Axiomatization of Rényi 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 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.

💡 Research Summary

The paper tackles a long‑standing problem in phase‑space formulations of quantum mechanics: how to define a real‑valued, operationally meaningful entropy when the underlying quasiprobability distribution can take negative values. Conventional Shannon and Rényi entropies become complex because the logarithm of a negative number is undefined in the real domain, and the naïve signed‑Shannon entropy fails basic physical requirements such as extensivity.

Building on the classic axiomatizations of Rényi (1961) and Daróczy (1963), the authors modify the axiom set to accommodate signed finite measures. The modified axioms are: (0) real‑valuedness, (2′) continuity for all non‑zero real components, (3) calibration H(½)=1, (4) extensivity under the direct product of measures, and (5′) a mean‑value property where the weights are the absolute values of the total masses. These changes preserve the spirit of the original axioms while ensuring that cancellation between positive and negative contributions is treated correctly.

The central result (Theorem 1) shows that the only entropy functional satisfying the revised axioms is the “signed Rényi α‑entropy”

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