保持量子态凸组合的Tsallis的映射

Maps on Quantum States Preserving Tsallis Entropy of Convex Combinations

摘要:

设Hm是维数为m的复希尔伯特空间，S(Hm?Hn)为复双体希尔伯特空间Hm?Hn上的量子态的全体，Ssep(Hm?Hn)为其中可分量子态构成的凸集.映射φ:S(Hm?Hn)→S(Hm?Hn)是满射，且φ(Ssep(Hm?Hn))=Ssep(Hm?Hn)． 若对于某个r∈R+\\{1}，满射φ保持量子态凸组合的Tsallis熵Sr(tρ+(1－t)σ)=Sr(tφ(ρ)+(1－t)φ(σ))对于任意的ρ、σ∈S(Hm?Hn)和任意的t∈［0,1］成立;那么在Hm、Hn上分别存在酉算子Um、Vn,使得φ(ρ)=(Um?Vn)ρ(Um?Vn)对于任意的ρ∈Ssep(Hm?Hn)成立.

Abstract:

Let Hm be the complex Hilbert space with dimension m, S(Hm?Hn) be all the quantum states acting on complex bipartite Hilbert space Hm?Hn and Ssep(Hm?Hn) be the convex set of comparable quantum states.φ:S(Hm?Hn)→S(Hm?Hn)be a surjective map andφ(Ssep(Hm?Hn)=Ssep(Hm?Hn)).For some r∈R+\\{1}, if φ satisfies Tsallis entropySr(tρ+(1－t)σ)=Sr(tφ(ρ)+(1－t)φ(σ))for any ρ,σ∈S(Hm?Hn) and for any t∈［0,1］, there exist unitary operators Um, Vn acting on Hm, Hn such that φ(ρ)=(Um?Vn)ρ(Um?Vn) for any ρ∈Ssep(Hm?Hn)．