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Projection methods for quantum channel construction. (English) Zbl 1327.81083

Summary: We consider the problem of constructing quantum 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 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.

MSC:

81P45 Quantum information, communication, networks (quantum-theoretic aspects)
94A40 Channel models (including quantum) in information and communication theory
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[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 · Zbl 1305.90341 · doi:10.1007/s10957-013-0488-0
[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 · Zbl 0801.47042 · doi:10.1007/BF01027691
[3] Bauschke, H., Borwein, J.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367-426 (1996). doi:10.1137/S0036144593251710 · Zbl 0865.47039 · doi:10.1137/S0036144593251710
[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 · doi:10.1364/JOSAA.19.001334
[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 · Zbl 1272.49027 · doi:10.1007/s11228-013-0239-2
[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 · Zbl 1259.90098
[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) · Zbl 0464.90086
[8] Borwein, J., Wolkowicz, H.: Regularizing the abstract convex program. J. Math. Anal. Appl. 83(2), 495-530 (1981) · Zbl 0467.90076 · doi:10.1016/0022-247X(81)90138-4
[9] Bregman, L.: The method of successive projection for finding a common point of convex sets. Sov. Math. Dokl. 6, 688-692 (1965) · Zbl 0142.16804
[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) · Zbl 1069.81506 · doi:10.1142/S0219749904000031
[11] Choi, M.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285-290 (1975) · Zbl 0327.15018 · doi:10.1016/0024-3795(75)90075-0
[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 · Zbl 1165.65014 · doi:10.1137/070688778
[13] Drusvyatskiy, D., Ioffe, A., Lewis, A.: Alternating projections and coupling slope (2014). arXiv:1401.7569 · Zbl 1338.49057
[14] Duffin, R.; Tucker, A. (ed.), Infinite programs, 157-170 (1956), Princeton · Zbl 0072.37603
[15] Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1, 211-218 (1936) · JFM 62.1075.02 · doi:10.1007/BF02288367
[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 · Zbl 1160.90495
[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 · Zbl 1275.65031
[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 · Zbl 1286.81032
[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 · Zbl 1394.94227 · doi:10.1109/TSP.2014.2339801
[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 · Zbl 1278.81020
[21] Kraus, K.: States, Effects, and Operations: Fundamental Notions of Quantum Theory. Lecture Notes in Physics, vol. 190. Springer, Berlin (1983) · Zbl 0545.46049 · doi:10.1007/3-540-12732-1
[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 · Zbl 1169.49030 · doi:10.1007/s10208-008-9036-y
[23] Lewis, A., Malick, J.: Alternating projections on manifolds. Math. Oper. Res. 33(1), 216-234 (2008). doi:10.1287/moor.1070.0291 · Zbl 1163.65040 · doi:10.1287/moor.1070.0291
[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 · Zbl 1236.15004 · doi:10.1080/03081087.2011.585987
[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 · Zbl 1151.47008 · doi:10.1016/j.jmaa.2008.08.016
[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 · Zbl 0426.65050 · doi:10.1137/0716071
[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 · Zbl 1167.81011 · doi:10.1007/s00220-009-0824-2
[28] Nielsen, M., Chuang, I. (eds.): Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) · Zbl 1049.81015
[29] Phan, H.M.: Linear convergence of the Douglas-Rachford method for two closed sets (2014). arXiv:1401.6509 · Zbl 1305.90341
[30] Watrous, J.: Distinguishing quantum operations having few kraus operators. Quantum Inf. Comput. 8, 819-833 (2008). http://arxiv.org/abs/0710.0902 · Zbl 1156.81015
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