When the state of a quantum system belongs to a N-dimensional Hilbert space, with N the power of a prime number, it is possible to associate to the system a finite field (Galois field) with N elements. In this paper, we introduce generalized Bell states that can be intrinsically expressed in terms of the field operations.These Bell states are in one to one correspondence with the N^2 elements of the generalised Pauli group or Heisenberg-Weyl group. This group consists of discrete displacement operators and provides a discrete realisation of the Weyl function.Thanks to the properties of generalised Bell states and of quadratic extensions of finite fields, we derive a particular solution for the Mean King's problem. This solution is in turn shown to be in one to one correspondence with a set of N^2 self-adjoint operators that provides a discrete realisation of the Wigner quasi-distribution.When the state of a quantum system belongs to a N-dimensional Hilbert space, with N the power of a prime number, it is possible to associate to the system a finite field (Galois field) with N elements. In this paper, we introduce generalized Bell states that can be intrinsically expressed in terms of the field operations.These Bell states are in one to one correspondence with the N^2 elements of the generalised Pauli group or Heisenberg-Weyl group. This group consists of discrete displacement operators and provides a discrete realisation of the Weyl function.Thanks to the properties of generalised Bell states and of quadratic extensions of finite fields, we derive a particular solution for the Mean King's problem. This solution is in turn shown to be in one to one correspondence with a set of N^2 self-adjoint operators that provides a discrete realisation of the Wigner quasi-distribution.

arXiv:quant-ph/0401037Jan 8, 20044 citations

The Heisenberg-Weyl-parity group its coherent states and a unified Wigner-Weyl function

A. Vourdas

The Heisenberg-Weyl group $HW(d)$ related to a $d$ -dimensional Hilbert space $H(d)$ , is enlarged into the Heisenberg-Weyl-parity group $HWP(d)$ that incorporates parity transformations. It consists of $2d^3$ elements, of which $d^3$ elements belong to the $HW(d)$ subgroup, and extra $d^3$ elements which are related through a Fourier transform with the former ones. It is shown that $HWP(d)$ is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. $HWP(d)$ is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space. $2d^2$ coherent states related to the $HWP(d)$ group are introduced, which consist of $d^2$ coherent states related to the $HW(d)$ subgroup, and extra $d^2$ coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the $2d^2$ coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the $d^2$ coherent states related to $HW(d)$ . One of the consequences of the $HWP(d)$ group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.The Heisenberg-Weyl group $HW(d)$ related to a $d$-dimensional Hilbert space $H(d)$, is enlarged into the Heisenberg-Weyl-parity group $HWP(d)$ that incorporates parity transformations. It consists of $2d^3$ elements, of which $d^3$ elements belong to the $HW(d)$ subgroup, and extra $d^3$ elements which are related through a Fourier transform with the former ones. It is shown that $HWP(d)$ is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. $HWP(d)$ is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.$2d^2$ coherent states related to the $HWP(d)$ group are introduced, which consist of $d^2$ coherent states related to the $HW(d)$ subgroup, and extra $d^2$ coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the $2d^2$ coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the $d^2$ coherent states related to $HW(d)$. One of the consequences of the $HWP(d)$ group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.

quant-pharXiv:2605.14820May 15, 20260 citations

The Discrete Phase Space For 3-Qubit And 2-Qutrit Systems Based On Galois Field

Mazlinda Zainy

Generally, quantum states are abstract states that carry probabilistic information of position and momentum of any dynamical physical quantity in quantum system. E.P.Wigner (1932) had introduced a function that can determine the combination of position and momentum simultaneously, and it was the starting point to define a phase space probability distribution for a quantum mechanical system using density matrix formalism. This function named as Wigner Function. Recently, Wootters (1987) has developed a discrete phase space analogous to Wigner’s ideas. The space is based on Galois field or finite field. The geometry of the space is represented by N ´ N point, where N denoted the number of elements in the field and it must be a prime or a power of a prime numbers. In this work, we study the simplest way to compute the binary operations in finite field in order to form such a discrete space. We developed a program using Mathematica software to solve the binary operation in the finite field for the case of 3-qubit and 2-qutrit systems. The program developed should also be extendible for the higher number of qubit and qutrit. Each state is defined by a line aq + bp = c and parallel lines give equivalent states. The results show that, there are 9 set of parallel lines for the 3-qubit system and 10 sets of parallel lines for 2-qutrit system. These complete set of parallel lines called a ‘striation’.Generally, quantum states are abstract states that carry probabilistic information of position and momentum of any dynamical physical quantity in quantum system. E.P.Wigner (1932) had introduced a function that can determine the combination of position and momentum simultaneously, and it was the starting point to define a phase space probability distribution for a quantum mechanical system using density matrix formalism. This function named as Wigner Function. Recently, Wootters (1987) has developed a discrete phase space analogous to Wigner’s ideas. The space is based on Galois field or finite field. The geometry of the space is represented by N ´ N point, where N denoted the number of elements in the field and it must be a prime or a power of a prime numbers. In this work, we study the simplest way to compute the binary operations in finite field in order to form such a discrete space. We developed a program using Mathematica software to solve the binary operation in the finite field for the case of 3-qubit and 2-qutrit systems. The program developed should also be extendible for the higher number of qubit and qutrit. Each state is defined by a line aq + bp = c and parallel lines give equivalent states. The results show that, there are 9 set of parallel lines for the 3-qubit system and 10 sets of parallel lines for 2-qutrit system. These complete set of parallel lines called a ‘striation’.

Nov 1, 20090 citations

On discrete Wigner transforms

Zhenning Cai, Jianfeng Lu, Kevin D. Stubbs

In this work, we derive a discrete analog of the Wigner transform over the space $(\mathbb{C}^p)^{\otimes N}$ for any prime $p$ and any positive integer $N$ . We show that the Wigner transform over this space can be constructed as the inverse Fourier transform of the standard Pauli matrices for $p=2$ or more generally of the Heisenberg-Weyl group elements for $p > 2$ . We connect our work to a previous construction by Wootters of a discrete Wigner transform by showing that for all $p$ , Wootters' construction corresponds to taking the inverse symplectic Fourier transform instead of the inverse Fourier transform. Finally, we discuss some implications of these results for the numerical simulation of many-body quantum spin systems.In this work, we derive a discrete analog of the Wigner transform over the space $(\mathbb{C}^p)^{\otimes N}$ for any prime $p$ and any positive integer $N$. We show that the Wigner transform over this space can be constructed as the inverse Fourier transform of the standard Pauli matrices for $p=2$ or more generally of the Heisenberg-Weyl group elements for $p > 2$. We connect our work to a previous construction by Wootters of a discrete Wigner transform by showing that for all $p$, Wootters' construction corresponds to taking the inverse symplectic Fourier transform instead of the inverse Fourier transform. Finally, we discuss some implications of these results for the numerical simulation of many-body quantum spin systems.

arXiv:1802.05834Feb 16, 20182 citations

The Heisenberg–Weyl-parity group its coherent states and a unified Wigner–Weyl function

Apostol Vourdas

The Heisenberg-Weyl group H W ( d ) related to a d -dimensional Hilbert space H ( d ) , is enlarged into the Heisenberg-Weyl-parity group H W P ( d ) that incorporates parity transformations. It consists of 2 d 3 elements, of which d 3 elements belong to the H W ( d ) subgroup, and extra d 3 elements which are related through a Fourier transform with the former ones. It is shown that H W P ( d ) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. H W P ( d ) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space. 2 d 2 coherent states related to the H W P ( d ) group are introduced, which consist of d 2 coherent states related to the H W ( d ) subgroup, and extra d 2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2 d 2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d 2 coherent states related to H W ( d ) . One of the consequences of the H W P ( d ) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner–Weyl function are discussed.The Heisenberg-Weyl group H W ( d ) related to a d -dimensional Hilbert space H ( d ) , is enlarged into the Heisenberg-Weyl-parity group H W P ( d ) that incorporates parity transformations. It consists of 2 d 3 elements, of which d 3 elements belong to the H W ( d ) subgroup, and extra d 3 elements which are related through a Fourier transform with the former ones. It is shown that H W P ( d ) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. H W P ( d ) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space. 2 d 2 coherent states related to the H W P ( d ) group are introduced, which consist of d 2 coherent states related to the H W ( d ) subgroup, and extra d 2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2 d 2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d 2 coherent states related to H W ( d ) . One of the consequences of the H W P ( d ) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner–Weyl function are discussed.

Annals of PhysicsMay 13, 20261 citation

Harmonic oscillators coupled by springs: Discrete solutions as a Wigner quantum system

Stijn F. Lievens, Nedialka I. Stoilova, Joris Van Der Jeugt

We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1∣M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1∣M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1∣M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1∣M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.

arXiv:hep-th/0606192Journal of Mathematical PhysicsNov 1, 200619 citations

The Heisenberg-Weyl group

HW(d)

related to a

d

-dimensional Hilbert space

H(d)

, is enlarged into the Heisenberg-Weyl-parity group

HWP(d)

that incorporates parity transformations. It consists of

2d^3

elements, of which

d^3

elements belong to the

HW(d)

subgroup, and extra

d^3

elements which are related through a Fourier transform with the former ones. It is shown that

HWP(d)

is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed.

HWP(d)

is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.

2d^2

coherent states related to the

HWP(d)

group are introduced, which consist of

d^2

coherent states related to the

HW(d)

subgroup, and extra

d^2

coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the

2d^2

coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the

d^2

coherent states related to

HW(d)

. One of the consequences of the

HWP(d)

group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.

In this work, we derive a discrete analog of the Wigner transform over the space

(\mathbb{C}^p)^{\otimes N}

for any prime

p

and any positive integer

N

. We show that the Wigner transform over this space can be constructed as the inverse Fourier transform of the standard Pauli matrices for

p=2

or more generally of the Heisenberg-Weyl group elements for

p > 2

. We connect our work to a previous construction by Wootters of a discrete Wigner transform by showing that for all

p

, Wootters' construction corresponds to taking the inverse symplectic Fourier transform instead of the inverse Fourier transform. Finally, we discuss some implications of these results for the numerical simulation of many-body quantum spin systems.