Projection methods in quantum information science

TL;DR

This paper explores projection-based algorithms for constructing quantum channels that transform specific sets of quantum states, providing empirical evidence and heuristics for solving large-scale positive semi-definite feasibility problems in quantum information science.

Contribution

It introduces projection methods like alternating projections and Douglas-Rachford for quantum state transformations, with practical heuristics and empirical validation.

Findings

Projection methods are effective for large-scale quantum channel problems.

Heuristics can find both high and low rank solutions.

Empirical evidence supports the use of these algorithms over traditional SDP methods.

Abstract

We consider the problem of constructing quantum operations or channels, if they exist, that transform a given set of quantum states $\{\rho_1, \dots, \rho_k\}$ to another such set $\{\hat\rho_1, \dots, \hat\rho_k\}$. 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.