一些迹不等式的若干结论

目的如果Si(i=1,…,n)是密度矩阵,π={πi}in=1是一个概率分布,并且A(0)≡∑ni=1πiSi12是可逆的,那么Tr{∑nj=1πjSj(logSj)2}-A(0)-1{∑nj=1πjH(Sj)}2]≥0,其中H(x)=-xlogx,这是Yanagi证明的不等式的一个推广。方法利用Caushy-Schwarz不等式,Jensen's不等式和迹的一些性质来证明。结果这些涉及矩阵和对数的不等式给出了由Yanagi提出的开放问题的部分解答。结论因为这些结论仅仅是特例,所以在此基础上可做进一步的研究。

On some trace inequalities

Aim It is proved that if Si(I=1,…,n) is density matrices, π={πi}ni=1 is a probability distribution and A(0)≡∑ni=1πiS(1)/(2)I is invertible, then Tr{∑nj=1πjSj(log Sj)2}-A(0)-1{∑nj=1πjH(Sj)}2]≥0, where H(x) =-xlog x, 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 properties of trace. Results The partial answers of the open problem posed by Yanagi were given by these inequalities related matrix logarithm.Conclusion Some further studies on this base will been done because the results of those trace inequalities is just one case.