Wigner entropy conjecture and the interference formula in quantum phase space

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 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 $\alpha \ge 1/2$.

v2 updated to match published version

References in corpus (18)

• Gaussian Quantum Information
• Negativity of the Wigner function as an indicator of nonclassicality
• Positive Wigner functions render classical simulation of quantum computation efficient
• Resource theory of quantum non-Gaussianity and Wigner negativity
• Measuring Gaussian quantum information and correlations using the Renyi entropy of order 2
• Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture
• Minimum output entropy of bosonic channels: a conjecture
• The Wigner Entropy Production Rate
• Experimental determination of irreversible entropy production in out-of-equilibrium mesoscopic quantum systems
• Proof of an entropy conjecture for Bloch coherent spin states and its generalizations
• Extending Hudson's theorem to mixed quantum states
• Measuring nonclassicality of bosonic field quantum states via operator ordering sensitivity
• Witnessing Wigner Negativity
• Quantum Wigner entropy
• Entropy-power uncertainty relations : towards a tight inequality for all Gaussian pure states
• Continuous majorization in quantum phase space
• Complex-valued Wigner entropy of a quantum state
• Wigner non-negative states that verify the Wigner entropy conjecture

Cited by in corpus (1)

• Upper bounds on the purity of Wigner positive quantum states that verify the Wigner entropy conjecture