[Paper Review] On the family of Wigner functions for N-level quantum system

A. Khvedelidze, Vahagn Abgaryan|arXiv (Cornell University)|Aug 20, 2017

Quantum Mechanics and Applications3 citations

TL;DR

This paper proposes a general method for constructing unitary non-equivalent Wigner quasiprobability distributions for N-level quantum systems using the Stratonovich-Weyl correspondence. By deriving an algebraic 'master equation' for the Stratonovich-Weyl kernel, the authors show that the Wigner functions depend on N−2 real parameters, leading to a family of phase-space representations indexed by the kernel's degeneracy structure on complex flag manifolds, with explicit constructions for qubits and qutrits.

ABSTRACT

The family of unitary non-equivalent Weyl-Stratonovich kernels determining the Wigner probability distribution function of an arbitrary N-level quantum system is constructed.

Motivation & Objective

• To address the non-uniqueness of Wigner quasiprobability distributions in finite-dimensional quantum systems.
• To establish a general framework for constructing Wigner functions using the Stratonovich-Weyl correspondence for arbitrary N-level systems.
• To analyze the algebraic structure of the Stratonovich-Weyl kernel and its dependence on N−2 real parameters.
• To demonstrate the method through explicit constructions for qubits (N=2) and qutrits (N=3).
• To connect the general formalism to known SU(2)-symmetric spin-j Wigner functions via reduction from flag manifolds.

Proposed method

• Derives an algebraic 'master equation' governing the Stratonovich-Weyl operator kernel Δ(Ω_N|ν) for N-level systems.
• Identifies the kernel's eigenvalue degeneracy structure, which determines the isotropy group H and the underlying complex flag manifold Ω_N = U(N)/H.
• Parametrizes the family of Wigner functions using N−2 real parameters ν = (ν₁,…,ν_{N−2}), leading to unitary non-equivalent representations.
• Constructs the Wigner function as W^{(ν)}_ρ(ϑ) = tr[ρ Δ(Ω_N|ν)], with Δ expressed via unitary transformations and a diagonal matrix of eigenvalues.
• Utilizes the Euler decomposition of SU(N) to express the kernel in terms of angles and adjoint representations, particularly for SU(3).
• Reduces the general flag manifold construction to 2D submanifolds to recover standard spin-1/2 and spin-1 Wigner functions.

Experimental results

Research questions

• RQ1 How can a family of unitary non-equivalent Wigner functions be systematically constructed for an N-level quantum system?
• RQ2 What algebraic constraints define the Stratonovich-Weyl kernel, and how does its solution space encode the ambiguity in the Wigner mapping?
• RQ3 How does the degeneracy structure of the kernel’s eigenvalues determine the geometry of the phase space manifold?
• RQ4 Can the standard Wigner functions for spin-1/2 and spin-1 systems be derived as special cases of this general framework?
• RQ5 What is the role of the N−2 real parameters ν in labeling distinct, physically inequivalent Wigner representations?

Key findings

• The Stratonovich-Weyl kernel Δ(Ω_N|ν) satisfies a set of algebraic equations that admit a solution space parameterized by N−2 real parameters ν₁,…,ν_{N−2}.
• The kernel’s eigenvalue degeneracy structure defines a complex flag manifold Ω_N = U(N)/H, where H = U(k(π₁)) × ⋯ × U(k(π_r)), with the partition (d₁,…,d_r) summing to N.
• For N=2 (qubit), the method reproduces the standard Wigner function on the 2-sphere, with the kernel uniquely determined up to unitary equivalence.
• For N=3 (qutrit), the full family of Wigner functions is constructed using the SU(3) adjoint representation, with explicit expressions for the 8-dimensional vector n^(8) in terms of Euler angles.
• The reduced Wigner function for spin-1/2 and spin-1 systems is recovered by restricting the flag manifold to 2D submanifolds, confirming consistency with known SU(2) symmetric constructions.
• The method establishes a complete correspondence between density operators and phase-space functions via the trace formula W^{(ν)}_ρ(ϑ) = tr[ρ Δ(Ω_N|ν)], with Δ(Ω_N|ν) unitary non-equivalent for different ν.