Quantum Information Processing

, Volume 14, Issue 8, pp 3075–3096 | Cite as

Projection methods for quantum channel construction

  • Dmitriy Drusvyatskiy
  • Chi-Kwong Li
  • Diane Christine Pelejo
  • Yuen-Lam Voronin
  • Henry Wolkowicz


We consider the problem of constructing quantum channels, if they exist, that transform a given set of quantum states \(\{\rho _1, \ldots , \rho _k\}\) to another such set \(\{\hat{\rho }_1, \ldots , \hat{\rho }_k\}\). 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.


Quantum channels Completely positive linear maps Alternating projection methods Douglas–Rachford method Choi matrix Semidefinite feasibility problem Large scale 



We would like to thank the editors and referees for their careful reading and helpful comments on the paper.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Dmitriy Drusvyatskiy
    • 1
  • Chi-Kwong Li
    • 2
  • Diane Christine Pelejo
    • 2
  • Yuen-Lam Voronin
    • 3
  • Henry Wolkowicz
    • 4
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of MathematicsCollege of William and MaryWilliamsburgUSA
  3. 3.Department of Computer ScienceUniversity of ColoradoBoulderUSA
  4. 4.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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