2024/11/08 by Zacharie Van Herstraeten, Nicolas J. Cerf, Van Herstraeten, Zacharie +1 · 1 citation
Computer Science · Mathematics · #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Quantum Information and Cryptography #Quantum Physics (quant-ph) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2411.05562
openalex publication_date 2024/11/08 · openalex created_date 2024/11/15 · openalex updated_date 2026/07/28
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 under convolution 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-Rényi entropy of beam-splitter states, albeit in a restricted range for the Rényi parameter α≥ 1/2.