Projection methods in quantum information science

5 We consider the problem of constructing quantum operations or channels, if they exist, that 6 transform a given set of quantum states {ρ1, . . . , ρk} to another such set {ρ̂1, . . . , ρ̂k}. In other 7 words, we must find a completely positive linear map, if it exists, that maps a given set of density 8 matrices to another given set of density matrices. This problem, in turn, is an instance of a 9 positive semi-definite feasibility problem, but with highly structured constraints. The nature of 10 the constraints makes projection based algorithms very appealing when the number of variables 11 is huge and standard interior point-methods for semi-definite programming are not applicable. 12 We provide emperical evidence to this effect. We moreover present heuristics for finding both 13 high rank and low rank solutions. Our experiments are based on the method of alternating 14 projections and the Douglas-Rachford reflection method. 15