Bures Contractive Channels on Operator Algebras

Abstract

In a unital C*-algebra with a faithful trace functional $\tau$, the set $D_\tau(A)$ of positive $\rho\in A$ of trace \tau(\rho)=1 is an algebraic analogue of the space of density matrices (the set of all positive matrices of a fixed dimension of unit trace). Motivated by the literature concerning the metric properties of the space of density matrices, the present paper studies the density space $D_\tau(A)$ in terms of the Bures metric. Linear maps on A that map $D_\tau(A)$ back into itself are positive and trace preserving, hence, they may be viewed as an algebraic analogue of a quantum channel, which are studied intensely in the literature on quantum computing and quantum information theory. The main results in this paper are: (i) to establish that the Bures metric is indeed a metric, (ii) to prove that channels induce nonexpansive maps of the density space $D_\tau(A)$, (iii) to introduce and study channels on A that are locally contractive maps (which we call Bures contractions) on the metric space $D_\tau(A)$, and (iv) to analyse Bures contractions from the point of view of the Frobenius theory of cone preserving linear maps. Although the focus is on unital C*-algebras, an important class of examples is furnished by finite von Neumann algebras. Indeed, several of the C*-algebra results are established by first proving them for finite von Neumann algebras and then proving them for C*-algebras by embedding a C*-algebra A into its enveloping von Neumann algebra $A^{**}$.