Quantum Wigner entropy

TL;DR

This paper introduces the Wigner entropy as a measure of quantum uncertainty for Wigner-positive states, proves a conjecture about its lower bound for certain states, and explores its implications in quantum optics and thermodynamics.

Contribution

It defines the Wigner entropy for quantum states, proves a lower bound conjecture for Wigner-positive states, and develops methods to construct and analyze such states.

Findings

Wigner entropy is invariant under symplectic transformations.

The lower bound of Wigner entropy is conjectured and proved for passive states.

A technique to generate Wigner-positive states using beam splitters is presented.

Abstract

We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by $\ln\pi +1$ within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural minimum-uncertainty states. This conjecture bears a resemblance with the Wehrl-Lieb conjecture, and we prove it over the subset of passive states of the harmonic oscillator which are of particular relevance in quantum thermodynamics. Along the way, we present a simple technique to build a broad class of Wigner-positive states exploiting an optical beam splitter and reveal an unexpectedly simple convex decomposition of extremal passive states. The Wigner entropy is anticipated to be a significant physical quantity, for example, in quantum optics where it allows us to establish a Wigner entropy-power inequality. It also opens a way towards stronger entropic uncertainty relations. Finally, we define the Wigner-R\'enyi entropy of Wigner-positive states and conjecture an extended lower bound that is reached for Gaussian pure states.