Characterizations of von Neumann Entropy and Tsallis p-Entropy on Quantum States

Abstract

Assume that $\mathcal {S}(H)$ is the set of all quantum states of a quantum system associating a separable complex Hilbert space H. Denote by S(ρ) and S_p(ρ) (p > 0 with p≠ 1) respectively the von Neumann entropy and Tsallis p-entropy of ρ. In this paper, a sufficient condition for two quantum states being equal is given by von Neumann entropy and Tsallis p-entropy, and we show that, for $\rho ,\sigma \in \mathcal {S}(H)$, $S(\alpha \rho +(1-\alpha )P)=S(\alpha \sigma +(1-\alpha )P)<\infty$ for all pure states $P\in \mathcal {P}_{1}(H)$ and $S_{p}(\alpha \rho +(1-\alpha )P)=S_{p}(\alpha \sigma +(1-\alpha )P)<\infty$ holds for all pure states $P\in \mathcal {P}_{1}(H)$ respectively implies ρ = σ, where α ∈ (0,1) is any fixed number. In addition, the maps on $\mathcal {S}(H)$ preserving the von Neumann entropy and Tsallis p-entropy of a convex combination are also respectively characterized.