Projection methods for quantum channel construction

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 $$\{\rho _1, \ldots , \rho _k\}$$ { ρ 1 , … , ρ k } to another such set $$\{\hat{\rho }_1, \ldots , \hat{\rho }_k\}$$ { ρ ^ 1 , … , ρ ^ k } . 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

Projection methods for quantum channel construction

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Publisher
Springer US
Copyright
Copyright © 2015 by Springer Science+Business Media New York
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-1024-y
Publisher site
See Article on Publisher Site

Abstract

We consider the problem of constructing quantum channels, if they exist, that transform a given set of quantum states $$\{\rho _1, \ldots , \rho _k\}$$ { ρ 1 , … , ρ k } to another such set $$\{\hat{\rho }_1, \ldots , \hat{\rho }_k\}$$ { ρ ^ 1 , … , ρ ^ k } . 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.

Journal

Quantum Information ProcessingSpringer Journals

Published: May 23, 2015

References

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