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
Article

Abstract

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

Keywords

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

Notes

Acknowledgments

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

References

  1. 1.
    Artacho, F.A., Borwein, J., Tam, M.: Recent results on Douglas-Rachford methods for combinatorial optimization problems. J. Optim. Theory Appl. 163, 1–30 (2013). doi: 10.1007/s10957-013-0488-0 CrossRefMATHGoogle Scholar
  2. 2.
    Bauschke, H., Borwein, J.: On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1(2), 185–212 (1993). doi: 10.1007/BF01027691 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bauschke, H., Borwein, J.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996). doi: 10.1137/S0036144593251710 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bauschke, H., Combettes, P., Luke, D.: Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization. J. Opt. Soc. Am. A 19(7), 1334–1345 (2002). doi: 10.1364/JOSAA.19.001334 MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Bauschke, H., Luke, D., Phan, H., Wang, X.: Restricted normal cones and the method of alternating projections: theory. Set-Valued Var. Anal. 21, 1–43 (2013). doi: 10.1007/s11228-013-0239-2 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Borwein, J., Sims, B.: The Douglas–Rachford algorithm in the absence of convexity. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optim. Appl., vol. 49, pp. 93–109. Springer, New York (2011). doi: 10.1007/978-1-4419-9569-8_6
  7. 7.
    Borwein, J., Wolkowicz, H.: Facial reduction for a cone-convex programming problem. J. Aust. Math. Soc. Ser. A 30(3), 369–380 (1980/81)Google Scholar
  8. 8.
    Borwein, J., Wolkowicz, H.: Regularizing the abstract convex program. J. Math. Anal. Appl. 83(2), 495–530 (1981)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bregman, L.: The method of successive projection for finding a common point of convex sets. Sov. Math. Dokl. 6, 688–692 (1965)MATHGoogle Scholar
  10. 10.
    Chefles, A., Jozsa, R., Winter, A.: On the existence of physical transformations between sets of quantum states. Int. J. Quantum Inf. 2, 11–21 (2004)CrossRefMATHGoogle Scholar
  11. 11.
    Choi, M.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975)CrossRefMATHGoogle Scholar
  12. 12.
    Demmel, J., Marques, O., Parlett, B., Vömel, C.: Performance and accuracy of LAPACK’s symmetric tridiagonal eigensolvers. SIAM J. Sci. Comput. 30(3), 1508–1526 (2008). doi: 10.1137/070688778 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Drusvyatskiy, D., Ioffe, A., Lewis, A.: Alternating projections and coupling slope (2014). arXiv:1401.7569
  14. 14.
    Duffin, R.: Infinite programs. In: Tucker, A. (ed.) Linear Equalities and Related Systems, pp. 157–170. Princeton University Press, Princeton (1956)Google Scholar
  15. 15.
    Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1, 211–218 (1936)CrossRefMATHGoogle Scholar
  16. 16.
    Elser, V., Rankenburg, I., Thibault, P.: Searching with iterated maps. Proc. Natl. Acad. Sci. 104(2), 418–423 (2007). doi: 10.1073/pnas.0606359104. http://www.pnas.org/content/104/2/418.abstract
  17. 17.
    Escalante, R., Raydan, M.: Alternating projection methods, Fundamentals of Algorithms, vol. 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2011). doi: 10.1137/1.9781611971941
  18. 18.
    Fung, C.H., Li, C.K., Sze, N.S., Chau, H.: Conditions for degradability of tripartite quantum states. Tech. Rep., University of Hong Kong (2012). arXiv:1308.6359
  19. 19.
    Hesse, R., Luke, D., Neumann, P.: Alternating projections and Douglas–Rachford for sparse affine feasibility. IEEE Trans. Signal Process. 62(18), 4868–4881 (2014). doi: 10.1109/TSP.2014.2339801 MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Huang, Z., Li, C.K., Poon, E., Sze, N.S.: Physical transformations between quantum states. J. Math. Phys. 53(10), 102, 209, 12 (2012). doi: 10.1063/1.4755846
  21. 21.
    Kraus, K.: States, Effects, and Operations: Fundamental Notions of Quantum Theory. Lecture Notes in Physics, vol. 190. Springer, Berlin (1983)CrossRefGoogle Scholar
  22. 22.
    Lewis, A., Luke, D., Malick, J.: Local linear convergence for alternating and averaged nonconvex projections. Found. Comput. Math. 9(4), 485–513 (2009). doi: 10.1007/s10208-008-9036-y MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lewis, A., Malick, J.: Alternating projections on manifolds. Math. Oper. Res. 33(1), 216–234 (2008). doi: 10.1287/moor.1070.0291 MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Li, C.K., Poon, Y.T.: Interpolation by completely positive maps. Linear Multilinear Algebra 59(10), 1159–1170 (2011). doi: 10.1080/03081087.2011.585987 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Li, C.K., Poon, Y.T., Sze, N.S.: Higher rank numerical ranges and low rank perturbations of quantum channels. J. Math. Anal. Appl. 348(2), 843–855 (2008). doi: 10.1016/j.jmaa.2008.08.016 MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Lions, P., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979). doi: 10.1137/0716071 MathSciNetADSCrossRefMATHGoogle Scholar
  27. 27.
    Mendl, C., Wolf, M.: Unital quantum channels—convex structure and revivals of Birkhoff’s theorem. Commun. Math. Phys. 289(3), 1057–1086 (2009). doi: 10.1007/s00220-009-0824-2 MathSciNetADSCrossRefMATHGoogle Scholar
  28. 28.
    Nielsen, M., Chuang, I. (eds.): Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)Google Scholar
  29. 29.
    Phan, H.M.: Linear convergence of the Douglas–Rachford method for two closed sets (2014). arXiv:1401.6509
  30. 30.
    Watrous, J.: Distinguishing quantum operations having few kraus operators. Quantum Inf. Comput. 8, 819–833 (2008). http://arxiv.org/abs/0710.0902

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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