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Projection methods in quantum information science

Yuen-Lam Cheung (vris.cheung***at***gmail.com)
Dmitriy Drusvyatskiy (ddrusvyatskiy***at***uwaterloo.ca)
Chi-Kwong Li (ckli***at***math.wm.edu)
Diane Pelejo (dcpelejo***at***gmail.com)
Henry Wolkowicz (hwolkowicz***at***uwaterloo.ca)

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 empirical 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.

Keywords: quantum operations, completely positive linear maps, alternating projection methods, Douglas-Rachford method, Choi matrix, semidefinite feasibility problem, large scale

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Other Topics (Other )

Citation: University of Waterloo, July/14

Download: [PDF]

Entry Submitted: 07/23/2014
Entry Accepted: 07/23/2014
Entry Last Modified: 07/28/2014

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