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Symmetry

On Families of Wigner Functions for N-Level Quantum Systems

Vahagn Abgaryan, A. Khvedelidze

Published Jun 4, 2021• 11 citations

Abstract

A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary N-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for the spectrum of the Stratonovich–Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich–Weyl kernels for a N=(2j+1)-level system, one can always identify the representative that realizes the so-called SU(2)-symmetric spin-j symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit.

DOI: 10.3390/sym13061013

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A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary N-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for the spectrum of the Stratonovich–Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich–Weyl kernels for a N=(2j+1)-level system, one can always identify the representative that realizes the so-called SU(2)-symmetric spin-j symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit.

For a symmetric

N

-quDit system described by a density matrix

ρ

, we construct a one-parameter

s

family

\mathcal{F}^{(s)}_ρ

of quasi-probability distributions through generalized Fano multipole operators and Stratonovich-Weyl kernels. The corresponding phase space is the complex projective

{C}P^{D-1}=U(D)/U(D-1)\times U(1)

, related to fully symmetric irreducible representations of the unitary group

U(D)

. For the particular cases

D=2

(qubits) and

D=3

(qutrits), we analyze the phase-space structure of Schrödinger

U(D)

-spin cat (parity adapted coherent) states and we provide plots of the corresponding Wigner

\mathcal{F}^{(0)}_ρ

function. We examine the connection between non-classical behavior and the negativity of the Wigner function. We also compute the generalized heat kernel relating two quasi-probability distributions

\mathcal{F}^{(s)}_ρ

and

\mathcal{F}^{(s')}_ρ

, with

t=(s'-s)/2

playing the role of ``time'', together with their twisted Moyal product in terms of a trikernel. In the thermodynamic limit

N\to\infty

, we recover the usual Gaussian smoothing for

s'>s

. A diagramatic interpretation of the phase-space construction in terms of Young tableaux is also provided.

The Weyl correspondence between classical and quantum observables is rigorously formulated for a linear mechanical system with a finite number of degrees of freedom. A multiplication of functions and a *-operation are introduced to make the Hilbert space of Lebesgue square-integrable complex-valued functions on phase space into a H*-algebra. The Weyl correspondence is realized as a *-isomorphism f → W(f) of this H*-algebra onto the H*-algebra of Hilbert-Schmidt operators on the Hilbert space of Lebesgue square-integrable complex-valued functions on configuration space. Moreover, the kernel of W(f) is exhibited in terms of a Fourier-Plancherel transform of f. Elementary properties of the Wigner quasiprobability density function and its characteristic function are deduced and used to obtain these results.

We present a general theory of quasiprobability distributions on phase spaces of quantum systems whose dynamical symmetry groups are (finite-dimensional) Lie groups. The family of distributions on a phase space is postulated to satisfy the Stratonovich - Weyl correspondence with a generalized traciality condition. The corresponding family of the Stratonovich - Weyl kernels is constructed explicitly. In the presented theory we use the concept of generalized coherent states, that brings physical insight into the mathematical formalism.

Abstract The Weyl‐Wigner‐Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. A one to one correspondence between operators in the Hilbert space and functions on the phase space is found. The expressions for the Stratonovich‐Weyl quantiser, star product and Wigner functions of such systems for arbitrary values of spin are obtained in detail. As examples the Landau levels and the corresponding Wigner functions for a spin nonrelativistic particle as well as the magnetic resonance for a spin nonrelativistic uncharged particle are analysed.