×

Decomposable Pauli diagonal maps and tensor squares of qubit maps. (English) Zbl 1500.81014

Summary: It is a well-known result due to E. Størmer [Acta Math. 110, 233–278 (1963; Zbl 0173.42105)] that every positive qubit map is decomposable into a sum of a completely positive map and a completely copositive map. Here, we generalize this result to tensor squares of qubit maps. Specifically, we show that any positive tensor product of a qubit map with itself is decomposable. This solves a recent conjecture by S. N. Filippov and K. Y. Magadov [J. Phys. A, Math. Theor. 50, No. 5, Article ID 055301, 21 p. (2017; Zbl 1357.81152)]. We contrast this result with examples of non-decomposable positive maps arising as the tensor product of two distinct qubit maps or as the tensor square of a decomposable map from a qubit to a ququart. To show our main result, we reduce the problem to Pauli diagonal maps. We then characterize the cone of decomposable ququart Pauli diagonal maps by determining all 252 extremal rays of ququart Pauli diagonal maps that are both completely positive and completely copositive. These extremal rays split into three disjoint orbits under a natural symmetry group, and two of these orbits contain only entanglement breaking maps. Finally, we develop a general combinatorial method to determine the extremal rays of Pauli diagonal maps that are both completely positive and completely copositive between multi-qubit systems using the ordered spectra of their Choi matrices. Classifying these extremal rays beyond ququarts is left as an open problem.
©2021 American Institute of Physics

MSC:

81P45 Quantum information, communication, networks (quantum-theoretic aspects)
47N50 Applications of operator theory in the physical sciences
46N50 Applications of functional analysis in quantum physics
22E70 Applications of Lie groups to the sciences; explicit representations

Software:

QETLAB; polymake; MPT
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Størmer, E., Positive linear maps of operator algebras, Acta Math., 110, 233-278 (1963) · Zbl 0173.42105 · doi:10.1007/BF02391860
[2] Woronowicz, S. L., Positive maps of low dimensional matrix algebras, Rep. Math. Phys., 10, 2, 165-183 (1976) · Zbl 0347.46063 · doi:10.1016/0034-4877(76)90038-0
[3] Tang, W.-S., On positive linear maps between matrix algebras, Linear Algebra Appl., 79, 33-44 (1986) · Zbl 0612.15012 · doi:10.1016/0024-3795(86)90290-9
[4] Choi, M.-D., Some assorted inequalities for positive linear maps on C^*-algebras, J. Oper. Theory, 4, 271-285 (1980) · Zbl 0511.46051
[5] Horodecki, M.; Horodecki, P.; Horodecki, R., Separability of mixed states: Necessary and sufficient conditions, Phys. Lett. A, 223, 1, 1-8 (1996) · Zbl 0984.81007 · doi:10.1103/physreva.54.1838
[6] Stormer, E., Decomposable positive maps on C^⋆-algebras, Proc. Am. Math. Soc., 86, 3, 402-404 (1982) · Zbl 0526.46054 · doi:10.2307/2044436
[7] Horodecki, P., Separability criterion and inseparable mixed states with positive partial transposition, Phys. Lett. A, 232, 5, 333-339 (1997) · Zbl 1053.81504 · doi:10.1016/s0375-9601(97)00416-7
[8] Horodecki, M.; Horodecki, P.; Horodecki, R., Mixed-state entanglement and distillation: Is there a ‘bound’ entanglement in nature?, Phys. Rev. Lett., 80, 24, 5239 (1998) · Zbl 0947.81005 · doi:10.1103/physrevlett.80.5239
[9] Horodecki, K.; Horodecki, M.; Horodecki, P.; Oppenheim, J., Secure key from bound entanglement, Phys. Rev. Lett., 94, 16, 160502 (2005) · doi:10.1103/physrevlett.94.160502
[10] Smith, G.; Yard, J., Quantum communication with zero-capacity channels, Science, 321, 5897, 1812-1815 (2008) · Zbl 1226.94011 · doi:10.1126/science.1162242
[11] Bäuml, S.; Christandl, M.; Horodecki, K.; Winter, A., Limitations on quantum key repeaters, Nat. Commun., 6, 6908 (2015) · doi:10.1038/ncomms7908
[12] Aubrun, G.; Szarek, S. J., Two proofs of Størmer’s theorem (2015)
[13] Gurvits, L., Classical complexity and quantum entanglement, J. Comput. Syst. Sci., 69, 3, 448-484 (2004) · Zbl 1093.81012 · doi:10.1016/j.jcss.2004.06.003
[14] Aubrun, G.; Szarek, S. J., Alice and Bob Meet Banach (2017), Am. Math. Soc.
[15] Filippov, S. N.; Yu Magadov, K., Positive tensor products of maps and n-tensor-stable positive qubit maps, J. Phys. A: Math. Theor., 50, 5, 055301 (2017) · Zbl 1357.81152 · doi:10.1088/1751-8121/aa5301
[16] Ruskai, M. B.; Szarek, S.; Werner, E., An analysis of completely-positive trace-preserving maps on M_2, Linear Algebra Appl., 347, 1-3, 159-187 (2002) · Zbl 1032.47046 · doi:10.1016/s0024-3795(01)00547-x
[17] Fujiwara, A.; Algoet, P., One-to-one parametrization of quantum channels, Phys. Rev. A, 59, 5, 3290 (1999) · doi:10.1103/physreva.59.3290
[18] Choi, M.-D., Completely positive linear maps on complex matrices, Linear Algebra Appl., 10, 3, 285-290 (1975) · Zbl 0327.15018 · doi:10.1016/0024-3795(75)90075-0
[19] Horodecki, M.; Shor, P. W.; Ruskai, M. B., Entanglement breaking channels, Rev. Math. Phys., 15, 6, 629-641 (2003) · Zbl 1080.81006 · doi:10.1142/s0129055x03001709
[20] Rockafellar, R. T., Convex Analysis (1970), Princeton University Press · Zbl 0229.90020
[21] Rudolph, O., A separability criterion for density operators, J. Phys. A: Math. Gen., 33, 21, 3951 (2000) · Zbl 0947.46054 · doi:10.1088/0305-4470/33/21/308
[22] Rudolph, O., Further results on the cross norm criterion for separability, Quantum Inf. Process., 4, 3, 219-239 (2005) · Zbl 1130.81016 · doi:10.1007/s11128-005-5664-1
[23] Chen, K.; Yang, L.; Wu, L., A matrix realignment method for recognizing entanglement, Quantum Inf. Comput., 3, 193-202 (2002) · Zbl 1152.81692 · doi:10.5555/2011534.2011535
[24] Lupo, C.; Aniello, P.; Scardicchio, A., Bipartite quantum systems: On the realignment criterion and beyond, J. Phys. A: Math. Theor., 41, 41, 415301 (2008) · Zbl 1192.81091 · doi:10.1088/1751-8113/41/41/415301
[25] Johnston, N.; Patterson, E., The inverse eigenvalue problem for entanglement witnesses, Linear Algebra Appl., 550, 1-27 (2018) · Zbl 1390.81076 · doi:10.1016/j.laa.2018.03.043
[26] Horodecki, P.; Lewenstein, M.; Vidal, G.; Cirac, I., Operational criterion and constructive checks for the separability of low-rank density matrices, Phys. Rev. A, 62, 3, 032310 (2000) · doi:10.1103/physreva.62.032310
[27] Herceg, M.; Kvasnica, M.; Jones, C.; Morari, M., Multi-parametric toolbox 3.0, 502-510 (2013), IEEE
[28] Gawrilow, E.; Joswig, M., polymake: A framework for analyzing convex polytopes, Polytopes—Combinatorics and Computation (Oberwolfach, 1997), 43-73 (2000), Birkhäuser: Birkhäuser, Basel · Zbl 0960.68182
[29] Assarf, B.; Gawrilow, E.; Herr, K.; Joswig, M.; Lorenz, B.; Paffenholz, A.; Rehn, T., Computing convex hulls and counting integer points with polymake, Math. Program. Comput., 9, 1, 1-38 (2017) · Zbl 1370.90009 · doi:10.1007/s12532-016-0104-z
[30] Christandl, M., PPT square conjecture
[31] Christandl, M.; Müller-Hermes, A.; Wolf, M. M., When do composed maps become entanglement breaking?, Ann. Henri Poincare, 20, 7, 2295-2322 (2019) · Zbl 1416.81025 · doi:10.1007/s00023-019-00774-7
[32] Kennedy, M.; Manor, N. A.; Paulsen, V. I., Composition of PPT maps, Quantum Inf. Comput., 18, 5-6, 472-480 (2018) · doi:10.26421/qic18.5-6-4
[33] Rahaman, M.; Jaques, S.; Paulsen, V. I., Eventually entanglement breaking maps, J. Math. Phys., 59, 6, 062201 (2018) · Zbl 1391.81037 · doi:10.1063/1.5024385
[34] Collins, B.; Yin, Z.; Zhong, P., The PPT square conjecture holds generically for some classes of independent states, J. Phys. A: Math. Theor., 51, 42, 425301 (2018) · Zbl 1407.81019 · doi:10.1088/1751-8121/aadd52
[35] Chen, L.; Yang, Y.; Tang, W.-S., Positive-partial-transpose square conjecture for n = 3, Phys. Rev. A, 99, 1, 012337 (2019) · doi:10.1103/physreva.99.012337
[36] Hanson, E. P.; Rouzé, C.; França, D. S., Eventually entanglement breaking Markovian dynamics: Structure and characteristic times, Ann. Henri Poincare, 21, 1517 (2020) · Zbl 1473.81042 · doi:10.1007/s00023-020-00906-4
[37] Størmer, E., Extension of positive maps into B(H), J. Funct. Anal., 66, 2, 235-254 (1986) · Zbl 0637.46061 · doi:10.1016/0022-1236(86)90072-8
[38] Skowronek, Ł.; Størmer, E.; Życzkowski, K., Cones of positive maps and their duality relations, J. Math. Phys., 50, 6, 062106 (2009) · Zbl 1216.46052 · doi:10.1063/1.3155378
[39] Müller-Hermes, A., Decomposability of linear maps under tensor powers, J. Math. Phys., 59, 10, 102203 (2018) · Zbl 1411.46044 · doi:10.1063/1.5045559
[40] Müller-Hermes, A.; Reeb, D.; Wolf, M. M., Positivity of linear maps under tensor powers, J. Math. Phys., 57, 1, 015202 (2016) · Zbl 1391.15108 · doi:10.1063/1.4927070
[41] DiVincenzo, D. P.; Mor, T.; Shor, P. W.; Smolin, J. A.; Terhal, B. M., Unextendible product bases, uncompletable product bases and bound entanglement, Commun. Math. Phys., 238, 3, 379-410 (2003) · Zbl 1027.81004 · doi:10.1007/s00220-003-0877-6
[42] Brualdi, R. A., Algorithms for constructing (0,1)-matrices with prescribed row and column sum vectors, Discrete Math., 306, 23, 3054-3062 (2006) · Zbl 1110.05102 · doi:10.1016/j.disc.2004.10.028
[43] Fulton, W., Young Tableaux: With Applications to Representation Theory and Geometry (1997), Cambridge University Press · Zbl 0878.14034
[44] Kostka, C., Uber den zusammenhang zwischen einigen formen von symmetrischen functionen, J. Reine Angew. Math., 1882, 89-123 · JFM 14.0112.02 · doi:10.1515/crll.1882.93.89
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.