Projection methods in quantum information science
This work provides a practical computational approach for quantum information scientists needing to find quantum channels for state transformations, though the results are empirical and incremental.
The paper addresses the problem of constructing quantum operations (channels) that transform one set of quantum states to another, formulated as a positive semi-definite feasibility problem. Empirical evidence shows that projection-based methods (alternating projections and Douglas-Rachford) are effective for large-scale instances where interior-point methods fail.
We consider the problem of constructing quantum operations or channels, if they exist, that transform a given set of quantum states $\{ρ_1, \dots, ρ_k\}$ to another such set $\{\hatρ_1, \dots, \hatρ_k\}$. In other words, we must find a {\em completely positive linear map}, if it exists, that maps a given set of density matrices to another given set of density matrices. This problem, in turn, is an instance of a positive semi-definite feasibility problem, but 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 semi-definite programming are not applicable. We provide emperical evidence to this effect. We moreover present heuristics for finding both high rank and low rank solutions. Our experiments are based on the \emph{method of alternating projections} and the \emph{Douglas-Rachford} reflection method.