Negativity of Wigner distribution function as a measure of incompatibility

TL;DR

This paper explores the relationship between the negativity of the Wigner distribution function and measurement incompatibility in quantum systems, showing how negativity varies with noise and system dimension.

Contribution

It establishes a connection between Wigner function negativity and incompatibility, extending analysis from qubits to higher-dimensional qudits.

Findings

Negativity increases as noise decreases, indicating higher incompatibility.

Maximum negativity occurs for maximally unbiased operators.

Negativity and incompatibility decrease with increasing system dimension.

Abstract

Measurement incompatibility and the negativity of quasiprobability distribution functions are well-known non-classical aspects of quantum systems. Both of them are widely accepted resources in quantum information processing. We acquaint an approach to establish a connection between the negativity of the Wigner function, a well-known phase-space quasiprobability distribution, of finite-dimensional Hermitian operators and incompatibility among them. We calculate the negativity of the Wigner distribution function for noisy eigenprojectors of qubit Pauli operators as a function of the noise and observe that the amount of negativity increases with the decrease in noise vis-\`a-vis the increase in the incompatibility. It becomes maximum for the set of maximally unbiased operators. Our results, although qualitatively, provide a direct comparison between relative degrees of incompatibility among a set of operators for different amounts of noise. We generalize our treatment for higher dimensional qudits for specific finite-dimensional Gell-Mann operators to observe that with an increase in the dimension of the operators, the negativity of their Wigner distribution, and hence incompatibility, decreases.