量子信息中的一些迹不等式  

Some Trace Inequalities of Quantum Information

作　　者：刘波[1] 

机构地区：[1]陕西教育学院数理工程系,陕西西安710061

出　　处：《陕西教育学院学报》2010年第1期98-101,共4页Journal of Shaanxi Institute of Education

摘　　要：为了证明对于Si是2阶密度矩阵,π={πi}in=1是概率分布,且矩阵A(s)≡∑i=1 n πiS11+si是可逆的,那么对任意0≤s≤1,H(x)=-xlogx,有Tr[A(s)s{∑j=1 n πjS11+sj(logS11+sj)2}-A(s)-1+s{∑j=1 n πjH(S11+sj)}2]≥0.可以利用cauchy-schwarz不等式,Jensen不等式和迹的一些性质来证明。结果表明这些涉及矩阵和对数的不等式给出了由K.Yanagi提出的开放问题的部分解答。因为这些结论仅仅是特例,所以在此基础上可以作进一步的研究。Aim： Supposebe Si be 2 × 2 is density matrix, π = {πi }in=1 is any a probability distribution,and A （s）≡^n∑i=1πuSu^1/1＋s is invertible, for any 0≤s≤1, H（x ） = - xlogx, and then Tr[A（s）^s{^n∑j=1πjSj^1/1＋s（logSj^1/1＋s）^2}-A（s）^-1＋s{n^∑j=1πjH（Sj^1/1＋s）}^2]≥0.which is a generalization of an inequality proved by K. Yanagi and others.Method： These problems are settled by applying Caushy-schwarz inequality, Jensen＇s inequality and some property of trace. Results： These inequalities related matrix logarithm give partial answer of the open problem posed by Yanagi. Conclusion： Some further studies on this base will been done because the results of those trace inequalities is just one case.

关 键 词：迹不等式 凹性 量子可靠性函数 

分 类 号：O174[理学—数学]