Wigner entropy conjecture and the interference formula in quantum phase space

• URL: http://arxiv.org/abs/2411.05562v1
• Date: Fri, 08 Nov 2024 13:37:05 GMT
• Title: Wigner entropy conjecture and the interference formula in quantum phase space
• Authors: Zacharie Van Herstraeten, Nicolas J. Cerf,
• Abstract summary: Wigner-positive quantum states have the peculiarity to admit a Wigner function that is a genuine probability distribution over phase space. We prove that this Wigner entropy conjecture holds true for a broad class of Wigner-positive states known as beam-splitter states.
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• Abstract: Wigner-positive quantum states have the peculiarity to admit a Wigner function that is a genuine probability distribution over phase space. The Shannon differential entropy of the Wigner function of such states - called Wigner entropy for brevity - emerges as a fundamental information-theoretic measure in phase space and is subject to a conjectured lower bound, reflecting the uncertainty principle. In this work, we prove that this Wigner entropy conjecture holds true for a broad class of Wigner-positive states known as beam-splitter states, which are obtained by evolving a separable state through a balanced beam splitter and then discarding one mode. Our proof relies on known bounds on the $p$-norms of cross-Wigner functions and on the interference formula, which relates the convolution of Wigner functions to the squared modulus of a cross-Wigner function. Originally discussed in the context of signal analysis, the interference formula is not commonly used in quantum optics although it unveils a strong symmetry exhibited by Wigner functions of pure states. We provide here a simple proof of the formula and highlight some of its implications. Finally, we prove an extended conjecture on the Wigner-R\'enyi entropy of beam-splitter states, albeit in a restricted range for the R\'enyi parameter $\alpha \geq 1/2$.

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