New York Journal of Mathematics
Volume 23 (2017) 1369-1393


Douglas Farenick and Mizanur Rahaman

Bures contractive channels on operator algebras

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Published: October 10, 2017
Keywords: C*-algebra, von Neumann algebra, faithful trace, positive linear map, completely positive linear map, quantum channel, Bures metric, fidelity, irreducible positive linear map, multiplicative domain, Schwarz map
Subject: Primary 46L05; Secondary 46L60, 81R15

In a unital C*-algebra with a faithful trace functional τ, the set Dτ(A) of positive ρ∈A of trace τ(ρ)=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 Dτ(A) in terms of the Bures metric. Linear maps on A that map Dτ(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 Dτ(A); (iii) to introduce and study channels on A that are locally contractive maps (which we call Bures contractions) on the metric space Dτ(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**.


The first author was supported in part by an NSERC Discovery Grant. The second author was supported part by a University of Regina Graduate Research Fellowship

Author information

Douglas Farenick:
Department of Mathematics & Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada

Mizanur Rahaman:
Department of Mathematics & Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada