 

Douglas Farenick and Mizanur Rahaman
Bures contractive channels on operator algebras view print


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 


Abstract
In a unital C*algebra with a faithful trace functional τ, the set D_{τ}(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 D_{τ}(A) in terms of the Bures metric. Linear maps on A
that map D_{τ}(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_{τ}(A); (iii) to introduce and study channels on A
that are locally contractive maps (which we call Bures contractions)
on the metric space D_{τ}(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^{**}.


Acknowledgements
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
douglas.farenick@uregina.ca
Mizanur Rahaman:
Department of Mathematics & Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada
mizanur1@gmail.com

