Abstract:
Some equalities and inequalities on expectations, variances, covariances, absolute variance and independence of matrices in a given density matrix are established. The following results are proved: (ⅰ) Two matrices A and B are ρ-independent if and only if Covρ(A，B)=0 if and only if Expρ(AB)=Expρ(A)Expρ(B)；(ⅱ) If the numerical ranges W(A) and W(B) of A and B are contained in disks with radius R and S, respectively, then |Expρ(AB)-Expρ(A)Expρ(B)|≤4RS and |Covρ(A，B)|≤4ω(A)ω(B), where ω(A), ω(B) are the numerical radius of A，B, respectively.
