Projection methods for quantum channel construction

We consider the problem of constructing quantum channels, if they exist, that transform a given set of quantum states {ρ1, . . . , ρk} to another such set {ρ̂1, . . . , ρ̂k}. In other words, we must find a completely positive linear map, if it exists, that maps a given set of density matrices to another given set of density matrices, possibly of different dimension. Using the theory of completely positive linear maps, one can formulate the problem as an instance of a positive semidefinite feasibility problem with highly structured constraints. The nature of the constraints makes projection based algorithms very appealing when the number of variables is huge and standard interior point-methods for semidefinite programming are not applicable. We provide empirical evidence to this effect. We moreover present heuristics for finding both high rank and low rank solutions. Our experiments are based on the method of alternating projections and the Douglas-Rachford reflection method. D. Drusvyatskiy Department of Mathematics, University of Washington, Seattle, WA [email protected] C-K. Li Department of Mathematics, College of William and Mary, Williamsburg, VA [email protected] D. Pelejo Department of Mathematics, College of William and Mary, Williamsburg, VA [email protected] Y.-L. Voronin Department of Computer Science, University of Colorado, Boulder, CO E-mail: [email protected] H. Wolkowicz Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada [email protected] 2 Dmitriy Drusvyatskiy et al.