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arXiv:0803.0591 · International Journal of Mathematics

RELATIVE ENTROPY FOR MAXIMAL ABELIAN SUBALGEBRAS OF MATRICES AND THE ENTROPY OF UNISTOCHASTIC MATRICES

Marie Choda

Published Aug 1, 2008• 11 citations
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Abstract

Let A and B be two maximal abelian *-subalgebras of the n × n complex matrices Mn(ℂ). To study the movement of the automorphisms of Mn(ℂ), we modify the Connes–Størmer relative entropy H(A|B) and the Connes relative entropy Hϕ(A|B) with respect to a state ϕ, and introduce the two kinds of the constant h(A|B) and hϕ(A|B). For the unistochastic matrix b(u) defined by a unitary u with B = uAu*, we show that h(A|B) is the entropy H(b(u)) of b(u). This is obtained by our computation of hϕ(A|B). The h(A|B) attains to the maximal value log n if and only if the pair {A, B} is orthogonal in the sense of Popa.

DOI
10.1142/s0129167x08004881

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Let A and B be two maximal abelian -subalgebras of the n × n complex matrices Mn(ℂ). To study the movement of the automorphisms of Mn(ℂ), we modify the Connes–Størmer relative entropy H(A|B) and the Connes relative entropy Hϕ(A|B) with respect to a state ϕ, and introduce the two kinds of the constant h(A|B) and hϕ(A|B). For the unistochastic matrix b(u) defined by a unitary u with B = uAu, we show that h(A|B) is the entropy H(b(u)) of b(u). This is obtained by our computation of hϕ(A|B). The h(A|B) attains to the maximal value log n if and only if the pair {A, B} is orthogonal in the sense of Popa.
This is an expository survey article based on a talk for the Spe- cial Session (Operator Algebra) in AMC2009. For two subalgebras A and B of a finite von Neumann algebra M, we discuss a modified version h(A|B) of the Connes-Stormer relative entropy H(A|B). As one of the most elementary examples, we pick two maximal abelian *-subalgebras A, B of the n × n com- plex matrices Mn(C), and show relations between h(A|B) and the entropy for unistochastic matrices. Also we show related results in the case of a pair of subfactors of a type II1 factor.
In this paper we compare the accessible information of quantum communication channels with the entropic content of finite-dimensional matrix algebras with respect to quantum states, as defined by Connes, Narnhofer, and Thirring. In particular, every Abelian n×n matrix algebra together with a density matrix define the input alphabet of a quantum communication channel whose accessible information equals the entropic content of the algebra with respect to the state. The cases n=2 and n=3 are concretely examined in connection with the problem of the best estimation.
A transition matrix can be constructed through the partial contraction of two given quantum states. We analyze and compare four different definitions of entropy for transition matrices, including (modified) pseudo entropy, SVD entropy, and ABB entropy. We examine the probabilistic interpretation of each entropy measure and show that only the distillation interpretation of ABB entropy corresponds to the joint success probability of distilling entanglement between the two quantum states used to construct the transition matrix. Combining the transition matrix with preceding measurements and subsequent non-unitary operations, the ABB entropy either decreases or remains unchanged, whereas the (modified) pseudo entropy and SVD entropy may increase or decrease. We further apply these entropy measures to transition matrices constructed from several ensembles: (i) pairs of independent Haar-random states; (ii) bi-orthogonal eigenstates of non-Hermitian random systems; and (iii) bi-orthogonal states in PT P T -symmetric systems near their exceptional points. Across all cases considered, the SVD and ABB entropies of the transition matrix closely mirror the behavior of the subsystem entanglement entropy of a single random state, in contrast to the (modified) pseudo entropy, which can exceed the bound of subsystem size, fail to scale with system size, or even take complex values.
An ensemble of random unistochastic (orthostochastic) matrices is defined by taking squared moduli of elements of random unitary (orthogonal) matrices distributed according to the Haar measure on U ( N ) (or O ( N )). An ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble. We study the distribution of complex eigenvalues of bistochastic, unistochastic and orthostochastic matrices in the complex plane. We compute averages (entropy, traces) over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices.