prove that Ef admits the representation [14] Ef (ˆ ̺) = † Kn,m;k,ℓ ̺ˆ Kn,m;k,ℓ (75) n,k=−sA m,ℓ=−sB involving the Kraus operators K̂n,m;k,ℓ : HA → HB , q K̂n,m;k,ℓ|iiA = f (n,m;k,ℓ) hi|n,m;k,ℓi. d A′ From the definition of K̂n,m;k,ℓ written in the form sB q X f (n,m;k,ℓ) |jiB hij|n,m;k,ℓi K̂n,m;k,ℓ|iiA = d (76) (77) j=−sB we get the relation B q † hi|K̂n,m;k,ℓ |jiB A q hj|K̂n,m;k,ℓ|iiA = whence = f (n,m;k,ℓ) d hij|n,m;k,ℓi, (78) f (n,m;k,ℓ) d hn,m;k,ℓ|iji (79) Density operators obtained through frame quantization 13 and consequently † hi|K̂n,m;k,ℓ = A We have (I⊗Ef )(|ΦihΦ|) = = = = = = because hnk| 1 dA 1 dA 1 dA 1 dA sA P f (n,m;k,ℓ) d hn,m;k,ℓ|iiA′ . i,j=−sA sA P i,j=−sA sA P (I⊗Ef )|iiA′hj|⊗|iiAhj| |iiA′hj|⊗Ef (|iiAhj|) sB P sA P i,j=−sA n,k=−sA m,ℓ=−sB sB sA sA P P P † |iiA′hj|⊗ K̂n,m;k,ℓ|iiAhj| K̂n,m;k,ℓ i,j=−sA n,k=−sA m,ℓ=−sB sA P i,j=−sA sA P i,j=−sA (80) (I⊗Ef )|iiihjj| i,j=−sA sA P 1 dA d 1 dA q f (n, m; k, ℓ) |iiA′hj|⊗ A′hi|n,m;k,ℓihn,m;k,ℓ|jiA′ |iiA′hj|⊗ A′hi|̺f |jiA′ = d1A ̺f |iiA′hj|⊗ A′hi|̺f |jiA′ |mℓi = = sA P i,j=−sA sA P hn|iihj|mihik|̺f |jℓi i,j=−sA δni δjm hik|̺f |jℓi = hnk|̺f |mℓi. So, up to a normalization, we have (I⊗Ef )(|ΦihΦ|) = ̺f . In addition, sB sA P P † Kn,m;k,ℓ Kn,m;k,ℓ |iiA n,k=−sA m,ℓ=−sB q sB sB sA P P † f (n,m;k,ℓ) P Kn,m;k,ℓ |biB hib|n,m;k,ℓi = d n,k=−sA m,ℓ=−sB b=−sB q sB sA sA sB P P P † f (n,m;k,ℓ) P = |aiAha|Kn,m;k,ℓ |biB hib|n,m;k,ℓi d = = = = = = = n,k=−sA m,ℓ=−sB sB sA P P n,k=−sA m,ℓ=−sB sB sA P P n,k=−sA m,ℓ=−sB sA sB P P b=−sB a=−sA sB sA P P b=−sB a=−sA sA sB P P b=−sB a=−sA sA sB P P b=−sB a=−sA sA P a=−sA f (n,m;k,ℓ) d f (n,m;k,ℓ) d b=−sB a=−sA sA P sB P b=−sB a=−sA sA sB P P b=−sB a=−sA |aiA hn,m;k,ℓ|abihib|n,m;k,ℓi |aiA hib|n,m;k,ℓihn,m;k,ℓ|abi |aiA hib|ˆ ̺f |abi |aiA hib|(I⊗E)(|ΦihΦ|)|abi |aiA hib| |aiA sA P j,ℓ=−sA sA P j,ℓ=−sA hi|jihℓ|aihb|Ef (|jiAhℓ|)|bi |aiA tr(Ef (|iiAha|)) = for any i ∈ RA , and consequently |jiA′hℓ|⊗Ef (|jiAhℓ|)|abi sA P a=−sA |aiA tr(|iiAha|) = sA P a=−sA |aiA δai = |iiA , Density operators obtained through frame quantization sA X sB X n,k=−sA m,ℓ=−sB † Kn,m;k,ℓ Kn,m;k,ℓ = IHA . 14 (81) 7. Concluding remarks The discrete coherent states (31) approximate well [6] the standard coherent states (20). In the case of this finite frame, the use of the frame quantization seems to lead to a remarkable discrete version of certain linear operators [6]. Particularly, the density operators defined in this way have some significant properties, and may describe quantum states useful in certain applications. References [1] Barker L, Candan Ç, Hakioğlu T, Kutay M A and Ozaktas H M 2000 The discrete harmonic oscillator, Harper’s equation, and the discrete fractional Fourier transform J. Phys. A: Math. 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