保持量子态凸组合的Tsallis的映射
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摘要
设H_m是维数为m的复希尔伯特空间,S(H_m?H_n)为复双体希尔伯特空间H_m?H_n上的量子态的全体,S_(sep)(H_m?H_n)为其中可分量子态构成的凸集.映射φ:S(H_m?H_n)→S(H_m?H_n)是满射,且φ(S_(sep)(H_m?H_n))=S_(sep)(H_m?H_n).若对于某个r∈R~+\1},满射φ保持量子态凸组合的Tsallis熵S~r(tρ+(1-t)σ)=S~r(tφ(ρ)+(1-t)φ(σ))对于任意的ρ、σ∈S(H_m?H_n)和任意的t∈[0,1]成立;那么在H_m、H_n上分别存在酉算子U_m、V_n,使得φ(ρ)=(U_m?V_n)ρ(U_m?V_n)~*对于任意的ρ∈S_(sep)(H_m?H_n)成立.
Let H_m be the complex Hilbert space with dimension m, S(H_m?H_n) be all the quantum states acting on complex bipartite Hilbert space H_m?H_n and S_(sep)(H_m?H_n) be the convex set of comparable quantum states.■be a surjective map and■For some r∈R~+\1}, if φ satisfies Tsallis entropy■for any ρ,σ∈S(H_m?H_n) and for any t∈[0,1], there exist unitary operators U_m, V_n acting on H_m, H_n such thatφ(ρ)=(U_m?V_n)ρ(U_m?V_n)~* for anyρ∈S_(sep)(H_m?H_n).
Let H_m be the complex Hilbert space with dimension m, S(H_m?H_n) be all the quantum states acting on complex bipartite Hilbert space H_m?H_n and S_(sep)(H_m?H_n) be the convex set of comparable quantum states.■be a surjective map and■For some r∈R~+\1}, if φ satisfies Tsallis entropy■for any ρ,σ∈S(H_m?H_n) and for any t∈[0,1], there exist unitary operators U_m, V_n acting on H_m, H_n such thatφ(ρ)=(U_m?V_n)ρ(U_m?V_n)~* for anyρ∈S_(sep)(H_m?H_n).
引文
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[11] FANG Xiaochun,LAO Yihui.Preservers for the Tsallis entropy of convex combinations of density operator[J].Advances in Mathematical Physics,2018.DOI:10.1155/2018/5296085.
[2] FOSNER A,HUANG Z J,LI C K,et al.Linear maps preserving Ky fan norms and Schatten norms of tensor products of matrices[J].SIAM Journal on Matrix Analysis and Applications,2013,34(2):673.
[3] FRIEDLAND S,LI C K,POON Y T,et al.The automorphism group of separable states in quantum information theory[J].Journal of Mathematical Physics,2011,54(2):042203.
[4] HOU J C,QI X F.Linear maps preserving separability of pure states[J].Linear Algebra and its Applications,2013,439(5):1245.
[5] HE K,YUAN Q,HOU J C.Entropy-preserving maps on quantum states[J].Linear Algebra and its Applications,2015,467(15):243.
[6] KARDER M,PETEK T.Maps on states preserving generalized entropy of convex combinations[J].Linear Algebra and its Applications,2017,532(1):86.
[7] HOU Jinchuan,LIU Liang.Quantum measurement and maps preserving convex combinations of separable states[J].Journal of Physics A:Mathematical and Theoretical,2012,45(20):205305.
[8] YANG Lihua,HOU Jinchuan.Quantum measurements and maps preserving strict convex combinations and pure states[J].Journal of Physics A:Mathematical and Theoretical,2013,46(19):195304.
[9] RAGGIO G A.Properties of q-entropies[J].Journal of Mathematical Physics,1995,36(9):4785.
[10] TSALLIS C.Possible generalization of boltzmann-gibbs statistics[J].Journal of Statistical Physics,1988,52(1/2):479.
[11] FANG Xiaochun,LAO Yihui.Preservers for the Tsallis entropy of convex combinations of density operator[J].Advances in Mathematical Physics,2018.DOI:10.1155/2018/5296085.