Axiomatization of Rényi Entropy on Quantum Phase Space
Adam Brandenburger, Pierfrancesco La Mura
TL;DR
The paper addresses the problem that standard entropies become ill-defined for signed phase-space representations in quantum mechanics. It axiomatically derives a unique one-parameter family, the signed Rényi entropy H_alpha, defined for signed measures, which retains real-valuedness and operational meaning. Key results include: a cancellation witness proving sensitivity to negative and positive mass, Schur-concavity for alpha>1, a quantum H-theorem under dephasing dynamics, and conservation of H2 under discrete Moyal-bracket (unitary) evolution. It also discusses a renormalized alpha=1 form and an interpretation of alpha as an inverse temperature, highlighting potential applications in quantum information and thermodynamics on phase space. Overall, the work provides a principled entropy construct tailored to signed phase-space descriptions of quantum systems.
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.