Skip to content

References

  1. Scott Aaronson. Lecture 6: mixed states. URL: https://www.scottaaronson.com/qclec/6.pdf. ↩

  2. Srinivasan Arunachalam, Abel Molina, and Vincent Russo. Quantum hedging in two-round prover-verifier interactions. 2017. arXiv:1310.7954. ↩

  3. Somshubhro Bandyopadhyay, Rahul Jain, Jonathan Oppenheim, and Christopher Perry. Conclusive exclusion of quantum states. Physical Review A, Feb 2014. URL: http://dx.doi.org/10.1103/PhysRevA.89.022336, doi:10.1103/physreva.89.022336. ↩

  4. Somshubhro Bandyopadhyay, Alessandro Cosentino, Nathaniel Johnston, Vincent Russo, John Watrous, and Nengkun Yu. Limitations on separable measurements by convex optimization. IEEE Transactions on Information Theory, 61(6):3593–3604, jun 2015. URL: http://dx.doi.org/10.1109/TIT.2015.2417755, doi:10.1109/tit.2015.2417755. ↩

  5. Francesco Barioli and Abraham Berman. The maximal CP-rank of rank \(k\) completely positive matrices. Linear Algebra and its Applications, 363:17–33, 2003. doi:10.1016/S0024-3795(02)00250-1. ↩

  6. V. P. Belavkin. Optimal distinction of non-orthogonal quantum signals. Radio Engineering and Electronic Physics, 20:39–47, 1975. ↩

  7. Charles H. Bennett and Stephen J. Wiesner. Communication via one- and two-particle operators on Einstein-P odolsky-Rosen states. Phys. Rev. Lett., 69:2881–2884, Nov 1992. URL: https://link.aps.org/doi/10.1103/PhysRevLett.69.2881, doi:10.1103/PhysRevLett.69.2881. ↩

  8. Charles H. Bennett, David P. DiVincenzo, Christopher A. Fuchs, Tal Mor, Eric Rains, Peter W. Shor, John A. Smolin, and William K. Wootters. Quantum nonlocality without entanglement. Physical Review A, 59(2):1070–1091, Feb 1999. URL: http://dx.doi.org/10.1103/PhysRevA.59.1070, doi:10.1103/physreva.59.1070. ↩

  9. Charles H. Bennett, David P. DiVincenzo, Tal Mor, Peter W. Shor, John A. Smolin, and Barbara M. Terhal. Unextendible product bases and bound entanglement. Physical Review Letters, 82(26):5385–5388, Jun 1999. URL: http://dx.doi.org/10.1103/PhysRevLett.82.5385, doi:10.1103/physrevlett.82.5385. ↩

  10. Erik G Boman, Doron Chen, Ojas Parekh, and Sivan Toledo. On factor width and symmetric H-matrices. Linear algebra and its applications, 405:239–248, 2005. doi:10.1016/j.laa.2005.03.029. ↩

  11. Heinz-Peter Breuer. Optimal entanglement criterion for mixed quantum states. Physical Review Letters, Aug 2006. URL: http://dx.doi.org/10.1103/PhysRevLett.97.080501, doi:10.1103/physrevlett.97.080501. ↩

  12. Dagmar Bruß and Asher Peres. Construction of quantum states with bound entanglement. Phys. Rev. A, 61:030301, Feb 2000. URL: https://link.aps.org/doi/10.1103/PhysRevA.61.030301, doi:10.1103/PhysRevA.61.030301. ↩

  13. Adán Cabello. \(n\)-particle \(n\)-level singlet states: some properties and applications. Phys. Rev. Lett., 89:100402, Aug 2002. URL: https://arxiv.org/abs/quant-ph/0203119. ↩

  14. A. G. Campos, D. Schmid, L. Mamani, R. W. Spekkens, and I. Sainz. No epistemic model can explain anti-distinguishability of quantum mixed preparations. arXiv preprint arXiv:2401.17980v2, 2024. Preprint. URL: https://arxiv.org/abs/2401.17980. ↩

  15. Daniel Cariello. Separability for weak irreducible matrices. 2013. arXiv:1311.7275. ↩

  16. D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P.H. Souto Ribeiro, and S. P. Walborn. Detection of entanglement in asymmetric quantum networks and multipartite quantum steering. Nature Communications, aug 2015. URL: http://dx.doi.org/10.1038/ncomms8941, doi:10.1038/ncomms8941. ↩

  17. Kai Chen and Ling-An Wu. A matrix realignment method for recognizing entanglement. 2003. arXiv:quant-ph/0205017. ↩

  18. Lin Chen and Dragomir Ž Đoković. Separability problem for multipartite states of rank at most 4. Journal of Physics A: Mathematical and Theoretical, 46(27):275304, jun 2013. URL: https://dx.doi.org/10.1088/1751-8113/46/27/275304, doi:10.1088/1751-8113/46/27/275304. ↩

  19. Jianxin Chen, Zhengfeng Ji, David Kribs, Norbert Lütkenhaus, and Bei Zeng. Symmetric extension of two-qubit states. Physical Review A, Sep 2014. URL: http://dx.doi.org/10.1103/PhysRevA.90.032318, doi:10.1103/physreva.90.032318. ↩

  20. Sung Je Cho, Seung-Hyeok Kye, and Sa Ge Lee. Generalized choi maps in three-dimensional matrix algebra. Linear Algebra and its Applications, 171:213–224, 1992. doi:https://doi.org/10.1016/0024-3795(92)90260-H. ↩

  21. Dariusz Chruściński and Andrzej Kossakowski. On the symmetry of the seminal horodecki state. Physics Letters A, 375(3):434–436, Jan 2011. URL: https://arxiv.org/abs/1009.4385, doi:10.1016/j.physleta.2010.11.069. ↩

  22. Richard Cleve, William Slofstra, Falk Unger, and Sarvagya Upadhyay. Strong parallel repetition theorem for quantum xor proof systems. 2008. arXiv:quant-ph/0608146. ↩

  23. Richard Cleve, Peter Hoyer, Ben Toner, and John Watrous. Consequences and limits of nonlocal strategies. 2010. arXiv:quant-ph/0404076. ↩

  24. Richard Cleve and Rajat Mittal. Characterization of binary constraint system games. In Automata, Languages, and Programming: 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I 41, 320–331. Springer, 2014. ↩

  25. Richard Cleve, Li Liu, and William Slofstra. Perfect commuting-operator strategies for linear system games. Journal of Mathematical Physics, 2016. URL: https://pubs.aip.org/aip/jmp/article-abstract/58/1/012202/317604/Perfect-commuting-operator-strategies-for-linear?redirectedFrom=fulltext, doi:10.1063/1.4973422. ↩

  26. Daniel Collins and Nicolas Gisin. A relevant two qubit bell inequality inequivalent to the chsh inequality. Journal of Physics A: Mathematical and General, 37(5):1775–1787, jan 2004. URL: http://dx.doi.org/10.1088/0305-4470/37/5/021, doi:10.1088/0305-4470/37/5/021. ↩

  27. Alessandro Cosentino. Positive-partial-transpose-indistinguishable states via semidefinite programming. Physical Review A, Jan 2013. URL: http://dx.doi.org/10.1103/PhysRevA.87.012321, doi:10.1103/physreva.87.012321. ↩

  28. Alessandro Cosentino and Vincent Russo. Small sets of locally indistinguishable orthogonal maximally entangled states. 2014. arXiv:1307.3232. ↩

  29. Alessandro Cosentino. Quantum state local distinguishability via convex optimization. 2015. URL: https://uwspace.uwaterloo.ca/handle/10012/9572. ↩

  30. G Mauro D’Ariano, Massimiliano F Sacchi, and Jonas Kahn. Minimax discrimination of two pauli channels. Physical Review A—Atomic, Molecular, and Optical Physics, 72(5):052302, 2005. arXiv:0507081. ↩

  31. David P. DiVincenzo, Peter W. Shor, John A. Smolin, Barbara M. Terhal, and Ashish V. Thapliyal. Evidence for bound entangled states with negative partial transpose. Physical Review A, May 2000. URL: http://dx.doi.org/10.1103/PhysRevA.61.062312, doi:10.1103/physreva.61.062312. ↩

  32. A. C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. Distinguishing separable and entangled states. Physical Review Letters, Apr 2002. URL: http://dx.doi.org/10.1103/PhysRevLett.88.187904, doi:10.1103/physrevlett.88.187904. ↩

  33. Andrew C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. Complete family of separability criteria. Physical Review A, 69(2):022308, 2004. doi:10.1103/PhysRevA.69.022308. ↩

  34. DSPRelated. Normalized dft. URL: https://www.dsprelated.com/freebooks/mdft/Normalized_DFT.html. ↩

  35. W. Dür, G. Vidal, and J. I. Cirac. Three qubits can be entangled in two inequivalent ways. Physical Review A, Nov 2000. URL: http://dx.doi.org/10.1103/PhysRevA.62.062314, doi:10.1103/physreva.62.062314. ↩

  36. Michael Elad. Sparse and redundant representations: from theory to applications in signal and image processing. Springer Science & Business Media, 2010. ↩

  37. Y.C. Eldar. A semidefinite programming approach to optimal unambiguous discrimination of quantum states. IEEE Transactions on Information Theory, 49(2):446–456, Feb 2003. URL: http://dx.doi.org/10.1109/TIT.2002.807291, doi:10.1109/tit.2002.807291. ↩

  38. Hamza Fawzi, João Gouveia, Pablo A Parrilo, Richard Z Robinson, and Rekha R Thomas. Positive semidefinite rank. Mathematical Programming, 153:133–177, 2015. URL: https://arxiv.org/abs/1407.4095. ↩

  39. N. Gisin. Hidden quantum nonlocality revealed by local filters. Physics Letters A, 210(3):151–156, 1996. doi:https://doi.org/10.1016/S0375-9601(96)80001-6. ↩

  40. Daniel M. Greenberger, Michael A. Horne, and Anton Zeilinger. Going beyond bell's theorem. 2007. arXiv:0712.0921. ↩

  41. Otfried Gühne and Géza Tóth. Entanglement detection. Physics Reports, 474(1):1–75, 2009. URL: https://arxiv.org/abs/0811.2803, doi:https://doi.org/10.1016/j.physrep.2009.02.004. ↩

  42. Tathagata Gupta, Shayeef Murshid, Vincent Russo, and Somshubhro Bandyopadhyay. Optimal discrimination of quantum sequences. arXiv preprint arXiv:2409.08705, 2024. ↩

  43. Tathagata Gupta, Shayeef Murshid, and Somshubhro Bandyopadhyay. Unambiguous discrimination of sequences of quantum states. Physical Review A, may 2024. URL: http://dx.doi.org/10.1103/PhysRevA.109.052222, doi:10.1103/physreva.109.052222. ↩

  44. Leonid Gurvits and Howard Barnum. Largest separable balls around the maximally mixed bipartite quantum state. Physical Review A, Dec 2002. URL: http://dx.doi.org/10.1103/PhysRevA.66.062311, doi:10.1103/physreva.66.062311. ↩

  45. Kil-Chan Ha and Seung-Hyeok Kye. Entanglement witnesses arising from exposed positive linear maps. 2011. arXiv:1108.0130. ↩

  46. Kwang-Chang Ha and Seung-Hyeok Kye. Positive maps and entanglement in qudits. Physical Review A, 84(2):022314, 2011. doi:10.1103/PhysRevA.84.022314. ↩

  47. William Hall. A new criterion for indecomposability of positive maps. Journal of Physics A: Mathematical and General, 39(46):14119, 2006. doi:10.1088/0305-4470/39/45/020. ↩

  48. Patrick Hayden, Kevin Milner, and Mark M. Wilde. Two-message quantum interactive proofs and the quantum separability problem. In 2013 IEEE Conference on Computational Complexity. IEEE, Jun 2013. URL: http://dx.doi.org/10.1109/CCC.2013.24, doi:10.1109/ccc.2013.24. ↩

  49. Teiko Heinosaari and Oskari Kerppo. Antidistinguishability of pure quantum states. Journal of Physics A: Mathematical and Theoretical, 51(36):365303, jul 2018. URL: https://arxiv.org/abs/1804.10457, doi:10.1088/1751-8121/aad1fc. ↩

  50. Teiko Heinosaari, Maria Anastasia Jivulescu, and Ion Nechita. Random positive operator valued measures. Journal of Mathematical Physics, Apr 2020. URL: http://dx.doi.org/10.1063/1.5131028, doi:10.1063/1.5131028. ↩

  51. Teiko Heinosaari and Mark Hillery. Can a qudit carry more information than a dit? arXiv preprint arXiv:2406.16566, 2024. URL: https://arxiv.org/abs/2406.16566. ↩

  52. Roland Hildebrand. Comparison of the ppt cone and the separable cone for 2-by-n systems. 2005. URL: https://membres-ljk.imag.fr/Roland.Hildebrand/coreMPseminar2005_slides.pdf. ↩

  53. Roland Hildebrand. Positive partial transpose from spectra. Phys. Rev. A, 76:052325, Nov 2007. URL: https://link.aps.org/doi/10.1103/PhysRevA.76.052325, doi:10.1103/PhysRevA.76.052325. ↩

  54. Roland Hildebrand. Semidefinite descriptions of low-dimensional separable matrix cones. Linear Algebra and its Applications, 429(4):901–932, 2008. URL: https://www.sciencedirect.com/science/article/pii/S0024379508001973, doi:https://doi.org/10.1016/j.laa.2008.04.018. ↩

  55. Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, 1985. doi:10.1017/CBO9780511810817. ↩

  56. Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Separability of mixed states: necessary and sufficient conditions. Physics Letters A, 223(1):1–8, 1996. URL: https://arxiv.org/abs/quant-ph/9605038, doi:https://doi.org/10.1016/S0375-9601(96)00706-2. ↩

  57. Pawel Horodecki. Separability criterion and inseparable mixed states with positive partial transposition. Physics Letters A, 232(5):333–339, Aug 1997. URL: http://dx.doi.org/10.1016/S0375-9601(97)00416-7, doi:10.1016/s0375-9601(97)00416-7. ↩

  58. Michal Horodecki and Pawel Horodecki. Reduction criterion of separability and limits for a class of protocols of entanglement distillation. 1998. arXiv:quant-ph/9708015. ↩

  59. Paweł Horodecki, Maciej Lewenstein, Guifré Vidal, and Ignacio Cirac. Operational criterion and constructive checks for the separability of low-rank density matrices. Phys. Rev. A, 62:032310, Aug 2000. URL: https://link.aps.org/doi/10.1103/PhysRevA.62.032310, doi:10.1103/PhysRevA.62.032310. ↩

  60. Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Operational criterion for the separability of low-rank density matrices. Physical Review Letters, 84(15):3494, 2000. ↩

  61. Zixin Huang and Mark M. Wilde. Semi-definite optimization of the measured relative entropies of quantum states and channels. 2025. URL: https://arxiv.org/abs/2406.19060, arXiv:2406.19060. ↩

  62. Lane P Hughston, Richard Jozsa, and William K Wootters. A complete classification of quantum ensembles having a given density matrix. Physics Letters A, 183(1):14–18, 1993. ↩

  63. Maria Anastasia Jivulescu, Nicolae Lupa, Ion Nechita, and David Reeb. Positive reduction from spectra. Linear Algebra and its Applications, 469:276–304, 2015. URL: https://www.sciencedirect.com/science/article/pii/S0024379514007666, doi:https://doi.org/10.1016/j.laa.2014.11.031. ↩

  64. Nathaniel Johnston and David W. Kribs. A family of norms with applications in quantum information theory. Journal of Mathematical Physics, Aug 2010. URL: http://dx.doi.org/10.1063/1.3459068, doi:10.1063/1.3459068. ↩

  65. Nathaniel Johnston. Norms and cones in the theory of quantum entanglement. 2012. arXiv:1207.1479. ↩

  66. Nathaniel Johnston. Separability from spectrum for qubit-qudit states. Phys. Rev. A, 88:062330, Dec 2013. URL: https://link.aps.org/doi/10.1103/PhysRevA.88.062330, doi:10.1103/PhysRevA.88.062330. ↩

  67. Nathaniel Johnston. Counting the possible orderings of pairwise multiplication. 2014. URL: http://njohnston.ca/2014/02/counting-the-possible-orderings-of-pairwise-multiplication/. ↩

  68. Nathaniel Johnston, Rajat Mittal, Vincent Russo, and John Watrous. Extended non-local games and monogamy-of-entanglement games. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2189):20160003, May 2016. URL: https://arxiv.org/abs/1510.02083. ↩

  69. Nathaniel Johnston, Shirin Moein, Rajesh Pereira, and Sarah Plosker. Absolutely k-incoherent quantum states and spectral inequalities for the factor width of a matrix. Physical Review A, 106(5):052417, 2022. ↩

  70. Nathaniel Johnston, Benjamin Lovitz, Vincent Russo, and Jamie Sikora. The complexity of quantum state classification. 2025. arXiv:X. ↩

  71. Nathaniel Johnston, Shirin Moein, and Sarah Plosker. The factor width rank of a matrix. Linear Algebra and its Applications, 716:32–59, 2025. doi:10.1016/j.laa.2025.03.016. ↩

  72. Nathaniel Johnston, Vincent Russo, and Jamie Sikora. Tight bounds for antidistinguishability and circulant sets of pure quantum states. Quantum, 9:1622, 2025. ↩

  73. Vishal Katariya and Mark M. Wilde. Geometric distinguishability measures limit quantum channel estimation and discrimination. Quantum Information Processing, Feb 2021. URL: http://dx.doi.org/10.1007/s11128-021-02992-7, doi:10.1007/s11128-021-02992-7. ↩

  74. Sumeet Khatri, Kunal Sharma, and Mark M. Wilde. Information-theoretic aspects of the generalized amplitude-damping channel. Phys. Rev. A, 102:012401, Jul 2020. URL: https://link.aps.org/doi/10.1103/PhysRevA.102.012401, doi:10.1103/PhysRevA.102.012401. ↩

  75. Tamara G. Kolda and Brett W. Bader. Tensor decompositions and applications. SIAM Review, 51(3):455–500, 2009. URL: https://epubs.siam.org/doi/10.1137/07070111X, doi:10.1137/07070111X. ↩

  76. Yeong-Cherng Liang and Andrew C. Doherty. Bounds on quantum correlations in bell-inequality experiments. Physical Review A, Apr 2007. URL: http://dx.doi.org/10.1103/PhysRevA.75.042103, doi:10.1103/physreva.75.042103. ↩

  77. Hoi-Kwong Lo and H. F. Chau. Why quantum bit commitment and ideal quantum coin tossing are impossible. 1997. arXiv:quant-ph/9711065. ↩

  78. Cosmo Lupo, Paolo Aniello, and Antonello Scardicchio. Bipartite quantum systems: on the realignment criterion and beyond. Journal of Physics A: Mathematical and Theoretical, 41(41):415301, Sep 2008. URL: https://arxiv.org/abs/0802.2019. ↩

  79. Keiji Matsumoto. Reverse test and quantum analogue of classical fidelity and generalized fidelity. 2010. arXiv:1006.0302. ↩

  80. Dominic Mayers. Unconditionally secure quantum bit commitment is impossible. Phys. Rev. Lett., 78:3414–3417, Apr 1997. URL: https://link.aps.org/doi/10.1103/PhysRevLett.78.3414, doi:10.1103/PhysRevLett.78.3414. ↩

  81. Caleb McIrvin, Ankith Mohan, and Jamie Sikora. The pretty bad measurement. 2024. arXiv:2403.17252. ↩

  82. J. A. Miszczak, Z. Puchała, P. Horodecki, A. Uhlmann, and K. Życzkowski. Sub– and super–fidelity as bounds for quantum fidelity. 2008. arXiv:0805.2037. ↩

  83. Abel Molina and John Watrous. Hedging bets with correlated quantum strategies. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468(2145):2614–2629, Apr 2012. URL: https://arxiv.org/abs/1104.1140. ↩

  84. Abel Molina, Thomas Vidick, and John Watrous. Optimal counterfeiting attacks and generalizations for wiesner's quantum money. 2012. arXiv:1202.4010. ↩

  85. Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr, Serge Fehr, and Marco Tomamichel. On quantum rényi entropies: a new generalization and some properties. Journal of Mathematical Physics, December 2013. URL: http://dx.doi.org/10.1063/1.4838856, doi:10.1063/1.4838856. ↩

  86. Miguel Navascués, Stefano Pironio, and Antonio Acín. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New Journal of Physics, 10(7):073013, Jul 2008. URL: http://dx.doi.org/10.1088/1367-2630/10/7/073013, doi:10.1088/1367-2630/10/7/073013. ↩

  87. Miguel Navascués. Pure state estimation and the characterization of entanglement. Physical Review Letters, Feb 2008. URL: http://dx.doi.org/10.1103/PhysRevLett.100.070503, doi:10.1103/physrevlett.100.070503. ↩

  88. Miguel Navascues, Gonzalo de la Torre, and Tamas Vertesi. Characterization of quantum correlations with local dimension constraints and its device-independent applications. Physical Review X, Jan 2014. URL: http://dx.doi.org/10.1103/PhysRevX.4.011011, doi:10.1103/PhysRevX.4.011011. ↩

  89. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, USA, 10th edition, 2011. ISBN 1107002176. ↩

  90. Maris Ozols. How to generate a random unitary matrix. 2009. URL: http://home.lu.lv/~sd20008/papers/essays/Random%20unitary%20[paper].pdf. ↩

  91. Asher Peres. Separability criterion for density matrices. Physical Review Letters, 77(8):1413–1415, Aug 1996. URL: http://dx.doi.org/10.1103/PhysRevLett.77.1413, doi:10.1103/physrevlett.77.1413. ↩

  92. Aby Philip, Soorya Rethinasamy, Vincent Russo, and Mark M. Wilde. Schrödinger as a quantum programmer: estimating entanglement via steering. 2023. arXiv:2303.07911. ↩

  93. PlanetMath. Commutant. URL: https://planetmath.org/commutant. ↩

  94. Matthew F. Pusey, Jonathan Barrett, and Terry Rudolph. On the reality of the quantum state. Nature Physics, 8(6):475–478, May 2012. URL: http://dx.doi.org/10.1038/nphys2309, doi:10.1038/nphys2309. ↩

  95. Quantiki. Entanglement of formation. URL: https://www.quantiki.org/wiki/entanglement-formation. ↩

  96. Quantiki. Trace distance. URL: https://www.quantiki.org/wiki/trace-distance. ↩

  97. Quantiki. Trace norm. URL: https://www.quantiki.org/wiki/trace-norm. ↩

  98. Swapan Rana, Preeti Parashar, Andreas Winter, and Maciej Lewenstein. Logarithmic coherence: operational interpretation of \(\\ensuremath \\ell \_1\)-norm coherence. Phys. Rev. A, 96:052336, Nov 2017. URL: https://link.aps.org/doi/10.1103/PhysRevA.96.052336, doi:10.1103/PhysRevA.96.052336. ↩

  99. Rigetti. Forest benchmarking. URL: https://github.com/rigetti/forest-benchmarking. ↩

  100. Vincent Russo. Extended nonlocal games. 2017. arXiv:1704.07375. ↩

  101. Stack Exchange Mathematics. Why do the columns of a unitary matrix form an orthonormal basis? URL: https://math.stackexchange.com/q/1688950. ↩

  102. Akshay Seshadri. Minimax fidelity estimation. URL: https://github.com/akshayseshadri/minimax-fidelity-estimation. ↩

  103. Akshay Seshadri, Martin Ringbauer, Thomas Monz, and Stephen Becker. Theory of versatile fidelity estimation with confidence. 2021. arXiv:2112.07947. ↩

  104. Akshay Seshadri, Martin Ringbauer, Rainer Blatt, Thomas Monz, and Stephen Becker. Versatile fidelity estimation with confidence. 2021. arXiv:2112.07925. ↩

  105. Jamie Sikora. Semidefinite programming in quantum theory (lecture series). Lecture Series, Perimeter Institute for Theoretical Physics; Lecture 2: "Semidefinite programs for nice problems and popular functions", 2019. URL: https://sites.google.com/site/jamiesikora/teaching/sdps-in-quantum-theory. ↩

  106. Stack Overflow Post. Check if a large matrix is diagonal matrix in python. URL: https://stackoverflow.com/questions/43884189/. ↩

  107. Denis Sych and Gerd Leuchs. A complete basis of generalized bell states. New Journal of Physics, 11(1):013006, Jan 2009. URL: https://dx.doi.org/10.1088/1367-2630/11/1/013006, doi:10.1088/1367-2630/11/1/013006. ↩

  108. Barbara M. Terhal, David P. DiVincenzo, and Debbie W. Leung. Hiding bits in bell states. Physical Review Letters, 86(25):5807–5810, jun 2001. URL: http://dx.doi.org/10.1103/PhysRevLett.86.5807, doi:10.1103/physrevlett.86.5807. ↩

  109. Marco Tomamichel, Serge Fehr, Jędrzej Kaniewski, and Stephanie Wehner. A monogamy-of-entanglement game with applications to device-independent quantum cryptography. New Journal of Physics, 15(10):103002, oct 2013. URL: http://dx.doi.org/10.1088/1367-2630/15/10/103002, doi:10.1088/1367-2630/15/10/103002. ↩

  110. Guifré Vidal and Rolf Tarrach. Robustness of entanglement. Phys. Rev. A, 59:141–155, Jan 1999. URL: https://link.aps.org/doi/10.1103/PhysRevA.59.141, doi:10.1103/PhysRevA.59.141. ↩

  111. John Watrous. Semidefinite programs for completely bounded norms. 2009. arXiv:0901.4709. ↩

  112. John Watrous. Theory of quantum information lecture notes. 2011. URL: https://cs.uwaterloo.ca/~watrous/TQI-notes/. ↩

  113. John Watrous. Simpler semidefinite programs for completely bounded norms. 2012. arXiv:1207.5726. ↩

  114. John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. URL: https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf, doi:10.1017/9781316848142. ↩

  115. Reinhard F. Werner. Quantum states with einstein-podolsky-rosen correlations admitting a hidden-variable model. Phys. Rev. A, 40:4277–4281, Oct 1989. URL: https://link.aps.org/doi/10.1103/PhysRevA.40.4277, doi:10.1103/PhysRevA.40.4277. ↩

  116. Stephen Wiesner. Conjugate coding. SIGACT News, 15(1):78–88, Jan 1983. URL: https://doi.org/10.1145/1008908.1008920, doi:10.1145/1008908.1008920. ↩

  117. Wikipedia. Anti-symmetric operator. URL: https://en.wikipedia.org/wiki/Anti-symmetric_operator. ↩

  118. Wikipedia. Bb84. URL: https://en.wikipedia.org/wiki/BB84. ↩

  119. Wikipedia. Bell state. URL: https://en.wikipedia.org/wiki/Bell_state. ↩

  120. Wikipedia. Bra-ket notation. URL: https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation. ↩

  121. Wikipedia. Brauer algebra. URL: https://en.wikipedia.org/wiki/Brauer_algebra. ↩

  122. Wikipedia. Bures distance. URL: https://en.wikipedia.org/wiki/Bures_metric#Bures_distance. ↩

  123. Wikipedia. Circulant matrix. URL: https://en.wikipedia.org/wiki/Circulant_matrix. ↩

  124. Wikipedia. Circular law. URL: https://en.wikipedia.org/wiki/Circular_law. ↩

  125. Wikipedia. Generalizations of pauli matrices. URL: https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices. ↩

  126. Wikipedia. Controlled not gate. URL: https://en.wikipedia.org/wiki/Controlled_NOT_gate. ↩

  127. Wikipedia. Commuting matrices. URL: https://en.wikipedia.org/wiki/Commuting_matrices. ↩

  128. Wikipedia. Comparison matrix. URL: https://en.wikipedia.org/wiki/Comparison_matrix. ↩

  129. Wikipedia. Concurrence (quantum computing). URL: https://en.wikipedia.org/wiki/Concurrence_(quantum_computing). ↩

  130. Wikipedia. Cyclic permutation. URL: https://en.wikipedia.org/wiki/Cyclic_permutation. ↩

  131. Wikipedia. Definite matrix. URL: https://en.wikipedia.org/wiki/Definite_matrix. ↩

  132. Wikipedia. Density matrix. URL: https://en.wikipedia.org/wiki/Density_matrix. ↩

  133. Wikipedia. Quantum depolarizing channel. URL: https://en.wikipedia.org/wiki/Quantum_depolarizing_channel. ↩

  134. Wikipedia. DFT matrix. URL: https://en.wikipedia.org/wiki/DFT_matrix. ↩

  135. Wikipedia. Diagonal matrix. URL: https://en.wikipedia.org/wiki/Diagonal_matrix. ↩

  136. Wikipedia. Diagonally dominant matrix. URL: https://en.wikipedia.org/wiki/Diagonally_dominant_matrix. ↩

  137. Wikipedia. Doubly stochastic matrix. URL: https://en.wikipedia.org/wiki/Doubly_stochastic_matrix. ↩

  138. Wikipedia. Fidelity of quantum states. URL: https://en.wikipedia.org/wiki/Fidelity_of_quantum_states. ↩

  139. Wikipedia. Gell-Mann matrices. URL: https://en.wikipedia.org/wiki/Gell-Mann_matrices. ↩

  140. Wikipedia. Generalizations of Pauli matrices. URL: https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices. ↩

  141. Wikipedia. Gram matrix. URL: https://en.wikipedia.org/wiki/Gram_matrix. ↩

  142. Wikipedia. Hadamard transform. URL: https://en.wikipedia.org/wiki/Hadamard_transform. ↩

  143. Wikipedia. Hermitian matrix. URL: https://en.wikipedia.org/wiki/Hermitian_matrix. ↩

  144. Wikipedia. Hilbert-schmidt operator. URL: https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator. ↩

  145. Wikipedia. Holevoś theorem. URL: https://en.wikipedia.org/wiki/Holevo%27s_theorem. ↩

  146. Wikipedia. Idempotent matrix. URL: https://en.wikipedia.org/wiki/Idempotent_matrix. ↩

  147. Wikipedia. Identity matrix. URL: https://en.wikipedia.org/wiki/Identity_matrix. ↩

  148. Wikipedia. Inner product space. URL: https://en.wikipedia.org/wiki/Inner_product_space. ↩

  149. Wikipedia. Kernel (linear algebra). URL: https://en.wikipedia.org/wiki/Kernel_(linear_algebra). ↩

  150. Wikipedia. Linear independence. URL: https://en.wikipedia.org/wiki/Linear_independence. ↩

  151. Wikipedia. Majorization. URL: https://en.wikipedia.org/wiki/Majorization. ↩

  152. Wikipedia. Quantum entanglement. URL: https://en.wikipedia.org/wiki/Quantum_entanglement. ↩

  153. Wikipedia. Measurement in quantum mechanics. URL: https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics. ↩

  154. Wikipedia. Quantum state - mixed states. URL: https://en.wikipedia.org/wiki/Quantum_state#Mixed_states. ↩

  155. Wikipedia. Mutually unbiased bases. URL: https://en.wikipedia.org/wiki/Mutually_unbiased_bases. ↩

  156. Wikipedia. Mutual coherence (linear algebra). URL: https://en.wikipedia.org/wiki/Mutual_coherence_(linear_algebra). ↩

  157. Wikipedia. Negativity. URL: https://en.wikipedia.org/wiki/Negativity_(quantum_mechanics). ↩

  158. Wikipedia. Nonnegative matrix. URL: https://en.wikipedia.org/wiki/Nonnegative_matrix. ↩

  159. Wikipedia. Normal matrix. URL: https://en.wikipedia.org/wiki/Normal_matrix. ↩

  160. Wikipedia. Orthogonality. URL: https://en.wikipedia.org/wiki/Orthogonality. ↩

  161. Wikipedia. Outer product. URL: https://en.wikipedia.org/wiki/Outer_product. ↩

  162. Wikipedia. Parity of a permutation. URL: https://en.wikipedia.org/wiki/Parity_of_a_permutation. ↩

  163. Wikipedia. Partial trace. URL: https://en.wikipedia.org/wiki/Partial_trace. ↩

  164. Wikipedia. Pauli matrices. URL: https://en.wikipedia.org/wiki/Pauli_matrices. ↩

  165. Wikipedia. Peres-horodecki criterion. URL: https://en.wikipedia.org/wiki/Peres%E2%80%93Horodecki_criterion. ↩

  166. Wikipedia. Permutation matrix. URL: https://en.wikipedia.org/wiki/Permutation_matrix. ↩

  167. Wikipedia. POVM. URL: https://en.wikipedia.org/wiki/POVM. ↩

  168. Wikipedia. Projection matrix. URL: https://en.wikipedia.org/wiki/Projection_matrix. ↩

  169. Wikipedia. Quantum state - pure states. URL: https://en.wikipedia.org/wiki/Quantum_state#Pure_states_of_wave_functions. ↩

  170. Wikipedia. Purity (quantum mechanics). URL: https://en.wikipedia.org/wiki/Purity_(quantum_mechanics). ↩

  171. Wikipedia. Rank (linear algebra). URL: https://en.wikipedia.org/wiki/Rank_(linear_algebra). ↩

  172. Wikipedia. Reduction criterion. URL: https://en.wikipedia.org/wiki/Reduction_criterion. ↩

  173. Wikipedia. Schmidt decomposition. URL: https://en.wikipedia.org/wiki/Schmidt_decomposition. ↩

  174. Wikipedia. Separable state. URL: https://en.wikipedia.org/wiki/Separable_state. ↩

  175. Wikipedia. Skew-hermitian matrix. URL: https://en.wikipedia.org/wiki/Skew-Hermitian_matrix. ↩

  176. Wikipedia. Square matrix. URL: https://en.wikipedia.org/wiki/Square_matrix. ↩

  177. Wikipedia. Stochastic matrix. URL: https://en.wikipedia.org/wiki/Stochastic_matrix. ↩

  178. Wikipedia. Symmetric matrix. URL: https://en.wikipedia.org/wiki/Symmetric_matrix. ↩

  179. Wikipedia. Tensor product. URL: https://en.wikipedia.org/wiki/Tensor_product. ↩

  180. Wikipedia. Totally positive matrix. URL: https://en.wikipedia.org/wiki/Totally_positive_matrix. ↩

  181. Wikipedia. Unitary matrix. URL: https://en.wikipedia.org/wiki/Unitary_matrix. ↩

  182. Wikipedia. Von neumann entropy. URL: https://en.wikipedia.org/wiki/Von_Neumann_entropy. ↩

  183. Jon Yard. Introduction to quantum information processing. URL: https://www.math.uwaterloo.ca/~jyard/qic710/F17/Qic710Lec11-2017.pdf. ↩

  184. Nengkun Yu, Runyao Duan, and Mingsheng Ying. Four locally indistinguishable ququad-ququad orthogonal maximally entangled states. Physical Review Letters, Jul 2012. URL: http://dx.doi.org/10.1103/PhysRevLett.109.020506, doi:10.1103/physrevlett.109.020506. ↩

  185. Cheng-Jie Zhang, Yong-Sheng Zhang, Shun Zhang, and Guang-Can Guo. Entanglement detection beyond the computable cross-norm or realignment criterion. Phys. Rev. A, 77:060301, Jun 2008. URL: https://link.aps.org/doi/10.1103/PhysRevA.77.060301, doi:10.1103/PhysRevA.77.060301. ↩