Unitarily inequivalent local and global Fourier transforms in multipartite quantum systems

A multipartite system comprised of n subsystems, each of which is described with ‘local variables’ in Z(d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {Z}}}(d)$$\end{document} and with a d-dimensional Hilbert space H(d), is considered. Local Fourier transforms in each subsystem are defined and related phase space methods are discussed (displacement operators, Wigner and Weyl functions, etc). A holistic view of the same system might be more appropriate in the case of strong interactions, which uses ‘global variables’ in Z(dn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {Z}}}(d^n)$$\end{document} and a dn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^n$$\end{document}-dimensional Hilbert space H(dn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(d^n)$$\end{document}. A global Fourier transform is then defined and related phase space methods are discussed. The local formalism is compared and contrasted with the global formalism. Depending on the values of d, n the local Fourier transform is unitarily inequivalent or unitarily equivalent to the global Fourier transform. Time evolution of the system in terms of both local and global variables, is discussed. The formalism can be useful in the general area of Fast Fourier transforms.


I. INTRODUCTION
Entanglement and stronger than classical correlations in multipartite systems, are fundamental concepts in quantum mechanics (e.g., [1]).Even if the various components of the system are physically located far from each other, strong correlations and strong interactions between them, weaken the concept of separate identity for each component .This motivates a comparison between the formalism of a multipartite system, with a holistic formalism of the same system that uses global quantities.
We consider a finite quantum system with variables in Z(d) where d is an odd integer, described by the d-dimensional Hilbert space H(d) (e.g.[2,3]).We also consider a multipartite system that consists of n of these systems (which are possibly located far from each other).In this system the positions and momenta take values in [Z(d)] n = Z(d) × ... × Z(d).The system is described with the d n -dimensional Hilbert space H = H(d) ⊗ ... ⊗ H(d).
In the case of strong correlations and strong interactions between the n components we introduce a holistic approach and regard this as one system with variables in Z(d n ) and d n -dimensional Hilbert space H(d n ).We note that • The Hilbert space H is isomorphic to the H(d n ), because they both have the same dimension.
• There is a bijective map between the sets [Z(d)] n and Z(d n ) given below in Eq.( 35) (in fact we can have many bijective maps between these two sets).However the [Z(d)] n as a ring is not isomorphic to the ring Z(d n ) (see Eq.(36) below).
With this in mind, we study the following: • We define a local Fourier transform F L in the phase space [Z(d)] n × [Z(d)] n of the system when considered as n-component system.We also define a global Fourier transform F G in the phase space Z(d n ) × Z(d n ) of the system when considered as a single system.This has been introduced briefly in a different context in ref. [4], and here it is studied as a problem in its own right and in connection with a global phase space formalism.We show that depending on the values of d, n the local Fourier transform is unitarily inequivalent (unitarily equivalent) to the global Fourier transform.By that we mean that there exists no unitary transformation U (there exists such a transformation U ) so that F G = U F L U † .This is discussed in section IV D and in proposition IV.4.
• Starting from an orthonormal basis of 'position states', we use local and global Fourier transforms to define local and global momentum states.Some of the local momentum states are the same as the global momentum states as discussed in proposition IV.3.We also define local position and momentum operators, and also global position and momentum operators.We do numerical calculations of the time evolution for the case where the Hamiltonian is expressed in terms of local variables and also for the case where the Hamiltonian is expressed in terms of global variables (section V B).For multipartite systems with strong interactions between the various components, it might be more appropriate to express the Hamiltonian in terms of the global variables.
• We define a local phase space formalism in [Z(d)] n × [Z(d)] n and a global phase space formalism in Z(d n ) × Z(d n ).Displacements, Wigner and Weyl functions, etc, are defined in these two cases.Density matrices which have only diagonal elements with respect to the position basis, have the same local and global Wigner function (proposition V.3).The difference between local and global Wigner functions, is contained entirely in the off-diagonal elements.
• Deviations of a density matrix ρ from the corresponding factorisable density matrix R(ρ) (defined in Eq.( 16)) are described with the matrices R L , R L and R G , R G .They describe classical and quantum correlations in the multipartite system described by ρ (section V E).
• Understanding of the relationship between global and local Fourier transforms and related phase space methods, might be useful in other areas like fast Fourier transforms.For n = 2 we show that the global Fourier transform can be expressed in terms of many local Fourier transforms (section IV E).This is similar to the Cooley-Tukey formalism in fast Fourier transforms [5][6][7].The general area of Fast Fourier transforms (in a quantum or even classical context) is a potential application of the present formalism.
• In the case that the local and global Fourier transform are unitarily inequivalent (Eq.(65) below), the concept of a multipartite system (and related concepts like entanglement) is fundamentally different from that of a single quantum system.But if they are unitarily equivalent (Eq.(64) below), the distinction between a multipartite system and a single system is weak.Unitary equivalence means that with a change of basis one concept is transformed to another, and consequently there is no fundamental difference between the two.In this case, further work is needed in order to clarify the correspondence between the two (especially of entanglement which is a concept applicable to a multipartite system but not to a single system).
In section 2 we review briefly the phase-space formalism for systems with finite Hilbert space [2,3].In section 3 we apply this to each component of a n-partite system, and this is the 'local formalism'.In section 4 we define the global Fourier transform and discuss for which values of d, n it is unitarily inequivalent to the local Fourier transform.In section 5 we present the global phase space formalism and compare and contrast it with the local formalism.In section 6, we present examples.We conclude in section 7 with a discussion of our results.

II. BACKGROUND
We consider a quantum system (qudit) with variables in the ring Z(d) of integers modulo d where d is an odd integer.H(d) is the d-dimensional Hilbert space describing this system.There are well known technical differences between quantum systems with odd dimension d and even dimension d (e.g., [8][9][10]).In this paper we consider systems with odd dimension d.
Let |X; j where j ∈ Z(d) be an orthonormal basis in H(d).The X in the notation is not a variable, it simply indicates 'position states'.The finite Fourier transform F is given by [11] Its trace is [2] We act with F on position states and get the dual basis The P in the notation is not a variable, it simply indicates 'momentum states'.Using the relation we show that F 2 is the parity operator around the origin: The phase space of this system is Z(d) × Z(d) and in it we introduce the displacement operators General displacement operators are the unitary operators The 2 −1 = d+1 2 is an integer in Z(d) with odd d, considered here.The D(α, β)ω(γ) form a representation of the Heisenberg-Weyl group.We note that The are d × d matrices which can be interpreted as position and momentum operators.The commutator [X , P] can be calculated (it is not i1) but it has no mathematical significance because the Heisenberg-Weyl group in this context is discrete, and the concept of generators is non-applicable.Hamiltonians can be written as functions of these operators as h(X , P).

A. Wigner and Weyl functions
The parity operator (around the point (γ, δ)) is defined as It is related to the displacement operators through the Fourier transform If ρ is a density matrix, we define the Wigner function W (γ, δ) and the Weyl function W (α, β) as: From Eq.( 11) follows immediately that they are related to each other through the Fourier transform: The following marginal properties of the Wigner function are well known for odd values of the dimension d (e.g., [2]):

A. Local Fourier transforms
We consider a n-partite system comprised of n components each of which is a qudit.This system is described with the d n -dimensional Hilbert space H = H(d) ⊗ ... ⊗ H(d).Positions and momenta take values in [Z(d)] n = Z(d) × ... × Z(d).If ρ is the density matrix of the system, we use the notation ρr = Tr i =r ρ; r = 0, ..., n − 1, (15) for the reduced density matrix describing the r-component of the system.We also define the corresponding factorisable density matrix and the correlator For factorisable density matrices R(ρ) = ρ and C(ρ) = 0. Below we compare quantities for ρ with the corresponding quantities for R(ρ).
The phase space of the system is [Z(d) × Z(d)] n and local displacement operators in it are defined as where r = 0, ..., n − 1, and n as an additive group The same is true for the X L ({β r }).Also Using the notation general local displacement operators are defined as The D L ({α r , β r })ω({γ r }) form a representation of the Heisenberg-Weyl group of displacements in the phase space [Z(d) × Z(d)] n .The local parity operator (around the point {γ r , δ r } in the phase space [Z(d) × Z(d)] n ) is defined as It is related to the local displacement operators through the Fourier transform The proof of this follows easily from Eq. (11).

C. Local Wigner and local Weyl functions in
If ρ is a density matrix, we define the local Wigner function W L ({γ r , δ r }|ρ) and the local Weyl function W L ({α r , β r }|ρ) as: From Eq.(31) follows immediately that they are related to each other through the Fourier transform:

IV. GLOBAL FOURIER TARNSFORMS
A. A bijective map between the non-isomorphic rings [Z(d)] n and Z(d n ) We consider a bijective map between [Z(d)] n and Z(d n ) as follows.We first take each j r ∈ Z(d) and j ∈ Z(d n ) in the 'periods' correspondingly (for odd d).We introduce the bijective map We then take each j r modulo d and the j modulo d n , and we get a bijective map from [Z(d)] n to Z(d n ).Numbers in Z(d n ) will be denoted with a 'hat', so that it is clear whether a number belongs to Z(d) or to Z(d n ).
The Hilbert space H is isomorphic to H(d n ) (a d n -dimensional Hilbert space describing systems with variables in Z(d n )).But the [Z(d)] n as a ring (with addition and multiplication componentwise), is not isomorphic to the ring Z(d n ) because addition and multiplication is different, and consequently our 'local formalism' is different from our 'global formalism'.Indeed The sum is different because j + k in Z(d n ) has the 'carry' rule and the r-component might be j r + k r + 1 rather than j r + k r .In contrast, there is no 'carry' rule in [Z(d)] n : Also multiplication in Z(d n ) is The corresponding multiplication in and with the bijective map in Eq.( 35) this corresponds to It is seen that in general j k = jk (but 1 k = k).
Example IV.1.We consider the elements of Z(3) in the 'period' [−1, 1] and the elements of Z( 9) in the 'period ' [−4, 4].A bijective map between [Z(3)] 2 and Z( 9) is as follows An example of addition that confirms Eq.( 36) is the following: An example of multiplication that confirms Eq.( 36) is the following: We can define a bijective map such that This is based on the Chinese remainder theorem, and has been used by Good [12] in the context of fast Fourier transforms (see also [5][6][7]).In a quantum context it has been use in [2,13] for factorisation of a quantum system into subsystems.Here we consider the case d 1 = ... = d n and then the bijective map of Eq.( 35) does not establish an isomorphism between the ring [Z(d)] n and the ring Z(d n ) (because of Eq.( 36)).

B. Dual notation
We use the following dual notation for position states, based on the bijective map in Eq.( 35): When local operators act on them we use addition and multiplication in [Z(d)] n , in connection with the phase space [Z(d) × Z(d)] n .When global operators (defined below) act on them we use addition and multiplication in Z(d n ), in connection with the phase space Analogous dual notation is used for all quantities.For example, the displacement operators in Eq.( 29) can be denoted as In some equations both notations appear together.

C. Global Fourier transforms
The global Fourier transform in H is defined as: The index G in the notation stands for global.It is easily seen that Acting with F G on the basis |X; j 0 , ..., j n−1 (which we also denote as |X; j ) we get the 'global momentum states': In the states |P G ; j 0 , ..., j n−1 , the coefficients ω d n [(j 0 d r + .. + j n−r−1 d n−1 )k r ] in the r-component depend on all j 0 , ..., j n−1 , and the term 'global' refers to this.Information from all components is needed, in order to determine these coefficients.In the local Fourier transform of Eq.( 21), the coefficients ω d (j r k r ) in the r-component depend only on j r .We note that and that Proposition IV.3.We take the elements of Z(d) and the elements of Z(d n ) in the 'periods' of Eq.( 34).Then (1) The parity operator around the origin is the same in both the local and global formalism: Here the matrix is in the position basis.
(2) For any n (3) For n = 2 we have the stronger result At least d of the global momentum states are equal to d of the local momentum states.
For later use we define the matrix elements of the correlator C(ρ): In both the 'local formalism' and the 'global formalism' the position states are the same and the momentum states are different.Consequently the E(P G ; j) is different from the corresponding C(P L ; j 0 , ..., j n−1 ).Then For factorisable density matrices C(X; j) = E(P G ; j) = 0.

D. Unitarily inequivalent local and global Fourier transforms
In this paper we use the following definition of unitary equivalence.Two square matrices A, B are called unitarily equivalent if there exists a unitary matrix U such that A = U BU † .Unitary equivalence is an equivalence relation, i.e., matrices which are unitarily inequivalent belong to different equivalence classes.It is known [14,15] that two normal d × d matrices A, B are unitarily equivalent if and only if We note that Specht's general theorem for unitary equivalence (e.g., [15]) reduces easily to the above criteria for the Fourier matrices which are unitary.Some authors call the above unitary similarity, and they use the term unitary equivalence for the case where there exist two unitary matrices U, V such that A = U BV † .
Proposition IV. 4. In an n-partite system that has Hilbert space with dimension d n , the Proof.The matrices F G and F L are unitary and therefore normal, and we use the criterion in Eq.( 63).We first note that We next compare Tr(F η G ) with Tr(F η L ) for η = 1, ..., d n .But For an n-partite system with dimension d n we get TrF L = (TrF ) n and using Eq.( 2) we get For the TrF G if d = 4m + 1, the d n = (4m + 1) n = 4M 1 + 1 and we get If d = 4m + 3 we consider two cases where n is an even number and an odd number.For even n we get In the case of Eq.( 64) there exists a d n × d n unitary matrix U such that F G = U F L U † , i.e., So if instead of the basis |X; j we choose the basis U |X; j as position states, then the local Fourier transform with respect to the new basis is the global Fourier transform with respect to the old basis F G = U F L U † .So in this case the global Fourier transform is not a new concept.However U is in general a global transformation (it cannot be written as U 1 ⊗ ... ⊗ U n ) and for this reason there is some merit in the study of the global Fourier transform even in this case.
The case of Eq.( 65) where global and local Fourier transforms are unitarily inequivalent is clearly the most interesting one.Then the global Fourier transform is a new concept.In any case, the formalism below is the same for both cases in Eqs(64), (65).

E. The global Fourier transform in terms of local Fourier transforms and applications in Fast Fourier transforms
The general idea of Fast Fourier transforms is to express the 'large' Fourier transform in a large Hilbert space, as an 'appropriate' combination of 'small' Fourier transforms in smaller Hilbert spaces.Performing the 'small' Fourier transforms instead of the 'large' Fourier transform, is computationally beneficial.The general formalism of this paper can be helpful in this direction.
As an example, we express in this section the global Fourier transform in terms of many local Fourier transforms.This is similar to the Cooley-Tukey formalism in fast Fourier transforms [5][6][7].We only consider the special case n = 2, and we do not discuss complexity issues.But we point out that understanding of the relationship between global and local Fourier transforms and related phase space methods, can be useful in other areas like fast Fourier transforms.
For the case n = 2 we get Let |s be a quantum state in where In this way the Fourier transform in a d 2 -dimensional space reduces to two Fourier transforms in d-dimensional spaces.

V. GLOBAL PHASE SPACE METHODS
The phase space is defined by the Fourier transform and for global Fourier transforms is Z(d n ) × Z(d n ).Displacement operators in it are defined as and Addition in Z(d n ) is used in these two equations, in contrast to Eqs(26), (25) where we have addition in [Z(d)] n .The X L ({β r }) in Eq.(25) can also be written as We have explained that j + β = j + β, and consequently Eqs(75), (77) are an example of the difference between the local and global formalism.Also the Z L ({α r }) in Eq.( 26) can also be written as Comparison of Eqs(76), (78) again shows the difference between the local and global formalism.Since Z G ( α)Z G ( γ) = Z G ( α + γ), the Z G ( α) form a representation of Z(d n ) as an additive group (which is not isomorphic to [Z(d)] n ).The same is true for the X G ( β).Also These relations should be compared and contrasted with Eq.( 27) for the local formalism.Global displacement operators are defined as Here Also These relations should be compared and contrasted to Also The global parity operator (around the point ( γ, δ) in the phase space In analogy to Eq.( 11) we find that the global parity operator is related to the global displacement operators through the Fourier transform Example V.1.We consider the case d = 3 and n = 2.In this case the global Fourier transform is unitarily inequivalent to the local Fourier transform.We work in the 'periods ' of Eq.(34).Let |X; k 0 , k 1 be the basis of position states.The globally Fourier transformed basis is The j r , k r take the values −1, 0, 1.The locally Fourier transformed basis is Then We next consider the local displacement operator X L (−1, 1) which acts on the states |X; 1, 0 and |P L ; 1, 0 as follows: and X L (−1, 1) acts on the state |P G ; 1, 0 as follows: |X; j 0 , j 1 X; j 0 , j 1 |P G ; 1, 0 We also consider the corresponding global displacement operator X G ( −1 + 3 • 1) = X G ( 2) which acts on the states |X; 1, 0 = |X; 1 and |P G ; 1, 0 = |P G ; 1 as follows: and X G ( 5) acts on the state |P L ; 1, 0 as follows: Eqs. ( 90), ( 91), ( 92) involve local displacements and should be compared and contrasted to Eqs. ( 93), ( 95), (94) correspondingly, that involve global displacements.

B. Local and global position and momentum operators and time evolution
We define local position and local momentum operators for the r-component of the system as The X , P have been defined in Eq.( 9).We can also define global position and global momentum operators as They all are d n × d n matrices which can be interpreted as position and momentum operators.In a multipartite system with weak interaction between the various parties, it can be argued that the local variables X (r) L are more physical operators and the Hamiltonian should be expressed in terms of them.But in the case of strong interactions between the parties, the global variables X G , P G might be better for a holistic simple description of the physical Hamiltonian with a good approximation.
Example V.2.We consider the case d = 3, n = 2 and the quantum state We also consider time evolution with the Hamiltonians The first uses the global momentum and position, and the second uses the local momenta and positions.Here (in the position basis) Using both notations X G and P G are At time t = 1 the state becomes From Eq.( 86) it follows that they are related to each other through the Fourier transform: The following marginal properties of the Wigner function follow immediately from Eq.( 14) (for odd values of the dimension d): We have already emphasized that in both the 'local formalism' and the 'global formalism' the position states are the same and the momentum states are different.Consequently P L ; γ 0 , ..., γ n−1 |ρ|P L ; γ 0 , ..., γ n−1 is different from P G ; γ|ρ|P G ; γ and the marginal properties in the second and third of these equations are different.

D. The difference between the local and global Wigner functions
We first consider states for which the local Wigner function is the same as the global Wigner function.
An arbitrary density matrix ρ can be written in the basis of position states, as the sum of a separable density matrix σ(ρ) that contains the d n diagonal elements (as in Eq.( 108)), and a Hermitian matrix τ (ρ) with trace zero that contains the d 2n − d n off-diagonal elements: τ (ρ) is not a density matrix but using Eqs(32), (105) we can define 'Wigner-like' functions for it.Then the Wigner function is written as a sum of two terms that correspond to the diagonal and off-diagonal part: and Then The difference between local and global Wigner functions, is related only to the off-diagonal elements of the density matrix (with respect to the position basis).
Proposition V.4.We consider the following separable density matrices Here the p( j) are probabilities.Then Proof.We first consider the density matrices and prove that Indeed and This proves Eq.( 123).Then and also Example V.5.We consider the density matrices Then In tables VII,VIII, IX and X we present the global Wigner function W G ( γ, δ|ρ), the global Weyl function W G ( α, β|ρ) and the matrices R G ( γ, δ) and R G ( α, β) for the density matrix ρ in Eq.( 137).The correlations in Eq.( 61) are We easily confirm that Eqs(107) hold for the global Wigner and Weyl function.
In general there is no simple relation that links the local with the global quantities.We see this by comparing the expectation values of the local observables X ⊗ 1, 1 ⊗ X , X ⊗ X , P ⊗ 1, 1 ⊗ P, P ⊗ P for the density matrix ρ in Eq.( 137), with the expectation values of the global observables X G , P G for the same density matrix (written in the 'global language' in Eq.( 140 (142) The X , P, X G , P G , have been given in Eqs(100), (101).
The results for the local observables are different from the global observables.For strongly correlated systems global quantities might be physically more relevant.
We note that an observable can be written in both the local and global formalism (using the map in Eq.(35)).For example, for the above system we consider the observable (Hermitian operator) which can also be written as O = a|X; 3 X; 3| + b|X; −2 X; −2| + c|X; −1 X; −1| Important physical quantities like the position can be defined locally like X ⊗ 1, 1 ⊗ X , X ⊗ X , or globally as X G (defined in Eq.( 97)).The same is true for local and global momenta.For strongly correlated systems the identity of each component becomes weak, and global quantities might be physically more appropriate for the description of these systems.

VII. DISCUSSION
In this paper we introduced local and global Fourier transforms and related phase space methods for multipartite systems.The multipartite system consists of n components, each of which is described with variables in Z(d) and with a d-dimensional Hilbert space H(d).In the global formalism we take a holistic view of the system and describe it with variables in [Z(d n )] and the d n -dimensional Hilbert space H.Even if the various components of the system are located far from each other, in the case of strong interactions and strong correlations between them they might loose their individual identity.In this case a holistic approach that uses global quantities, might be more appropriate.
In the local formalism the phase space is [Z(d) × Z(d)] n , and in the global formalism We have explained that although the map in Eq.( 35) is bijective, the ring [Z(d)] n is not isomorphic to the ring n )] (because of Eq.( 36)).The heart of the formalism is the local and global Fourier transforms.We have shown that for some values of d, n they are unitarily inequivalent to each other (proposition IV.4).
We have compared and contrasted the local phase space formalism with the global phase space formalism.Examples of this are: • Some of the local momentum states are the same as the global momentum states (proposition IV.3).
• Density matrices which have only diagonal elements with respect to the position basis, have the same local and global Wigner function (proposition V.3).The difference between local and global Wigner functions, is contained entirely in the off-diagonal elements.
• We have calculated the time evolution in terms of both local variables and also global variables (section V B) • Classical and quantum correlations have been described in the local formalism with the matrices R L , R L and in the global formalism with the matrices R G , R G .
The formalism could be used in the general area of Fast Fourier transforms (in a quantum or even classical context).For example, a link between the present formalism (in some special cases) and the Cooley-Tukey formalism has been discussed in section IV E.
The work is a contribution to the various approaches for multipartite systems.Unitary equivalence between the local and global Fourier transform (Eq.( 64)), implies that the distinction between the concept of a multipartite system and that of a single system is weak.Unitary inequivalence (Eq.( 65) ) implies that the concept of a multipartite system is fundamentally different from that of a single quantum system.

)
Therefore if Tr(F G ) = Tr(F L ) the F G , F L are unitarily equivalent, and if Tr(F G ) = Tr(F L ) the F G , F L are unitarily inequivalent.