[Paper Review] Geometric approach to the discrete Wigner function

A. B. Klimov, Claudio Muñoz+1 more|ArXiv.org|May 13, 2006

Spectral Theory in Mathematical Physics3 citations

TL;DR

This paper presents a geometric approach to the discrete Wigner function using finite field structures for quantum systems of prime power dimension. By labeling Hilbert space states and operators directly with elements of a Galois field, it derives explicit expressions for the Wigner kernel in both odd- and even-characteristic cases, resolving the algebraic origin of non-uniqueness in Wigner function representations and showing covariant transformations under displacement, rotation, and squeezing operations via unitary operators.

ABSTRACT

We analyze the Wigner function constructed on the basis of the discrete rotation and displacement operators labeled with elements of the underlying finite field. We separately discuss the case of odd and even characteristics and analyze the algebraic origin of the non uniqueness of the representation of the Wigner function. Explicit expressions for the Wigner kernel are given in both cases.

Motivation & Objective

• To develop a geometric formulation of the discrete Wigner function based on finite field structures for quantum systems of dimension $ d = p^n $.
• To resolve the algebraic origin of the non-uniqueness in Wigner function representations by analyzing the role of field basis choices.
• To derive explicit expressions for the Wigner kernel in both odd- and even-characteristic finite fields.
• To establish a consistent mapping between quantum states and phase-space structures using Stratonovich-Weyl postulates and finite group operations.
• To demonstrate how unitary transformations—displacement, rotation, and squeezing—induce covariant transformations on the Wigner function.

Proposed method

• Uses generalized position and momentum operators $ Z_eta $, $ X_eta $ defined over $ GF(d) $, with $ d = p^n $, and labeled by field elements.
• Defines the Wigner function via a Weyl-ordered symbol using the finite Fourier transform $ F $, satisfying $ F X_eta F^ op = Z_eta $, with $ F^4 = I $ for $ p \neq 2 $.
• Introduces rotation operators $ V_ u $, $ U_ u $, and squeezing operators $ S_ u $, constructed from symplectic transformations on the finite phase space.
• Derives the Wigner kernel $ f( heta, heta') $ via consistency conditions from the Stratonovich-Weyl postulates and character properties $ \chi(\alpha) = \exp[2\pi i \cdot \mathrm{tr}(\alpha)/p] $.
• Analyzes the action of $ V_ u $, $ U_ u $, and $ S_ u $ on the Wigner function, showing $ W_{V_ u \rho V_ u^ op}(\alpha,\beta) = W_\rho(\alpha, \beta - \nu\alpha) $ for odd $ p $.
• Uses the trace operation $ \mathrm{tr}(\theta) = \theta + \theta^p + \cdots + \theta^{p^{n-1}} $ to define additive characters and ensure unitarity and closure of the generalized Pauli group.

Experimental results

Research questions

• RQ1 What is the algebraic origin of the non-uniqueness in discrete Wigner function representations for finite-dimensional quantum systems?
• RQ2 How do rotation and displacement operators labeled by finite field elements affect the Wigner function in odd- and even-characteristic systems?
• RQ3 What are the explicit forms of the Wigner kernel $ f(\theta, \theta') $ in both odd- and even-characteristic finite fields?
• RQ4 How do unitary operations such as $ V_\nu $, $ U_\nu $, and $ S_\xi $ induce covariant transformations on the Wigner function?
• RQ5 Can the discrete Wigner function be consistently defined using finite field structures without relying on basis-dependent factorizations?

Key findings

• The Wigner function is constructed via a Weyl-ordered symbol using finite Fourier transform, ensuring consistency with Stratonovich-Weyl postulates.
• For odd characteristic $ p \neq 2 $, the Wigner function transforms covariantly under $ V_\nu $: $ W_{V_\nu \rho V_\nu^ op}(\alpha,\beta) = W_\rho(\alpha, \beta - \nu\alpha) $.
• For odd characteristic, the $ U_\nu $ operator induces $ W_{U_\nu \rho U_\nu^ op}(\alpha,\beta) = W_\rho(\alpha - \nu\beta, \beta) $, showing dual covariance.
• The squeezing operator $ S_\xi $ acts as $ W_{S_\xi \rho S_\xi^ op}(\alpha,\beta) = W_\rho(\xi\alpha, \xi^{-1}\beta) $, confirming symplectic invariance.
• The Wigner kernel $ f(\theta, \theta') $ is uniquely determined by consistency conditions and field trace properties, with explicit values such as $ f(\theta, \theta^2) = 0 $, $ f(\theta, \theta^3) = \theta^2 $, etc.
• The construction avoids basis-dependent factorizations by directly labeling Hilbert space states and operators with elements of $ GF(d) $, resolving ambiguity in phase-space labeling.