Quantum Wigner entropy

• URL: http://arxiv.org/abs/2105.12843v2
• Date: Fri, 31 Dec 2021 02:10:32 GMT
• Title: Quantum Wigner entropy
• Authors: Zacharie Van Herstraeten and Nicolas J. Cerf
• Abstract summary: We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. We conjecture that it is lower bounded by $lnpi +1$ within the convex set of Wigner-positive states. The Wigner entropy is anticipated to be a significant physical quantity, for example, in quantum optics.
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• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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