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Marie Choda

Publications and source records attributed to Marie Choda.

7 recordsLinked to original sources
arXiv2016

Operational extreme points and Cuntz's canonical endomorphism

Based on the fact that the Cuntz algebra $O_n$ is generated by the operators consisting of a finite operatorional partition, we study the notion of operational extreme points (which we introduce here) by using several completely positive maps on $O_n$. As a typical example, we show that the Cuntz's canonical endomorphism $Phi_n$ is an operational extreme point in the set of completely positive maps on $O_n$ and that it induces a completely positive map which is extreme but not operational extreme, etc.

math.OA↗
arXiv2016

Operational extreme points of unital completely positive maps

Two notions for linear maps (operational convex combinations and operational exetreme points) are introduced. The set S of ucp maps on the n times n matrix algebra is the operational convex combinations of the identity map. An operational extreme point of S is an extreme point of S but the converse does not hold, and every automorphism is an operational extreme point of S.

math.OA↗
arXiv2014

Around Shannon's Interpretation for Entropy-preserving Stochastic Averages

We give various characterizations for a positive unital Tr-preserving map on a matrix algebra to preserve the von Neumann entropy of a state. Among others, it is given by that the map behaves as a *-automorphism. This is also equivalent to that the entropy of the stochastic matrix arising from the map and the state is zero.

math.OA↗
arXiv2010

von Neumann entropy and relative position between subalgebras

We give a numerical characterization of mutual orthogonality (that is, complementarity) for subalgebras. In order to give such a characterization for mutually orthogonal subalgebras $A$ and $B$ of the $k \times k$ matrix algebra $M_k(\mathbb{C})$, where $A$ and $B$ are isomorphic to some $M_n(\mathbb{C})$ $(n \leq k)$, we consider a density matrix which is induced from the pair $\{A, B\}$. We show that $A$ and $B$ are mutually orthogonal if and only if the von Neumann entropy of the density matrix is the maximum value $2\log n$, which is the logarithm of the dimension of the subfactors.

math.OA↗
arXiv2010

Conjugate Pairs of Subfactors and Entropy for Automorphisms

Based on the fact that, for a subfactor $N$ of a II$_1$ factor $M,$ the first non-trivial Jones index is 2 and then $M$ is decomposed as the crossed product of $N$ by an outer action of ${\mathbb{Z}}_2,$ we study pairs $ \{N, uNu^* \}$ from a view point of entropy for two subalgebras of $M$ with a connection to the entropy for automorphisms, where the inclusion of II$_1$ factors $ N \subset M$ is given as $M$ is the crossed product of $N$ by a finite group of outer automorphisms and $u$ is a unitary in $M.$

math.OA↗
arXiv2008

Relative entropy for maximal abelian subalgebras of matrices and the entropy of unistochastic matrices

Let $A$ and $B$ be two maximal abelian *-subalgebras of the $n\times n$ complex matrices $M_n(\mathbb{C}).$ To study the movement of the inner automorphisms of $M_n(\mathbb{C}),$ we modify the Connes-St$ø$rmer relative entropy $H(A | B)$ and the Connes relative entropy $H_ϕ(A | B)$ with respect to a state $ϕ,$ and introduce the two kinds of the constant $h(A | B)$ and $h_ϕ(A | B).$ 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 $h_ϕ(A | B).$ 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.

math.OA↗