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Journal of Shaanxi Normal University

Some properties of expectations and variances of matrices

Meng Hong-bing

Published Jan 1, 2010• 0 citations
OpenAlex

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(AB)=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.

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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(AB)=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.
The following conclusions are established in this paper. The supremum of the absolute value of the absolute variances of an operator in a rank-one projection is equal to the square of the distance from the operator to the scalar operators; a density operator is faithful if and only if it is injective; the strong convergence on the closure of the range of a density operator ρ implies the ρ-a. s. convergence of the sequence; a sequence of operators is strongly convergent if it is ρ-a. s. convergent for every density operator ρ; if ρ is a faithful density operator, then a sequence operators is strongly convergent to A if and only if it is uniformly bounded and convergent to A a. s. [ρ].
A general theorem is proved on the density matrix of quantum statistics. Let the density matrix be ρ{M}=exp[−β ∑ l=1MHl],where the operators Hl's are not always commutable. ρ{M} can be expanded in series of the form ρ{M}=(ΣM≧ ∑ i=1nmi, n≧0Σ{m1}M···{mn}M)ρ*{m1}···ρ*{mn}ρ̄{M−m1−···−mn},where {m1}M, ··· {mn}M are subsets of operators chosen from the set {M} and operators belonging to different subsets are commutable. The summation is over all the possible choices of the sets {m1}, ··· {mn}. ρ̄{k} is defined by ρ̄{k}=[Π{k}exp(−βHi)]s,where suffix s means the symmetrization by changing the order of the products. And ρ*{m} is defined by ρ*{m}=(Σ{k}m)(−)m−kρ{k}ρ̄{m−k},which is proved to be O(βm+1). Some possible applications are briefly discussed.
The (Fisher) information matrix (I2p) for a p×p density matrix (ρ) is taken to be the (2p2+p)-dimensional information matrix of a 2p-variate real normal distribution with an a priori zero mean vector and a covariance matrix (Σ2p), the two diagonal blocks of which are formed from the symmetric (real) component of ρ, and the two off-diagonal blocks from the skew-symmetric (imaginary) part of ρ. The inverse square root ‖nI2p‖−1/2 (where n is the number of observations) serves as a unitary invariant prior distribution over (the precision matrices) Σ2p−1 and ‖nI2p‖1/2 over Σ2p—and consequently over the p×p (mixed and pure) density matrices. The availability of these priors should facilitate the process of state determination. An uncertainty principle for the estimation of ρ is formulated. The determinant of Σ2p proves for integral p≥1 to equal (2−p‖ρ‖)2.
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The classical Bayes rule computes the posterior model probability from the prior probability and the data likelihood. We generalize this rule to the case when the prior is a density matrix (symmetric positive definite and trace one) and the data likelihood a covariance matrix. The classical Bayes rule is retained as the special case when the matrices are diagonal. In the classical setting, the calculation of the probability of the data is an expected likelihood, where the expectation is over the prior distribution. In the generalized setting, this is replaced by an expected variance calculation where the variance is computed along the eigenvectors of the prior density matrix and the expectation is over the eigenvalues of the density matrix (which form a probability vector). The variances along any direction is determined by the covariance matrix. Curiously enough this expected variance calculation is a quantum measurement where the co-variance matrix specifies the instrument and the prior density matrix the mixture state of the particle. We motivate both the classical and the generalized Bayes rule with a minimum relative entropy principle, where the Kullbach-Leibler version gives the classical Bayes rule and Umegaki’s quantum relative entropy the new Bayes rule for density matrices. 1
A block matrix [AXX∗B]\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right][AX∗​XB​] is positive partial transpose (PPT) if both [AXX∗B]\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right][AX∗​XB​] and [AX∗XB]\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right][AX​X∗B​] are positive semi-definite. This class is significant in studying the separability criterion for density matrices. The current paper presents new relations for such matrices. This includes some equivalent forms and new related inequalities that extend some results from the literature. In the end of the paper, we present some related results for positive semi-definite block matrices, which have similar forms as those presented for PPT matrices, with applications that include significant improvement of numerical radius inequalities.