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arXiv:2410.15976

Axiomatization of Rényi Entropy on Quantum Phase Space

Adam M. Brandenburger, Pierfrancesco La Mura

Published Oct 21, 2024• 0 citations
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Abstract

Phase-space versions of quantum mechanics -- from Wigner's original distribution to modern discrete-qudit constructions -- represent some states with negative quasi-probabilities. Conventional Shannon and Rényi entropies become complex-valued in this setting and lose their operational meaning. Building on the axiomatic treatments of Rényi (1961) and Daróczy (1963), we develop a conservative extension that applies to signed finite phase spaces and identify a single admissible entropy family, which we call signed Rényi $α$-entropy (for a free parameter $α\ge 0$). The obvious signed Shannon candidate is ruled out because it violates extensivity. We prove four results that bolster the usefulness of the new measure. (i) It serves as a witness of the presence of cancellation, detecting the coexistence of positive and negative weight in a signed measure. (ii) For $α> 1$, it is Schur-concave, delivering the intuitive property that mixing increases entropy (iii) The same parametric family obeys a quantum H-theorem, namely, that under de-phasing dynamics entropy cannot decrease. (iv) The $2$-entropy is conserved under discrete Moyal-bracket dynamics, mirroring conservation of von Neumann entropy under unitary evolution on Hilbert space. We also comment on interpreting the Rényi order parameter as an inverse temperature. Overall, we believe that our investigation provides good evidence that our axiomatically derived signed Rényi entropy may be a useful addition to existing entropy measures employed in quantum information, foundations, and thermodynamics.

DOI
10.48550/arxiv.2410.15976

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Phase-space versions of quantum mechanics -- from Wigner's original distribution to modern discrete-qudit constructions -- represent some states with negative quasi-probabilities. Conventional Shannon and Rényi entropies become complex-valued in this setting and lose their operational meaning. Building on the axiomatic treatments of Rényi (1961) and Daróczy (1963), we develop a conservative extension that applies to signed finite phase spaces and identify a single admissible entropy family, which we call signed Rényi ααα-entropy (for a free parameter α≥0α\ge 0α≥0). The obvious signed Shannon candidate is ruled out because it violates extensivity. We prove four results that bolster the usefulness of the new measure. (i) It serves as a witness of the presence of cancellation, detecting the coexistence of positive and negative weight in a signed measure. (ii) For α>1α> 1α>1, it is Schur-concave, delivering the intuitive property that mixing increases entropy (iii) The same parametric family obeys a quantum H-theorem, namely, that under de-phasing dynamics entropy cannot decrease. (iv) The 222-entropy is conserved under discrete Moyal-bracket dynamics, mirroring conservation of von Neumann entropy under unitary evolution on Hilbert space. We also comment on interpreting the Rényi order parameter as an inverse temperature. Overall, we believe that our investigation provides good evidence that our axiomatically derived signed Rényi entropy may be a useful addition to existing entropy measures employed in quantum information, foundations, and thermodynamics.
The R\'enyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of R\'enyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of R\'enyi entropies and the corresponding relative entropies that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities and a limited duality relation. We conclude that the treatment of these entropies is technically challenging and requires sophisticated tools from linear algebra. We share several conjectured properties which we were unable to prove.
The Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.
Abstract It is common knowledge that the Wigner function of a quantum state may admit negative values, so that it cannot be viewed as a genuine probability density. Here, we examine the difficulty in finding an entropy-like functional in phase space that extends to negative Wigner functions and then advocate the merits of defining a complex-valued entropy associated with any Wigner function. This quantity, which we call thecomplex Wigner entropy, is defined via the analytic continuation of Shannon’s differential entropy of the Wigner function in the complex plane. We show that the complex Wigner entropy enjoys interesting properties, especially its real and imaginary parts are both invariant under Gaussian unitaries (displacements, rotations, and squeezing in phase space). Its real part is physically relevant when considering the evolution of the Wigner function under a Gaussian convolution, while its imaginary part is simply proportional to the negative volume of the Wigner function. Finally, we define the complex-valued Fisher information of any Wigner function, which is linked (via an extended de Bruijn’s identity) to the time derivative of the complex Wigner entropy when the state undergoes Gaussian additive noise. Overall, it is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.
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 .