Quantum Fourier transform, Heisenberg groups and quasiprobability distributions

This paper aims to explore the inherent connection among Heisenberg groups, quantum Fourier transform and (quasiprobability) distribution functions. Distribution functions for continuous and finite quantum systems are examined first as a semiclassical approach to quantum probability distribution. This leads to studying certain functionals of a pair of"conjugate"observables, connected via the quantum Fourier transform. The Heisenberg groups emerge naturally from this study and we take a rapid look at their representations. The quantum Fourier transform appears as the intertwining operator of two equivalent representation arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra. The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We then study the reconstruction of Wigner function from the marginal distributions via inverse Radon transform giving explicit formulas. We consider applications of our approach to quantum information processing and quantum process tomography.


Introduction
Quasiprobability distribution functions (or simply distribution functions) on a quantum system provide an alternative and equivalent description of quantum states. We will discuss three possible approaches to distribution functions. The first approach is essentially Wigner's original approach [1] and it attempts to give a "phase-space" description of quantum states. The state of a quantum system determines the probability distributions of its observables. It is possible to completely specify the state by giving the distributions of functions of a pair of conjugate nondegenerate observablesf andĝ. The most well-known example is the position-momentum pair. Thus corresponding to a (mixed) state ρ we associate a real function W pp, q : ρq of "c-number" variables which have the same information content as the state and give the correct marginals. Now for conjugate observables the expectation values of the functions φpf ,ĝq will specify the state completely (ignoring the questions of operator ordering). In email:{manas,schmuel}@cs.york.ac.uk classical probability theory these expectation values are generated by the characteristic function. The characteristic function of (classical) random variables X 1 , X 2 , . . . , X n with joint probability distribution F px 1 , . . . , x n q " F pxq is given by [2] r F ptq " where t¨x is the usual Euclidean scalar product of two real vector x " px 1 ,¨¨¨, x n q T and t " pt 1 ,¨¨¨, t n q T where T denotes transpose. If the probability distribution is given by a probability density ppxq then the characteristic function is simply the Fourier transform of ppxq. Another way of viewing the characteristic function is to note thatF ptq " xe it¨X y, the expectation value of the complex random variable e it¨X . Then assuming the existence of a probability density it is given by the inverse Fourier transformation ppxq " In the form (1) it is suitable for a "quantum" extension [3]. Of course, in the quantum case the observables (or quantum random variables) X i will not commute in general and we have the problem of interpreting the function p as a joint probability distribution. However for a set of compatible or commuting observables a joint distribution is unambiguously defined. For incompatible observables we may take (1) as the definition of joint probability distribution. The values that are obtained on joint measurement of these observables constitute the joint spectrum which, in general, may have both continuous and discrete segments. In the finite-dimensional case we have a finite spectrum and hence the integral has to be replaced by a sum. But there are problems in interpreting this as a function in a classical phase space [4]. Alternatively, we can work with finite Fourier transform. Such a transform is defined over a finite abelian group. From the structure of such groups we may focus on the group Z N " Z{N Z, the additive group of integers modulo N . We can thus take our "configuration space" to be Z N . This in turn forces us to consider observables which take "values" in the set t0, 1, . . . , N´1u where N must now be identified with the dimension of the Hilbert space. Since we are concerned only with probability distributions of the possible outcomes this is really not a restriction. Operationally, we can always calibrate our instruments to yield these outcomes. The "position" and "momentum" variables are both identified with Z N and the corresponding unitary representations respectively act multiplicatively and additively on the "position space". So we have two representation of Z N . But they do not constitute a representation of the "phase space" Z NˆZN as the latter is a commutative group. The simplest possible noncommutative extension is a central extension [5]. After some restrictions due to finiteness of the dimension we get a Heisenberg group. Now let us approach the problem of distribution function from another perspective. The state of the system, ρ is a positive definite operator with trace 1. In infinite dimensions they belong to a special class called trace class operators.
In finite dimensions every operator is clearly trace class. Trace class operators admit a Hilbert space structure. Thus for two such operators A, B define pA, Bq " TrpA : Bq. In finite dimensions this introduces the familiar Frobenius or Hilbert-Schmidt norm. If we restrict to hermitian operators we get a real Hilbert space K. Pick an orthonormal basis tA i : i " 1, 2, . . . , u in K. We can write the state ρ " ÿ i W i pρqA i with W i pρq " TrpρA i q Clearly W i are real. If we also demand that ř i W i " 1 then we have a quasiprobability distribution over the index set I pi P Iq. We want this index set to have a classical interpretation and a natural choice is the phase space. Then, i " px, zq is a pair of indices 1 . Henceforth, we assume this and write Apx, zq instead of i. The works [6,7] follow this approach to distribution function (see also [8] for a review in the finite-dimensional case). Thus the choice of distribution is equivalent to choice of special bases. The operators Apx, zq are called phase-point operators. Actually, they have to satisfy some extra conditions. We will see that the phase point operators correspond to certain sets (called Wigner sets) in the group algebra of the Heisenberg group.
There is yet another view of distribution function which has origins in signal analysis. A signal may be represented in the time domain as f ptq or frequency domain asf pωq (Fourier transform). Thus we represent the signal in terms of "elementary" harmonic signals and the coefficients give the representation in frequency domain. But it can also be represented by other elementary nonharmonic signals with minimum uncertainty. This was Gabor's seminal idea [9]. Unlike harmonic signals Gabor's elementary signals are localized in time and frequency domains. This joint time-frequency domain is the analogue of phase space. How do we generate these elementary signals? Starting with a "reasonable" initial signal say a Gaussian function in the time domain we apply a sequence of two operators, multiplication and translation, which are "diagonal" in the time and frequency domain respectively. The resulting sequence of functions are used to represent an arbitrary signal. Now to represent a vector in some space we don't need a basis, any set of vectors that span the space will do. In finite dimensional Hilbert space such sets define frames [10]. An example of such overcomplete sets is the set of coherent states in quantum optics. The frame-theoretic approach to distribution functions was recently proposed in [11]. We will not go into details of frame theory but mention that frames are a generalization of orthonormal bases in Hilbert space. In this context, Gabor's elementary functions constitute Gabor-Weyl-Heisenberg (GWH) frames. Most distribution functions are examples of such frames although there are some exceptions. The GWH-frames are generated by applying sequence of translation and multiplication operators to (continuous) signals creating a function in the time-frequency domain (the phase space!). These operators generate a discrete Heisenberg group.
Finally, we come to another significant property of the Wigner function, a particularly important distribution function. Let W px, pq be the continuous Wigner function with x and p representing the classical position and momentum variables. Then the marginals ř x W px, pq and ř p W px, pq are probability distributions of the quantum momentum and position operator respectively. Further, if we sum W px, pq along some line ax`bp " 0 then the resulting function is the probability distribution of a quantum operator "orthogonal" to cx`dp where pc, dq is a vector orthogonal to pa, bq in the x´p plane. We will make these definitions precise later. Call this the Radon property. This is an important property and can be used to invert the transform. We will see that the Radon property is related to the transformation properties of distribution function under some automorphisms of the Heisenberg group. We note that the Radon property is very useful in the practical problem of reconstruction of states and processes. We prove a general Radon property which gives us a lot of freedom in our choice of possible measurements.
The brief (and incomplete) survey of the approaches to distribution functions in the preceding paragraphs indicates that a lot of work has been done in this area 2 . Besides their theoretical significance distribution functions have application in state tomography [13], statistical mechanics and quantum optics [14]. It is also intimately connected with the theory of coherent states. The GBH type operators after complexification and some algebra give rise to the familiar displacement operators. The coherent states are the orbits of Weyl-Heisenberg group (henceforth only Heisenberg group) [15]. In this work we mainly focus on the finite-dimensional case. This case presents some difficulties absent in the continuous case. The finite dimensional case is also significant for quantum information processing [16,17].
We have considered Z NˆZN as the basic model of finite "phase space". In the literature other phase spaces have been considered (see [11] for a discussion and references). The intrinsic structure of these phase spaces may have interesting bearing on the corresponding distribution functions. In particular, some authors have considered finite field F N with N elements instead of Z N (see e.g. [18]). This is only possible if N " p n is a power of some prime p. Now Z p and F p coincide. In general, the additive groups of F p n and Z pˆ¨¨¨ˆZp (n factors) are isomorphic. We can consider the Heisenberg groups over the latter (by central extension). We do not follow this here as the paper is already quite large. However, note that the analogy between what the authors in [18] call quantum nets and Wigner sets defined in this paper. More precisely, quantum nets correspond those Wigner sets which are permuted by the action of the automorphism group SLp2, Z N q. The paper is quite self-contained. We give most of the proofs. Some of the results are known but were derived using different approaches. Let us first note some of the main contributions of the present work.
1. We use the Heisenberg groups as the basic approach to distribution functions. As we have seen in the preceding paragraphs this is the unify-ing thread tying the different approaches and perspectives on distribution function.
2. Heisenberg group has been used in the literature context of distribution functions. But here we use discrete Heisenberg groups and family of finite quotient groups thereof. We define these groups abstractly in terms of generators and relations. Thus we can consider different representations (irreducible and reducible) and operators between representations.
3. Our treatment of the Heisenberg groups is defined in terms of generators and relations. This makes the computations and proofs easier. Further, we don't need the language of (pseudo) phase spaces. 4. We show that the existence of distribution functions is equivalent to certain sets in the group algebras of the Heisenberg groups. This provides us with powerful methods of representation theory. We list some of the outcomes by the use of these methods.
(a) The analysis of marginal distributions become transparent. They correspond to an invariance up to permutations under certain groups of automorphisms (e.g. SLp2, Z N q) of the Heisenberg groups.
(b) Demanding this invariance we get unique distribution functions in odd dimensions.
(c) We also infer that it is impossible to retain full invariance and other properties like hermiticity and linear independence even dimension. We therefore have three possible strategies: i. drop the requirement of invariance, ii. drop the requirement of independence or hermiticity or iii. require invariance under a smaller set of transformations. We discuss all three and give some alternative candidates for distribution functions in even dimensions.
(d) Our analysis via the automorphism groups obviates the need for ad hoc hypothesis and guess work.
We note again that although the results mentioned in (a), (b) and parts of (c) are known our approach via automorphisms is different.

5.
We give explicit formulas in most cases. The close connection with finite Radon transform is made clear. It is used to derive the formulas for state reconstruction. The inversion formulas in the case o dimension " 2 n the formulas appear to be new. 6. We explore applications to quantum computing and information. The fact that Weyl-Heisenberg groups describe the kinematics of quantum systems is known [19,20]. We show that the dynamics is described by (unitary) automorphisms of the group algebra. The case of unitary automorphisms of the group itself is analyzed in [21]. The latter correspond to the Clifford group and to go beyond it ("non-classical" dynamics) we have to consider the group algebra. We also illustrate the utility of the Heisenberg groups in quantum circuits. We give an application to quantum process tomography.
We now give a summary of the paper. In Section 2 we discuss the quantum Fourier operators. The quantum Fourier transform (QFT) is another form of finite Fourier transform [22]. Consider the two representations of Z N acting as multiplication and translation respectively. The QFT connects the two. The generators of the two representations give us the basic operators: Z and X. We can consider these as the finite-dimensional analogue of unitary operator generated by "position" and "momentum" respectively. In Section 3 we review (continuous) quasi-probability distribution functions or simply distribution functions. The continuous distribution functions are somewhat easier to deal with because the "infinitesimal" generators satisfy simple commutation rules (the Heisenberg relations). In Section 4 we come to one of our main themes the finite distribution functions. We list a set of properties, satisfied by the continuous Wigner function, and demand that any distribution function must satisfy them. In particular, we give examples of discrete Wigner functions. Here we encounter the difficulties when the dimension is even. We also derive explicit formulas for the phase-point operators. The odd-dimensional case (apart from some constants) is essentially same as Wootters' [7] operators in prime dimension.
In Section 5 we study the Heisenberg groups. We start with the continuous version as it has been studied well in connection with Fourier transforms [23]. We then look at discrete and finite Heisenberg groups, their structure, representation and automorphisms, all of which play important role in our study of the finite distribution functions. We show that there is a one-to-one correspondence between distribution functions in dimension N and certain sets tApx, zqu, (called Wigner sets) in the group algebra of the Heisenberg group H in that dimension. The representation of these sets are the phase-point operators. A slight generalization of the Wigner sets may be used to define Weyl-Heisenberg frames. The group SLp2, Z N q of 2ˆ2 matrices in Z N with determinant 1 induce automorphisms on the Heisenberg group H. Thus for each M P SLp2, Z N q we define an automorphism σ M of H. These automorphisms, in turn, determine the marginal properties of the Wigner functions. Thus if W px, zq " W pζq is a distribution function then the functions are the marginals. Qpzq is called a simple marginals if it is the probability distribution (in the given state) of an observableẑ M defined by e iẑM " σ M pZq. A similar definition can be given for P pxq. We show the necessary and sufficient condition for the existence of simple marginals for all members of SLp2, Z N q.
Thus the requirement that all marginals be simple determine the distribution function up to isomorphism. In the case of odd dimensions for the Wigner function all marginals are simple. This can be neatly expressed as follows.
Let A M px, zq " ApM´1x, M´1zq. Then tA M px, zqu is also a Wigner set. An analogous result is called Clifford invariance in [24]. This is not true in even dimensions. We investigate three alternatives by weakening our requirements. First, we do not demand that the phase-point operators form a basis. Now they constitute a frame. This is the most common approach (see for example, [25,16]). The marginal conditions are simple but at the expense of losing orthogonality of bases. We show that this is similar to the case of spin-1/2 representation in the sense that a complete "rotation" does not preserve the values of the functions involved. More precisely, we get functions which are not periodic on Z N . However, they have period 2N . So we go over to Z 2NˆZ2N as the phase space. Next we drop the requirement that the marginals be simple in the above sense. It is still possible to compute the marginals in terms of the probability distribution of the observableẑ M . We indicate how explicit formulas can derived to compute this. Finally, since it is not possible to satisfy simple marginal conditions on all of SLp2, Z N q we consider certain subsets adequate for inversion, that is, computing the state from the marginal data. We give such a subset in dimension N " 2 K . A similar construction from a different perspective was done in [26] but we present our formulas in an explicit functional form.
In the section 5.1 we explore the fact that the definition of marginals is a Radon transform of W in the sense of [27,28,29]. We then give the inversion formulas in several cases. The inversion formulas for odd dimensions were given in [13]. Our derivation, however, is more general and applicable to any finite distribution function. The important point is we can invert these transform and recover the Wigner function and hence the quantum state. In the next section we discuss some applications to quantum information processing. We provide some simple relations between standard quantum gates and operators representing the Heisenberg groups which will prove useful for implementing the state and process determination schemes using phase-point operators. We give formulas for quantum process tomography using phase-point operators. In the final section we discuss some more possible applications and future work.

Quantum Fourier Operators
Let G " Z{N Z be the additive group of integers modulo-N . There are two obvious representations of G on an N -dimensional Hilbert space H. Let g be a generator of G. Suppose φ : G Ñ UpHq is faithful representation of G by unitary operators where UpHq is the set of unitary operators on H. If φpgq " Z then we must have Z N " 1 since the order of G is N and the representation is faithful. The eigenvalues of Z are N th roots of unity. Let t|iy : i " 0,¨¨¨, N´1u be the corresponding eigenvectors such that Z |iy " ω i |iy where ω is a primitive N th root. Call it the the computational basis B c . There is another representation φ 1 of G defined by φ 1 pgq " X where X |iy " |i`1 pmod N qy. We can think of φ as the multiplicative and φ 1 as the additive representations. Z and X represent multiplication and translation operators resp. Clearly, X is unitary and there is basis B f in which it is diagonal. The quantum Fourier transform (QFT) is the unitary map connecting the two representations taking B c Ñ B f . The eigenvalues of X are also roots of unity as X N " I. Since φ 1 is also faithful the diagonalization of X yields Z fixing the ordering. Hence there exists a unitary operator Ω such that Ω : XΩ " Z The explicit form of Ω in the computational basis is easy to compute. Thus, if α " ř i x i |iy is and eigenvector then Xα " uα implies x 0 " ux 1 , . . . , x N´2 " ux N´1 , and x N´1 " ux 0 . This yields after normalization So the quantum Fourier transform (QFT) is the map We note that we follow the convention of mathematicians in the definition of discrete or finite Fourier transform. In the quantum information literature the usual definition is with a positive sign in the exponent which is our inverse Fourier transform. Now X and Z can be expressed as X " e ix and Z " e iẑ (4) wherex andẑ are the hermitian generators of the respective unitary rotation. Moreover, their eigenstates are (discrete) Fourier transforms of each other. This is reminiscent of position and momentum observables which also have the property that their (generalized) eigenstates are (continuous) Fourier transforms of each other. We may therefore regard the observablesx andẑ as conjugate "dynamical variables". This terminology is further justified by the following observation which is crucial for our calculations of quasi-probability distributions.
This is most easily derived by applying both sides to vectors in the computational basis B c . We observe that the unitary operators e iap and e ibq corresponding to translations in (continuous) position and and momentum space respectively obey a similar relation. Suppose now that H is a product space, that is, As a simple application of the basic relation (2) we show that the Fourier transform of product states in the computational basis are also product states and generalize a computationally useful formula. Proof. Observe that there is an implicit ordering of the product states. Thus if j " ř m´1 r"0 d r j r is the representation of a positive integer 0 ď j ď d m´1 then the state |jy " |j m´1 y b¨¨¨b |j 1 y b |j 0 y " |j m´1 y¨¨¨|j 1 y |j 0 y " |j m´1¨¨¨j0 y where we suppress the tensor product symbol in the last two relations. Further, we write |0y for |0¨¨¨0y From the definition of QFT xk m´1¨¨¨k0 |Ω|j m´1¨¨¨j0 y " x0|X´kΩ|j m´1¨¨¨j0 y " x0|ΩΩ : X´kΩ|j m´1¨¨¨j0 y " x0|ΩZ´k|j m´1¨¨¨j0 y " ω´k j x0|Ω|j m´1¨¨¨j0 y " ω´k j ? N A direct computation shows that for the state Since this is true for all 0 ď k ď N´1, Ω|jy " |ψ j y.

Distribution functions in quantum systems
One of the motivating factor's for distribution functions in Wigner's work [1] was the construction of a quantum analogue of Liouville density in classical phase space. Following this approach suppose we want a "joint" distribution function of the operators X and Z. More precisely, we seek hermitian operatorŝ x and andẑ such that X " e ix and Z " e iẑ (6) and then try to find distribution functions associated with the observablesx andẑ. We will do our computations in the computational basis B c in which Z is diagonal. It is easy to findẑ. Thuŝ Of courseẑ is only determined modulo 2πk. From (2) and (3) we have These entries of the matrixx imply that it is a hermitian circulant matrix. But a general linear combination ux`vẑ is not a circulant but a Toeplitz matrix.
There are efficient algorithms for finding the eigenvalues and eigenvectors of such matrices. So in principle we can compute expressions like xe ipux`vẑq y for real or integer u, v using the standard diagonalization procedure. It is feasible to find analytic expressions, however, in low dimensions only. We will tackle the problem by different approach. Let us briefly review the continuous case first.

The Wigner distribution
In the case of canonically conjugate variables like position and momentum a number of quasi-probability distributions are possible, each corresponding to a an operator ordering prescription. This is facilitated by the fundamental commutation relation rq,ps " i between the position and momentum operators. Taking traces it is clear that such a relation is not possible in finite dimensions. So in finite dimension it is not clear how to prescribe ordering of operators. Moreover, there is some ambiguity in defining hermitian generators themselves. For example, for integers a and b, x 1 "x`2πaI andẑ 1 "ẑ`2πbI are also infinitesimal generators for X and Z respectively but their linear combinations give rise to different set of unitary operators. The problem of non-uniqueness is essentially the same as the one that arises in defining roots and logarithms of complex numbers. Hence, we restrict to the principal branch of the logarithm as evident in the definition of x andẑ.

Continuous Wigner distribution
The inversion formula of a characteristic function of classical probability is different for continuous and discrete probability distributions. In finite-dimensional quantum systems or more generally in the discrete part of the spectrum of a quantum observable we should use a formula analogous to that for discrete distributions [3]. But, this gives a quasi-probability distribution which may not have the desired properties [4]. The problem seems to be rooted in the noncommutativity of quantum observables. The continuous Wigner distribution is defined by In this equation and the rest of the paper, unless the limits are explicitly stated, the real integrals are over the whole real line. Further we use the notation r " px, yq T for a real vector in 2 dimensions. The following theorem gives some of the important properties of the continuous Wigner distribution. First, we make the dependence on the state ρ (mixed state, in general) explicit when necessary: W c px, z : ρq. We give a simple proof of a well-known results in the appendix.
where |zy are generalized eigenvectors ofẑ. We have a similar relation for the other marginal. We also have the following results on general marginal distribution. Let R be an orthogonal matrix of order 2 representing a rotation. Let where |z 1 y are generalized eigenvectors ofẑ 1 and the variables x and z are considered as functions of x 1 , z 1 . A similar result holds for the conjugate observablê x.
The continuous version of distribution function of discrete observables is problematic. First, we want the marginals to resemble classical discrete probability distributions so we have delta functions at the isolated points. To justify the later we have to integrate over some domain of the continuous variable and this causes some problems interpreting these as probability distributions. Some authors have attempted to tackle these problems by focusing on continuous families of discrete observables like spin direction. These approaches seem somewhat unnatural to us. Discrete distributions must be characterized by a discrete measure (e.g. the counting measure) and thus the integrals must be replaced by sums. In particular, for finite systems we must have finite sums. This is the avenue we will explore in this paper. Finally, let us mention an important point. The Wigner distribution and some other related probability and quasi-probability distributions are sometimes interpreted as joint probability distributions of incompatible observables. Clearly, any measurement of such distribution must give unsharp values of these observables, consistent with the uncertainty principle. The Arthrus-Kelley scheme [30,31] is an example. For discrete observables the concept of joint distribution of noncommuting observables is difficult even for fuzzy measurements.

Discrete Quasiprobability Distributions
The Wigner and other distribution functions are an alternative to the density matrix formulation of quantum theory and are given by distribution function W py : ρq with y representing classical parameters. Expectation values of any quantum mechanical quantity that can be computed in a given state ρ can be computed from W py : ρq. Hence we have a correspondence between quantum observables and "classical" observables along with an ordering prescription. Since, the density matrix provides a maximal description of a quantum system so does W py : ρq. We thus have an alternative semiclassical picture. In some situations the latter may be easier to determine experimentally. In any case, such quasiprobability distributions provide a useful tool for visualization.

Properties of Distribution Functions
Let us make precise the requirements we impose on distribution functions. Let W py : ρq be a distribution function associated with a quantum state ρ of a quantum system S and y is a real vector representing a finite set of "classical" parameters. Let H be the system Hilbert space and SpHq the set of states, that is, the convex set of normalized positive trace-class operators.
R1. W py : ρq is a continuous real function on SpHq that preserves convex combinations: if ρ 1 , ρ 2 P SpHq and 0 ď s ď 1 then It is nondegenerate in the sense that at no point in phase space W py : ρq is identically 0 for all ρ.
R2. For two states ρ and ρ 1 Part of the above requirement is that we define the appropriate measure dy which also fixes the constant K. The constant K " 2π for continuous systems and K " N for finite system of dimension N . This constant equals the volume of a "phase space cell". With respect to this measure we also demand normalization condition ż W py : ρqdy " 1 Note that this is a nontrivial requirement as this implies that the left side of the above equation must be independent of the quantum state.
R3. For any observable A on S there is a real functionÃpyq such that the expectation value (in state ρ) The first requirement is that W must be real. If we try to impose nonnegativity however it becomes too stringent. As W py : ρq is convex linear on states in finite dimensional spaces it has a unique extension to a liner functional (for fixed y) on KpHq the set of bounded hermitian operators (observables). In infinite dimensions we need some delicate continuity arguments. Henceforth, we will assume linearity of W py : ρq. In these specifications for distribution function we have not mentioned marginals. We will discuss them shortly. What are the characteristics of the parametric vector y? If it is to be somehow identified with generators of classical observables its dimension must be related to degrees of freedom. The third item in the list gives a clue. A physical system, whether classical or quantum, is completely characterized by the set of observables O. Often O has more structure, in particular, it is an algebra. The difference between quantum and classical algebra of observables is that the former is noncommutative. These algebras have minimal sets of generators. For example, the observable algebra of a classical system with N degrees of freedom is generated by generalized coordinates tq i : i " 1, . . . , N u and the conjugate momenta tp i : i " 1, . . . , N u. The corresponding quantum algebra is also generated by the operatorsq i andp i which do not commute. In the finite dimensional case we have no classical analogue. But we will be guided by this example. We have already discussed the close analogy between the finite-dimensional unitary operators X, Z and the continuous operator e ip , e iq . We show next thatx andẑ are actually generators of the complex algebra of observables in the appropriate dimension.
Proposition 1. Let the dimension of the system Hilbert space be N andx and z be as given in (8) and (7) respectively. The completion of the complex algebra generated byx andẑ equals M n pCq, the algebra of complex matrices of order N .
Proof. The completion of algebra means that we include the limits of convergent sequences. In particular, X " e ix and Z " e iẑ are in the completion. We show that X and Z generate M n pCq. Let ω " e 2πi{N and Z k " ω´kZ. It is easy to see that where D k pijq " δ ij δ jk is the diagonal matrix with 1 in the kth row (and column) and 0's everywhere. We also see that X j D k " E j`k,k where E ij are the elementary matrices with 1 in the ijth place and 0's everywhere else. Note that j`k is to be considered mod N . Thus every elementary matrix is generated byx andẑ. As the elementary matrices constitute a basis for M n pCq the proof is complete.
We mention that the assertion of the proposition was essentially proved by J. Schwinger [20] in different way. The continuous quasiprobability distribution functions can be written as Here f is a scalar or c-number function which is usually interpreted as an operator ordering prescription. The Wigner distribution function is a special case corresponding to Weyl ordering. All this is possible because of the simple commutation properties of the observablesq i andp i . We have observed that the unitary operators X and Z have multiplicative relations very similar to e ip and e iq (see equation (5)). This analogy extends further provided a and b are integers. Of course, the second formula is valid for all real a and b but the first fails if both are non-integer. This provides another reason to construct a discrete version of distributions functions. Henceforth we will restrict to finite dimensional spaces mostly. Since the operatorx andẑ can be used as generators we will assume the "phase space" spanned by y " px, zq is 2-dimensional. Now we can state the marginal conditions corresponding to the "axes".
We seek a finite distribution function similar to the form (14) above. For a state ρ in a finite quantum system of dimension N define with ω " e 2πi{N and 0 ď j, k ď N´1 integers (17) Call the functions f in the above expression ordering functions. To compute the expectation values we need the following matrix elements in computational basis.
To see the implications of the reality condition R1 it is sufficient to verify it for pure states. Hence for ρ " |αy xα| In the last step we use X N " Z N " I. Since this must hold for all state vectors α we have We will see later that the condition of nondegeneracy is automatically satisfied. Next we consider R2. Let ρ " ř ρ jk |jy xj| k and ρ 1 " ř ρ 1 jk |jy xj| k. Then using where we have used (19) in the last step. According to R2 this should be equal to Trpρρ 1 q{N " p ř jk ρ jk ρ 1 kj q{N for all choices of density matrices ρ and ρ 1 . This is possible if |f pm, nq| 2 is a constant independent of m and n. A straightforward computation yields |f pm, nq| " 1{N 2 . Setting f pm, nq " gpm, nq{N 2 we may write gpm, nq " ω βpm,nq . We now prove existence and properties of distribution functions satisfying the conditions R1-R4.

Theorem 2. For every density matrix ρ in an N -dimensional Hilbert space and
Then there exist functions f pm, nq such that the corresponding W satisfies R1,R2 and R4. Moreover, for any W satisfying these conditions there are unique hermitian operatorsâpx, zq such that following hold.
Proof. We have observed that functions f pm, nq " ω βpm,nq {N 2 satisfying the relations (19) provide a distribution function W px, z : ρq that satisfies conditions R1 and R2. To see the implications of condition R4 on marginals we observe that ÿ For the last expression to be equal to Trpρ |zy xz|q, the probability of finding the system in an eigenstate ofẑ with eigenvalue 2πz{N , we must have f p0, nq " 1{N 2 for all n. Computing the trace in the Fourier transformed basis |jy " Ω |jy we conclude that the second condition in R3 yields f pm, 0q " 1{N 2 . Assuming βpm, nq to be a real polynomial in m and n we conclude that it must be of the form βpm, nq " mnαpm, nq. More generally, we may take βpm, nq " γpm, nqm nαpm, nq with γp0, nq " γpm, 0q " 0. With this choice of βpm, nq the first set of equations in (19) are satisfied. The second set gives the following requirement on the function α. mnpαpm, nq´1q`pN´mqpN´nqαpN´m, N´nqγ pm, nq`γpN´m, N´mq " 0 mod N Note that we do not require α or γ to be integer-valued or symmetric. There exist (real) functions satisfying equation (24) for all 0 ď m, n ď N´1. Simple solutions to these equations are given below.
where ν mn satisfies |ν mn | " 1 and ν N´m,M´m " p´1q m`n ν mn A particular choice of ν satisfying (26) is Other choices of ν mn will be given later when we impose more conditions on the distribution functions. Finally, suppose the functions f in the definition of W px, z : ρq satisfy the reality conditions (19) and the marginal condition f pm, 0q " f p0, nq " 1{N 2 . Then it is easy to see that for the incoherent state I{N W px, z : The distribution function is nondegenerate at each point in phase space. This proves the existence of solutions to equations (24) and hence quasiprobability distributions satisfying R1,R2 and R4 in all finite dimensions.
Observe that the map Ξpx, zq : ρ Ñ W px, z : ρq is real and can be uniquely extended to a linear map on all hermitian operators. That is, Ξ is a linear functional on K H , the linear space of hermitian operators on the system Hilbert space H. K H is a real Hilbert space with respect to the scalar product ă A, B ą" TrpABq, A, B P K H . Since Ξpx, zq is nondegenerate at each point there exists a unique nonzeroâpx, zq P K H such that W px, z : ρq "ă apx, zq, ρ ą" Trpâpx, zqρq. So the first of the equations in (21a) holds. The condition R2 and (21a) together imply ă ρ, ρ 1 ą" Trpρρ 1 q " ÿ xz W px, z : ρqW px, z : ρ 1 q " ÿ xz W px, z : ρqTrpâpx, zqρ 1 q " Trpp ÿ xz W px, z : ρqâpx, zqqρ 1 q "ă ÿ xz W px, z : ρqâpx, zq, ρ 1 ą Since this is true for all positive definite operators ρ 1 with trace 1 we conclude that the second of the equation in (21a) must hold. Now using this expansion of ρ in the operatorsâpx, zq and that fact R2 again we conclude that the equations (21b) hold. Finally, to prove that condition R3 also holds observe that any hermitian operator T can be written in the form T " b 1 ρ 1´b2 ρ 2 , with b 1 , b 2 ą 0 and ρ 1 , ρ 2 density matrices. Then, The distribution functions corresponding to f 0 will be called (finite) Wigner functions. In case of odd dimension there is one such function. But in even dimensions we have to be more careful in our choices.

Explicit formulas
The mere existence of "orthonormal" hermitian operators likeâpx, zq which span the (real) space of observables is simply a statement about the existence of orthonormal bases in any Hilbert space. Two characteristics distinguishâpx, zq: first the marginal distributions associated with them (R4) and second the way they were derived via the quantum Fourier transform. Our next task is to find explicit forms for these operators. Let be a quasiprobability distribution satisfying R1-R4. We have indicated explicit dependence on the ordering function f . From this it follows that the phase-point operators are given bŷ The fact thatâpx, z : f q form an orthonormal operator basis can be verified directly. The name "phase-point operator" derives from the fact that px, zq may be considered as a "point" in a finite phase space. The quasiprobability distribution W px, z : ρ, f q are simply the coefficients in the expansion of ρ in the basis tâpx, z : f qu. We will compute these operators in the "computational" basis t|jy " |j mod N yu, that is, the eigenbasis of the operator Z. Then X m " ř j |j`my xj| and X m Z n " ř j ω jn |j`my xj|. A straightforward calculation then giveŝ apx, z : f q kl " xk|âpx, z : f q|ly " ω´p k´lqx ÿ n f pk´l, nqω npl´zq In particular, the diagonal terms are easy.
Now using the formulas (25) for f pm, nq in the formula we get the following two cases for N . First for N odd, Apart from ordering and normalization these are precisely the phase-point operators found in [7] for prime dimensions. Note that we do not require the dimension N to be prime. If N " 2r is even the calculation is a bit more involved as the corresponding expression for f 0 pm, nq in (25) is not "homogeneous". We now havê apx, z : f 0 q kl " Trp|ly xk|âpx, z : f qq " ω´p k´lqx ÿ n ν mn ω npk`l´zq{2 Evaluating these sums is not difficult but one has to be careful about the signs. For the choice of ν mn given in (27) we get apx, zq kl " So we see that the quasiprobability functions given above are much more complicated in even dimension. More importantly, the phase-point operators given by (33) is more sparse than the one (35) for even dimension. This, in turn, implies that in general quasiprobability distributions are sparser in the odd dimension and "computationally simpler". Let us illustrate with an example. Suppose a quantum circuit or protocol is supposed to produce a state |by in the computational basis. Because of noise and imperfections we actually get a state (possibly mixed) which lies in the state space corresponding to the subspace K spanned by t|b˘iy : i ď au. From the formulas (33) and (35) it is easy to see that the number of nonzero entries W px, zq in the odd case is Opaq and in the even case it is Opa 2 q. From the duality between X and Z this is also true if the computational basis is replaced by its Fourier transform. Since finding quasiprobability distribution equivalent to determining the state does it mean that odd dimensions are tomographically "better"? That we should look at qutrits too?

Heisenberg groups
In this section we turn to our main theme: the Heisenberg groups and their close connections with Fourier transforms and distribution functions (see [23] for this connection in the continuous case). There are families of continuous and discrete Heisenberg groups. Although our primary focus will be on the discrete Heisenberg groups we first take a look at the continuous Wigner function from a different perspective. We start with the (continuous) n-dimensional Heisenberg group H n whose group manifold is R 2n`1 . Using vector notation we write the elements as pp, q, tq where p and q are vectors in R n and t is a real number. The reader can easily recognize the "phase space" behind this notation. The group multiplication is defined by pp, q, tqpp 1 , q 1 , t 1 q " pp`p 1 , q`q 1 , t`t 1`p p¨q 1´q¨p1 q{2q where the¨denotes the usual scalar product. The symplectic structure is apparent in the above definition. By changing the parametrization of the group pp, q, tq Ñ pp, q, t´pq{2q " pp 1 , q 1 , t 1 q we get the multiplication law of the (polar) Heisenberg group [23].
Note that the element p0, 0, tq is in the centre of the group. Let us restrict to n " 1 for simplicity. The Lie group H 1 is generated by the Lie algebra h 1 with generators tp, q, λu with brackets rp, qs " λ, and rλ, ps " rλ, qs " 0. One constructs the Poisson structure on the dual space h1 in a natural way. The Heisenberg group plays a fundamental role in quantum mechanics. The Stonevon Neumann theorem asserts that the standard representation of position and momentum are essentially unique. In other words, the Schroedinger picture pp, q, tq Ñ γpp, q, tq " e 2πit e 2πippp`qqq withqψpq 1 q " q 1 ψpq 1 q andpψpqq "´i Bψpqq Bx is the unique representation of Heisenberg group under some conditions of continuity. Here ψ is the wave function in one dimension. Mathematically, it lives in the space H " L 2 pRq of complex square integrable functions (we ignore the technical difficulties arising due to the unboundedness of the operators). Since, the elements p0, 0, tq are in the centre it is often sufficient to consider only elements of the form γpp, qq " γpp, q, 0q " e 2πippp`qqq . This is the reduced Heisenberg group. Let tψ α pxqu be a basis in H. The matrix elements in this basis are given by V αα 1 pp, qq " xψ α |γpp, qq|ψ α 1 y " xψ α |e 2πippp`qqq |ψ α 1 y " ż ψ α puqe 2πippp`qqq ψ α 1 pvqdudv These are precisely matrix elements of the Fourier transform of the phasepoint operators in the continuous case. In particular, V αα p0, qq yields Fourier transforms of the position probability density corresponding to the state ψ α . Similarly, using the momentum representation we get the other marginal for V αα 1 pp, 0q. Since the basis was arbitrary we conclude that the Wigner function W pp, qq " ż xγpu, vqye´2 πippu`qvq dudv when integrated over the strip between q " c 1 and q " c 2 gives the probability of the particle in (pure) state ψ to have its position between c 1 and c 2 . Explicitly, ż c2 c1 dq ż 8 8 W pp, q : |ψyqdp yields the probability that the position observable has value between c 1 and c 2 and similarly for the momentum observable. This is easily seen by expanding |ψy in the position basis. What do we get if we integrate over an arbitrary strip, not necessarily parallel to the p or q axes, say the lines ap`bq " c 1 and ap`bq " c 2 ? The answer is well-known and is discussed in [7] and [32]. But we look at it from a different perspective. First, put ap`bq " p 1 . This defines a family of parallel lines p 1 " c in the p-q plane. Another line cp`dq " q 1 does not belong to this family if and only if ad´bc ‰ 0. Thus, the matrix ζ "ˆa b c d˙i s invertible and defines a change of coordinate in the phase plane.
Then the form up`vq " u 1 p 1`v1 q 1 where pu 1 , v 1 q " pu, vqζ. This in turn defines a transformation on the Lie algebra generated byp,q: γpu, vq " e 2πipup`vqq " e 2πipu 1p1`vq1 q ,ˆp 1 q 1˙" ζˆp qİ f the transformationp Ñp 1 ,q Ñq 1 were an automorphism then thep 1 andq 1 have the same commutation relation asp andq. This will happen if and only if det ζ " ad´bc " 1. But then if we change the variable of integration to p 1 , q 1 the measure remains unchanged (| det ζ| " 1). We can now carry over the argument from the case of axes marginals and conclude that integration of W pp, q : |ψyq over a strip between ap`bq " c 1 and ap`bq " c 2 gives the probability that the observablep 1 " ap`bq will have value lying between c 1 and c 2 . Let us observe that ζ P SLp2, Rq " Spp1, Rq where SLpn, Rq is the group of nˆn real matrices with determinant 1 and Sppn, Rq is the real symplectic group of order n. In general for Sppn, Rq is a subgroup of the automorphism group of H n and is different from SLp2n, Rq. Now we turn to the discrete Heisenberg group H. We define a presentation of the group in terms of generators and defining relations [5]. H is generated by tx, z, γu. The defining relations are zx " γxz, γx " xγ and γz " zγ The advantage of this approach is that any map φ from the generators of a group H to another group K which satisfies the same defining relations as above can be uniquely extended to a group homomorphisms H Ñ K. A simple realization of the group over integers is given by the set Z 3 . The multiplication is defined by pj 1 , k 1 , t 1 qpj 2 , k 2 , t 2 q " pj 1`j2 , k 1`k2 , t 1`t2`j1 k 2 q The generators are z " p1, 0, 0q, x " p0, 1, 0q, and γ " p0, 0, 1q. If we specialize to Z N , the integers modulo N we get corresponding Heisenberg group H N with generators X, Z, and γ and the relations X N " Z N " γ N " e (identity), γX " Xγ, γZ " Zγ and ZX " γXZ (38) Since H N is a finite group its finite-dimensional representations are unitary and completely reducible. Let φ be a representation of H or H N on a vector space V of finite dimension. We say that the central element γ acts maximally if the order of φpγq is dimpV q. The following theorem characterizes representation of H pH N q and their relation to QFT. 2. φ and φ 1 are unitarily equivalent: φ 1 " ΩφΩ : and Ω is the quantum Fourier operator.
3. Any unitary irreducible representation ψ of the full discrete Heisenberg group H in V in which the order of γ, opγq " dim pV q " K is equivalent to an irreducible faithful representation of H K .
Proof. Assume first that γ acts maximally. Since φ is irreducible and γ is in the centre it must act as a constant (Schur's lemma). As order of γ is N , γ " ωI where ω is a primitive N th root of unity. Since H N is finite we may assume the representations to be unitary. Let α be an eigenvector of φpZq with eigenvalue c. As Z N " e, c must be an N th root of 1. Consider the set S " tα, φpXqα, . . . , φpX N´1 qαu. As φpZqφpX k qα " φpγ k qφpX k qφpqα " cω k φpX k qα φpX k qα " φpXq k α, k " 0, 1, . . . , N´1 are eigenvector of φpZq with eigenvalue cω k . These eigenvalues are distinct roots of 1 and hence S is linearly independent and a basis of V . We can reason similarly for φpXq. The converse is trivial. If γ k " I for k ă N then φ cannot be faithful. Next we recall some facts from the theory of characters associated with representation of a group [33]. If ρ is a representations of a finite group G on a finite-dimensional vector space V , the character χ ρ is a scalar function on G defined by χ ρ pgq " Trpρpgqq. It is constant on conjugacy classes. If we have two characters χ ρ and χ ρ 1 corresponding to representations ρ and ρ 1 then their scalar product is defined as pχ ρ , χ ρ 1 q " p1{N q ř gPG χ ρ pgqχ ρ 1 pgq. It is a fundamental result that two irreducible representations ρ and ρ 1 are (unitarily) equivalent if and only if pχ ρ , χ ρ 1 q ‰ 0. We apply this to the representation φ and φ 1 of H N . First, observe that since ZX m Z n Z´1 " γ m X m Z n , χ φ pX m Z n q " ω m χ φ pX m Z n q which is possible iff either m " 0 or χ φ pX m Z n q " 0. Conjugating with X we conclude that χ φ is nonzero only on the centre of H N . Hence, to prove equivalence of φ and φ 1 it suffices to show the scalar product of χ φ and χ φ 1 is non-zero. But φ and φ 1 have the same effect on the center (generated by γ) of H N . Hence pχ ρ , χ ρ 1 q " p1{N q ř k χ φ pγ k qχ φ 1 pγ k q " 1. Since φ and φ 1 are equivalent there exists a unitary map Ω : V Ñ V such that φ 1 pgq " Ω : φpgqΩ. In particular, φ 1 pZq " φpτ Zq " φpXq " Ω : φpZqΩ. Now let t|jy : j " 0, . . . , u be a complete set of eigenvectors of φpZq with φpZq |jy " ω j |jy and similarly let t|jyu be an eigenbasis of φpXq. Then φpZq " ř j ω j |jy xj| and φpXq " ř j ω j |jy xj|. Observing that t|jy xk| : j, k " 0, . . . , N´1u form a basis the space of operators on V it is easy to check that Ω " ř j |jy xj|. We have also seen that φpXq |jy " |j`1y. From these and the normalization xj|0y " 1 we get xj|Ω|ky " ω´j k { ? n. We have proved item 2.
To prove the last assertion we again start with an eigenvector α with eigenvalue a of ψpzq. Note that we can no longer assume that a is an Kth root of 1. However, the hypotheses that order of ψpγq is K implies that α, ψpxqα, . . . , ψpx K´1 qα are eigenvectors of ψpzq with distinct eigenvalues aψpγq j , j " 1, . . . , K´1. They must be then independent. This implies x K α " α and hence x K " 1. Hence the eigenvalues of x must be Kth roots of 1. Interchanging the role of x and z we conclude that a must be a primitive Kth root of 1 and the assertion follows.
Note that the condition on γ (maximal order) is necessary in case of the group H and H N . For example, let N " 3, ρpγq "´I, ρpZq |0y " |0y , ρpZq |1y "´|1y, ρpZq |2y " |2y and ρpXq the cyclic permutation. Then ρ is an irreducible representation of H 9 . We see the connection between representations of the Heisenberg groups and the QFT. For a vector α we writeα " Ωα for its Fourier transform. The Plancherel formula ||α|| 2 " ||α|| 2 is simply stating that the Fourier operator Ω is unitary. We also note that since the representations of the group H is equivalent to H N when γ acts maximally it will be sufficient to consider H N in a fixed representation space. However, when we are dealing with different representations (for example, taking tensor products) we have to deal with the full Heisenberg group. The condition that opγq " dimpV q is special case of general irreducible representations of H. It is sufficient for our purposes and will be implicitly assumed. Henceforth, for a fixed representation ρ we simply write the action of a group element g as gα instead of ρpgqα if the context is clear. Now we turn our attention to distribution functions. We have seen that the distribution functions can be given an alternative characterization in terms of phase-point operators. The formula (30) for these operators implies that they are linear combination of the unitary operators of the group. Thus, we look for them in the group algebra. Recall that for a group G the group algebra CpGq over complex numbers is the set of formal finite linear combinations ř i c i g i , g i P G and c i P C. The algebra product is defined as ÿ Any representation of the group is a representation of the group algebra and vice versa. Now for a unitary representation ρ of G on a finite-dimensional vector space the character χ ρ of the representation induces a scalar product on CpGq. Thus This is indeed a scalar product on CpGq. The resulting norm coincides with the Hilbert-Schmidt norm on the corresponding operators on V . Call an element µ P CpGq self-adjoint if µ˚" µ. Let G " H N or H. Since the central element γ acts as a scalar we write elements of CpGq in the form ř m,n c mn X m Z n . For a representation φ of H N with γ acting as ωI consider the following elements We demand that the set G " tApx, zqu be mutually orthogonal, self-adjoint and satisfy the following: the elements P pxq " ř z Apx, zq and Qpzq " ř x W px, zq are projections, that is, P pxq 2 " P pxq and P pzq 2 " P pzq. We call such a set of elements in CpGq a Wigner set. We have the following theorem. Proof. The proof is similar to that of Theorem 2. We only sketch some of the basic arguments since we are dealing with group algebras. First, the self-adjoint property implies conditions like (19) with c mn in place of f pm, nq since the tX m Z n u are independent in CpH N q. Let us compute the scalar product of two elements from G. Assuming now self-adjointness we have In deriving the second step we use the fact that TrpφpX j Z k qq " 0 unless j " k " 0 mod N . The last expression will be proportional to δ xx 1 δ zz 1 if |c mn | 2 " K, a constant. We will fix K shortly. Hence, we assume that c mn " Kω bmn . For the last requirement we have This would be possible if c 0n " K " 1{N for all n. We have already proved the existence of functions satisfying these conditions in Theorem 2. The fact that W px, z : ρq " TrpApx, zqρq is real follows from self-adjointness. The orthogonality property implies R2 in Section 4.1: Finally, the property about marginals is equivalent to showing that P pzq and P pxq represent projections on |zy and |xy respectively. We can deduce this directly from the fact that φpZq " ÿ j ω j |jy xj| and φpXq " ÿ j ω j |jy xj| The proof of the converse is straightforward.
To prove the last statement let c 1 mn " c mn ω am`bn . Then The assertion follows from this and the proof is complete.
We see the correspondence between orthogonal sets in the group algebra CpH N q and distribution functions. We have seen that the QFT arises out of a particular automorphism τ of the Heisenberg group. So we expect the general automorphisms of H N and H to contain more structure and information relating to QFT and distribution functions. It is easy to see that any two representations of H N in which γ acts maximally and has the same value are equivalent. In particular, if σ is an automorphism of H N that fixes γ and φ is an arbitrary representation then φ and φ¨σ are equivalent. If tApx, zq : 0 ď x, z ď N´1u is a Wigner set then tσApx, zqu is also a Wigner set. So we can generate new Wigner sets by automorphisms. As the value of σ on X and Z determines it on CpH N q let σpXq " X a Z b and σpZq " X c Z d . We must have for σ to be an automorphism. The second condition implies that the matrix M σ "ˆa b c dḣ as determinant 1. That is, M σ P SLp2, Z N q, the set of matrices with entries in the ring Z N and determinant 1 mod N . Conversely, given M P SLp2, Z N q, N odd we can define an automorphism σ M as above. If N is even this simple definition of σ M does not work in general. For example, if ab is odd then . This is reminiscent of half-integral representation of rotation group (SU p2q actually). Hence for even N we have an automorphism of H rather than H N . Note that in this case for any M P SLp2, Z N q, pσ M pgqq 2N " 1, g P H N . There is however a proper subgroup of SLp2, Z N q which induces an automorphism of H N . Alternatively, for even N define the function sgnpuq " # 0, u P Z and u even 1, u odd Now we can define the automorphism σ M on H N , M P SLp2, Z N q for odd N and for even N we define it on the representation space.
Let Apx, zq be a Wigner set and φ a representation of H N . Then we have seen that a distribution function is defined by This means that if φ and φ 1 are equivalent representations connected by a a unitary operator U and W and W 1 are the corresponding distribution functions then W px, z : ρq " W 1 px, z : U : ρU q In particular, if M P SLp2, Z N q then it induces an equivalent representation φ M . In case N is odd φ M is the representation that is given via the automorphism generated by M . In even dimension φ M is defined by (40) above. Now let us look at other "marginals" of a distribution function W px, z : ρq. One way of constructing such marginals is via a finite Radon transform [27,28]. Thus for f : Z NˆZN Ñ C define "lines" S ab " px, zq P Z NˆZN : ax`bz " 0 mod N, gcd pa, b, N q " 1 ( The condition gcd pa, b, N q " 1 ensures that the "line" ax`bz " t has a solution for all t P Z N . The "coordinate axes", for example, correspond to the sets S 10 and S 01 . The function x W pz 1 q in (42) is a Radon transform [28] of the distribution function W px, zq and each pair pa, bq P Z NˆZN such that gcd pa, bq is invertible in Z N , defines such a transform. Let x 1 " ax`bz. Let pc, dq P Z NˆZN such that the matrix SLp2, Z N q and set z 1 " cx`dz. In the following it will be convenient to use vector notation. Thus ξ ξ ξ " px, zq T P Z NˆZN is a two-dimensional "vector" 3 . We will also occasionally use the component notation: ξ ξ ξ " pξ 1 , ξ 2 q T . So the distribution functions and phase-point operators may be written as W pξ ξ ξ : ρq and Apξ ξ ξq respectively. Then a marginal with respect to the second component is given by W pMξ ξ ξ : ρq Here z 1 " Mξ ξ ξ 2 . In analogy with the continuous case we require that x W pz 1 : ρq be a probability distribution with respect to z 1 . More precisely, in the representation φ M´1 of the Heisenberg group corresponding to the automorphism induced by M´1, x W pz 1 ; ρq gives probability distribution of the quantum observ-able´i ln φ M´1 pZq in the state ρ. However, there is a sharp difference between the distribution functions in even and odd dimensions. The general marginal condition holds in odd dimensions for the Wigner distribution function defined by (25) but not for even dimensions. For even dimension we have more complicated formulas for the marginals of the Wigner function. In fact we will show that in this case no distribution function satisfying conditions R1-R4 in subsection 4.1 will satisfy the general marginal condition for all M P SLp2, Z N q.
be a Wigner set and W px, z : ρq " TrpApx, zqρq be the corresponding distribution function.
We say that simple marginal condition (with respect to M ) is satisfied if the marginals are probability distribution in the eigenbasis of the operator φ M pZq " φpσ M pZqq " φpX c Z d q. Then the following statements hold. 3. If the dimension N ą 2 is even it is not possible to satisfy the marginal condition for all M P SLp2, Z N q.

In odd dimension Wigner functions are given by
up to translations. In that case x W pz 1 : ρq " xα z 1`c d{2 |ρ|α z 1`c d{2 y where |α j y are the eigenvectors of σ M pZq.
Proof. Since gcd pa, b, N q " 1 there exist integers c, d and k such that ad´bck N " 1 and hence ad´bc " 1 mod N . For a matrix M let M 1 " pM T q´1. The first assertion in the list is relatively straightforward. If the equation (44) is satisfied then W M px, zq " W pM´1x, M´1zq is a distribution function. The condition R1 (reality) is clear, R2 follows from the state transformation equation (41) and (44) gives the marginal condition R4. From the correspondence between distribution functions and Wigner sets the first assertion is clear.
Suppose the marginal conditions are satisfied for some M P SLp2, Z N q given above. Using the formula we get setting ζ ζ ζ " pm, nq T P Z NˆZN We require that the operator T " ř n gpnqZ 1n be a projection: it must be hermitian and satisfy T 2 " T . Hence, we must have ÿ m,n Since Z 1 like Z has no repeated eigenvalue its minimum polynomial is the characteristic polynomial λ N´1 . Hence the operators I, Z 1 , Z 12 , . . . , Z 1N´1 are linearly independent and we must have ÿ m gpmqgpk´mq " gpkq But the left side is the convolution of the function g with itself. Taking (finite) Fourier transform of both sides we getg 2 "g. Thengpmq " 1 or 0. This implies that gpnq " ÿ i ω tin where t i P Z N are the values at whichg " 1. But from condition R2 we infer that |gpxq| " 1 and since R1 implies that gp0q " 1 we conclude that there must be exactly one term in the above sum: Putting n " 1 this implies that f pc, dq " ω tpc,dq whenever gcdpc, d, N q " 1.
Hence we rewrite the above equation as f pcn, dnq " f pc, dq n ω cdnpn´1q{2 and so f pcn, nq " f pc, 1q n ω cnpn´1q{2 (48) Now suppose N is even. By the definition of Wigner sets they must be independent since the operators are mutually orthogonal. Consequently, the function f must be periodic with period N and since f pc, 1q " ω tpc,1q , tpc, 1q must be an integer. Putting n " N in the second equation in (48) and noting that f px, 0q " 1, @x P Z we get a contradiction when c is odd for the right side iś 1. Hence it is not possible to have Wigner sets satisfying all simple marginal conditions.
Next suppose that N is odd. Then 2 has an inverse pN`1q{2 in Z N . It is an easy verification that the function f pm, nq " ω mnpN`1q{2 satisfies the functional relation (19). To prove uniqueness we assume that tpm, nq can be extended to all Z and that it can be expressed as a polynomial in m and n with integer coefficients (which may depend upon N ). Since f pm, 0q " f pn, 0q " 1 we may assume that the polynomial is of the form tpm, nq " ω mnra0`gpm,nqs where a 0 is a constant and gpm, nq is a polynomial without constant term. Then we have f pcn, nq " ω cn 2 pa0`gpcn,nqq " ω ncpa0`gpc,1qq ω cnpn´1q{2 Since this must be satisfied for all n we must have a 0 " pN`1q{2 and g " 0 mod N . This proves uniqueness up to linear terms.
The last statement is easily derived from the above proof of existence and uniqueness of distribution function satisfying all the marginal conditions for odd N .
We note that a similar relation holds for the marginal distribution over x when we average over the variable z. In fact, satisfaction of marginal conditions under the full SLp2, Z N q for one variable implies the same for other. In even dimensions there exist no distribution function satisfying all marginal conditions. Therefore, we have to relax some of the conditions of the theorem to get the marginal distributions. Let us recall why the marginal conditions are desirable. One of the main reasons is that by determining sufficient number of marginal distributions we can reconstruct the state. The simple marginal condition stated in the Theorem 5 is satisfied (see (44) and the statement that follows it) the marginal distribution corresponds to probabilities for a complete projective measurement in a suitable basis. In even dimension we have three options.
1. We do not require that the Wigner set be independent. Then the representation of the Heisenberg group H N need not be irreducible. This was the approach adopted in [25].
2. We drop the conditions that the marginals be simple type. As will be shown next we can still determine the "marginal" distributions from the measurement probabilities.
3. We do not demand that the marginal condition be satisfied for the full SLp2, Z N q but only for a subset. We show that in case N " 2 K there is such a subset and the marginal distribution for it are sufficient to reconstruct the distribution function.
We start with the first option [25,16]. Since the operators Apx, zq are no longer independent the function f (as a function on Z) is not required to be periodic and the labels px, zq can take any integer values. A minimal extension is obtained by looking at the basic recurrence relations (48). The problematic factor ω cnpn´1q{2 is periodic with a period 2N (as function on Z). The same relations then suggest that we take f pm, nq " ω mn {2 where ω 1{2 is a primitive 2N th root of 1. Hermiticity of phase-point operators then require that we now define them as Because of redundancy these operators are not uniquely determined (up to linear factors). But we can modify the proof in Theorem 5 for odd dimension to determine the possible solutions in this case.
Next we look at option 2. We defined a family of distribution functions say W px, z : ρ, νq in the even case in (25) depending on some function ν. The function ν is arbitrary apart from the condition (26). Let W 0 denote the special case when ν is given by (27). Of course, W 0 does not satisfy the marginal condition but the results below show how it may be computed from the measurement probabilities.
where X x \ is the greatest integer ď x. In particular, for W 0 with ν mn given by (27) we have The proof is given in the appendix. Observe that the if N " 2 k , K ą 1 then N 1 " N {u is always even. We can also write the appropriate formulas for the case N 1 odd. We avoid doing so as they are even more complicated. We can also simplify the trigonometric sums in (49). However, note that if we know the probabilities xj|ρ|jy then in principle the Radon transformŴ pz : ρ, M q can be computed by evaluating these sums. In case of odd dimension the expressions for the marginals are simpler but we still have to estimate the probability distribution in the basis t|α j yu defined above. If we have these the probabilities doing the sums in the even case is routine. So, is there a deeper reason for imposing the marginal conditions on the distribution function? Two possible reasons could be simplicity and some theoretical insight.
We consider the second option listed above for dimension N " 2 k only. Thus we aim to construct a distribution function which satisfies the marginal conditions for only a subset of SLp2, Z N q. The theorem below gives an explicit formula for this important case. Thus let L 1 Ă SLp2, Z N q be a subset consisting of the following matrices. If M P L 1 then each row has at least one entry " 1 and if the diagonal entry is ‰ 1 it is even.

51)
where the expressions like pmn´1q are first computed modulo N in the residue class t0, . . . , N´1u and then treated as an integer. txu denotes the largest integer less than or equal to x. Then W 1 satisfies the conditions R1-R4 and for every M P L 1 , W 1 satisfies a simple marginal condition with respect to the variable x: x W 1 pz 1 : ρ, νq " ÿ , c " 1 and d even Proof. We first note that the notation n´1 makes sense in the ring Z N since every odd n is invertible. The reality condition R 1 is seen from the following simple observation. For 0 ď m, n ă N let mn´1 " k 1 N`n 1 , k 1 ě 0 and 0 ď n 1 ă N . Since pN´mqpN´nq´1 " pmn´1q mod N we get ppN´mqpN´nq´1qpN´nq " pm´1´k 1 qN`n 1 This implies that k 1 has same (opposite) parity as X ppN´mqpN´nq´1qpN´nq \ if m is odd (even). Hence We can argue similarly for m odd and n even. Hence the reality condition (19) is satisfied. It is clear that W 1 is normalized. The other conditions easily follow from the definition and the analysis of these conditions in Section 4.1. Finally, the simple marginal condition with respect to x is seen to be satisfied as follows. From Theorem 5 and Proposition 2 we note that we have to consider pairs of the form pcn, dnq, where pc, dq is the second row of M , in the calculation of the marginals. Using the notation of Proposition 2 we set As the matrices belong to L 1 we consider two cases. If the diagonal element d " 1 then the only terms in the sum yielding W 1 that contribute to the marginal are indexed by ppcnq, nq where n runs through Z N and pcnq is calculated mod N . The case c " 0 is already covered. If c ‰ 0 then from (49) x W 1 pz 1 : ρ, νq " ÿ ν cn,n X cn N \ ω ppc`sgnpcqq{2`j´z 1 qn`ÿ n even ν cn,n ω ppc`sgnpcqq{2`j´z 1 qn" ÿ j ρ jj ÿ n ω ppc`sgnpcqq{2`j´z 1 qn " @ z 1´c`s gnpcq 2ˇˇρˇˇz 1´c`s gnpcq 2

D
For the case c " 1 and d even the terms in which n is odd drop out from the sum for x W 1 pz 1 : ρ, νq and the proof is similar to the first case.
We note that the subset L 1 of matrices from SLp2, Z N q cannot be extended arbitrarily preserving the property of simple marginals. For example, if we admit matrices with c " 1 and d odd then we get a factor of sgnp X pd´1qn The two need not be equal. However, as we will see below the set L 1 is sufficient to determine W 1 .

Inverse Radon transform and state determination
In the previous section we have seen that the finite Wigner distribution function enjoys a rich variety of marginal properties. We can use this to determine the former. This is equivalent to inverting a finite set of Radon transforms. From the distribution function we can determine the state. The invertibility of the Radon transforms also shows that the Wigner distribution function is unique up to a translation. In the rest of the section W px, zq will denote the Wigner distribution function. Replacing the matrix M´1 by The problem is to reconstruct W px, zq from x W pz : ρ, M q. Call the later the Radon transform of W with respect to the matrix M . The idea is that x W pz : ρ, M q is the probability distribution of the observable´i ln pX´cZ a q in the odd case. In case of even dimensions it can be computed from the distributions. Assuming that these distributions can be approximately determined experimentally we can reconstruct W and hence ρ. We have seen that in odd dimension N there is a distribution function satisfying simple marginal conditions for every M P SLp2, Z N q and in dimension N " 2 k we have only a subset of SLp2, Z N q with simple marginal conditions. We give explicit formulas for these two cases. First some notation. For a subset S of some set let χ S denote the indicator function: χ S pxq " 1 if x P S and 0 otherwise. In the rest of the section we use the boldface vector notation to denote a member of Z NˆZN and other non-bold letters to denote "scalars" belonging to Z N . For example, Now we take the finite Fourier transform of the above equation in the group Z NˆZN [22]. Recall that the Fourier transform of a complex function f puq on Z NˆZN byf is a function on the dual group pZ NˆZN q˚: ω´p µ1u1`µ2u2q f puq Using the fact that x W " x W χ Sz and that the Fourier transform of a convolution is a product and vice versa we have (suppressing ρ and M ) where F ptq " Ă W pct,´atq and r F is its Fourier transform in Z N . In proving the above we use the following facts: r χ S 1 1 pMq pνq ‰ 0 iff aν 1`c ν 2 " 0 and the solution to the congruence equation aν 1`c ν 2 " 0 mod N is given by the set tpct,´atq : t P Z N . This follows from a similar result for linear Diophantine equations [34] and the fact that gcdpa, c, N q " 1. We also use det M " ad´bc " 1 in the las but one step. The factor ? N appears because of the normalization used in our definition of FFT. It now follows that This formula is valid for any distribution function. Let now N be odd. Combining this with the equation (44) in Theorem 5 we get (54). Similarly, when N " 2 K and M´1 P L 1 we obtain (55). Note that the formulas in (52) are valid under the assumption that M P L 1 (see the footnote 4 above). We next show that it is always possible to find a, c P Z N such that gcd pa, c, N q " 1 and the "lines" tpct,´atq : t P Z N u cover the "plane" Z NˆZN in the two cases above. When N is odd this is obvious. If N " 2 k consider px, yq P Z NˆZN . For 0 ă j ă N let h j denote the highest power of 2 that divides j, that is, j{2 hj is an odd integer. If h x ě h y then we put a " 1 and c "´2 hx´hy py{2 hy q´1 where the inverse is evaluated in Z N and we assume that y ‰ 0. Then px, yq " pct,´atq for t "´y. If h x ă h y then put c " 1 and a "´2 hy´hx px{h x q´1. We have therefore shown that in all these cases the Radon transforms together can be inverted for from the values x W pµ 1 , µ 2 q so obtained we can take the inverse Fourier transform and the last assertion of the theorem is proved.
We can thus recover any distribution function W px, z : ρq and consequently the state ρ from the Radon transform data which are in turn probability distribution of measurement in appropriate bases (see (53)). The theorem shows the existence of an inverse transform corresponding to the set of Radon transforms of W , each corresponding to an element M in the group SLp2, Z N q. But we do not need all the Radon transforms. What is an optimal subset Q Ă SLp2, Z N q that suffices to determine the state uniquely from probability distributions corresponding to measurements in appropriate bases? This question can only be satisfactorily answered in the context of prior information about the state. One can show that without any such information the cardinality of Q is OpN q. Even then we have a lot of freedom. We can use our choices so as to ensure optimal measurement. Recall from Theorem 5 that the Radon transforms are given by probability distribution (corresponding to a state ρ) in the basis that diagonalizes the unitary operator X c Z d . The only condition imposed on the pair pc, dq P Z NˆZN is that gcd pc, d, N q " 1. We can often compute this basis explicitly. Then we can use quantum circuits to transform our original "computational basis" to the required basis. A criterion for the choice of pc, dq could be those that minimizes the size of the circuit. For example, if N " 6, the choice c " 3, d " 2 leads to a particularly simple basis. The analysis becomes simpler if the dimension N is a prime power. We aim to address these issues in future.

Distribution functions and quantum information
In this section we discuss some potential applications of distribution functions in quantum information processing (QIP). This is a developing area and we only sketch how our formalism may prove useful in various areas in QIP. For this it is best to view the distribution function as coefficients in the expansion of the state in some orthonormal basis in the space of operators, in particular, the basis consisting of phase-point operators. First we generalize to automorphism groups of the group algebra CpH N q: a linear isomorphism T : CpH N q Ñ CpH N q such that T pxyq " T pxqT pyq is bijective. It is sufficient to check the last condition for the generators X, Z and γ. We will consider only those automorphisms for which T pγq " γ. Then T pXq, T pZq and γ generate a group isomorphic to H N provided T pXq N " T pZq N " 1. In particular if c P CpH N q is invertible then the map T pxq " cxc´1 is an automorphisms satisfying these conditions. Such automorphisms are called inner. Further call an inner automorphism unitary if c´1 " c˚(see (39) for the definition of the˚operation). We can prove the following.
Proposition 3. If T pxq " cxc´1 is a unitary inner automorphism and φ is representation of H N then there is a unitary operator U c such that φpT pxqq " U c φpxqU´1 c . Conversely for any unitary operator U on the representation space of H N there is a unitary inner automorphism T U such that U φpxqU´1 " φpT U pxqq. Thus there is a one-to-one correspondence between the set UpH N q of unitary inner automorphisms on CpH N q and quantum dynamics on the representative Hilbert space.
This result is neither difficult nor surprising given the fact that the H N completely characterizes the kinematics of the system. It does however give us an alternative description and algebraic tools to study the dynamics. Thus we can study the effect of unitary operations on distribution functions [16] using these transformations. Note however that we allow reducible representations now. The set of automorphisms of the group H N is a subgroup of UpH N q.
In this work, we have concentrated on irreducible representations of H N in which γ acts maximally. By dropping the last assumption we can get all finite dimensional representations. The order of φpγq in the representation φ is the dimension. We can then use the products of these representations (actually we need some extra structure) to study unitary gates. We aim to explore this in future. Let us note some interesting relations in the case N " 2 n . If u P H N we will denote by φ k the representation in which γ 2 k " 1. Let σ i , i " 1, 2, 3 denote the Pauli matrices and I r the identity matrix of order r. Then where C "¨1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0‹ ‹ ‚ and S "ˆ1 0 0 iȧ re a CNOT gate (C) and the phase gate (S) respectively [35]. We also note that in the general case φ n pXq is the cyclic shift operator. It can be efficiently constructed, for example, using full adder circuits with n`1 ancillary qubits. Similarly, φ n pZq can be constructed using appropriate controlled phase gates as in the quantum Fourier transform. We also observe that iterating the simple relations φ k pX 2 q " φ k´1 pXq b I 2 and φ k pZ 2 q " I 2 b φ k´1 pXq we obtain the interesting relations φ n pX 2 k q " φ n´k pXq b I 2 k and φ n pZ 2 k q " I 2 k b φ n´k pZq (57) These relations can be used to devise more efficient implementations. We conclude this section with a discussion of potential application of these constructions to quantum process tomography [36]. A quantum process is characterized by a completely positive map T acting on the operators on the system Hilbert space. If we have a complete set of phase-point operators tApx, zqu then T is determined by its action on these. Let us assume that the dimension is odd so that we have a set of phase-point operators satisfying the full set of marginal conditions. Using Theorem 5 we can prove the following. Proof of Proposition 2. We will prove the second formula only. The proof is similar for the first formula. Using the induced automorphism given in (40)  ω´p abmpm´1q{2`cdnpn´1q{2`pm`nqpcd`sgnpcdq{2qq xσ M pXq m σ M pZq n yω´n z 1 δ m0 " 1 N ÿ n ω ppc`dqnq 2 N {4 ω pcnqpdnq{2 ω´s gnpcdqn{2´cdnpn´1q{2 xσ M pZq n yω´n z 1 .
Here we use that fact that cn, dn ‰ 0 mod N for any odd n since N {u is even. We have to consider two cases separately suppose first that cd is odd.