Axiomatization of R\'enyi Entropy on Quantum Phase Space

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 R\'enyi entropies become complex-valued in this setting and lose their operational meaning. Building on the axiomatic treatments of R\'enyi (1961) and Dar\'oczy (1963), we develop a conservative extension that applies to signed finite phase spaces and identify a single admissible entropy family, which we call signed R\'enyi $\alpha$-entropy (for a free parameter $\alpha \ge 0$). 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 $\alpha > 1$, 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 R\'enyi order parameter as an inverse temperature. Overall, we believe that our investigation provides good evidence that our axiomatically derived signed R\'enyi entropy may be a useful addition to existing entropy measures employed in quantum information, foundations, and thermodynamics.