References

Articles

  1. 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.

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

  3. Nathaniel Johnston. QETLAB: A MATLAB toolbox for quantum entanglement. https://github.com/nathanieljohnston/QETLAB. doi:10.5281/zenodo.44637.

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

  5. Rigetti. Forest benchmarking. https://github.com/rigetti/forest-benchmarking.

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

  7. 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.

  8. 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.

  9. 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.

  10. Wikipedia. Quantum depolarizing channel. https://en.wikipedia.org/wiki/Quantum_depolarizing_channel.

  11. Wikipedia. Partial trace. https://en.wikipedia.org/wiki/Partial_trace.

  12. Wikipedia. Peres-horodecki criterion. https://en.wikipedia.org/wiki/Peres.

  13. 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.

  14. Wikipedia. Reduction criterion. https://en.wikipedia.org/wiki/Reduction_criterion.

  15. Riley J Murray. PR: have kron support non-constant expressions in either argument from CXPY: a python-embedded modeling language for convex optimization problems. https://github.com/cvxpy/cvxpy/issues/457#issue-309891424.

  16. 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.

  17. Wikipedia. Controlled not gate. https://en.wikipedia.org/wiki/Controlled_NOT_gate.

  18. Wikipedia. Cyclic permutation. https://en.wikipedia.org/wiki/Cyclic_permutation.

  19. Wikipedia. DFT matrix. https://en.wikipedia.org/wiki/DFT_matrix.

  20. Wikipedia. Gell-Mann matrices. https://en.wikipedia.org/wiki/Gell-Mann_matrices.

  21. Wikipedia. Generalizations of Pauli matrices. https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices.

  22. John Watrous. Theory of quantum information lecture notes. https://cs.uwaterloo.ca/ watrous/TQI-notes/, 2011.

  23. Wikipedia. Generalizations of pauli matrices. https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices.

  24. Wikipedia. Hadamard transform. https://en.wikipedia.org/wiki/Hadamard_transform.

  25. Wikipedia. Pauli matrices. https://en.wikipedia.org/wiki/Pauli_matrices.

  26. Akshay Seshadri. Minimax fidelity estimation. https://github.com/akshayseshadri/minimax-fidelity-estimation.

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

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

  29. Wikipedia. Inner product space. https://en.wikipedia.org/wiki/Inner_product_space.

  30. Wikipedia. Outer product. https://en.wikipedia.org/wiki/Outer_product.

  31. Wikipedia. Tensor product. https://en.wikipedia.org/wiki/Tensor_product.

  32. Wikipedia. Gram matrix. https://en.wikipedia.org/wiki/Gram_matrix.

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

  34. Wikipedia. Circulant matrix. https://en.wikipedia.org/wiki/Circulant_matrix.

  35. Wikipedia. Commuting matrices. https://en.wikipedia.org/wiki/Commuting_matrices.

  36. Wikipedia. Density matrix. https://en.wikipedia.org/wiki/Density_matrix.

  37. Wikipedia. Diagonal matrix. https://en.wikipedia.org/wiki/Diagonal_matrix.

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

  39. Wikipedia. Diagonally dominant matrix. https://en.wikipedia.org/wiki/Diagonally_dominant_matrix.

  40. Wikipedia. Hermitian matrix. https://en.wikipedia.org/wiki/Hermitian_matrix.

  41. Wikipedia. Idempotent matrix. https://en.wikipedia.org/wiki/Idempotent_matrix.

  42. Wikipedia. Identity matrix. https://en.wikipedia.org/wiki/Identity_matrix.

  43. Wikipedia. Linear independence. https://en.wikipedia.org/wiki/Linear_independence.

  44. Wikipedia. Normal matrix. https://en.wikipedia.org/wiki/Normal_matrix.

  45. Wikipedia. Permutation matrix. https://en.wikipedia.org/wiki/Permutation_matrix.

  46. Wikipedia. Definite matrix. https://en.wikipedia.org/wiki/Definite_matrix.

  47. Wikipedia. Projection matrix. https://en.wikipedia.org/wiki/Projection_matrix.

  48. Wikipedia. Square matrix. https://en.wikipedia.org/wiki/Square_matrix.

  49. Wikipedia. Symmetric matrix. https://en.wikipedia.org/wiki/Symmetric_matrix.

  50. Wikipedia. Totally positive matrix. https://en.wikipedia.org/wiki/Totally_positive_matrix.

  51. Wikipedia. Unitary matrix. https://en.wikipedia.org/wiki/Unitary_matrix.

  52. Wikipedia. Majorization. https://en.wikipedia.org/wiki/Majorization.

  53. 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.

  54. Quantiki. Trace norm. https://www.quantiki.org/wiki/trace-norm.

  55. 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.

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

  57. Wikipedia. Measurement in quantum mechanics. https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics.

  58. Wikipedia. POVM. https://en.wikipedia.org/wiki/POVM.

  59. 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.

  60. Vincent Russo. Extended nonlocal games. 2017. arXiv:1704.07375.

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

  62. 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.

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

  64. 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.

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

  66. Wikipedia. Anti-symmetric operator. https://en.wikipedia.org/wiki/Anti-symmetric_operator.

  67. Wikipedia. Parity of a permutation. https://en.wikipedia.org/wiki/Parity_of_a_permutation.

  68. 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.

  69. Wikipedia. Circulant matrix. https://en.wikipedia.org/wiki/Circulant_matrix.

  70. Stanford University CCRMA. Normalized dft. https://ccrma.stanford.edu/ jos/st/Normalized_DFT.html.

  71. Wikipedia. Circular law. https://en.wikipedia.org/wiki/Circular_law.

  72. 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.

  73. Maris Ozols. How to generate a random unitary matrix. http://home.lu.lv/ sd20008/papers/essays/Random, 2009.

  74. Wikipedia. Bures distance. https://en.wikipedia.org/wiki/Bures_metric#Bures_distance.

  75. Wikipedia. Fidelity of quantum states. https://en.wikipedia.org/wiki/Fidelity_of_quantum_states.

  76. Wikipedia. Holevoś theorem. https://en.wikipedia.org/wiki/Holevo.

  77. Wikipedia. Hilbert-schmidt operator. https://en.wikipedia.org/wiki/Hilbert.

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

  79. 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.

  80. Quantiki. Trace distance. https://www.quantiki.org/wiki/trace-distance.

  81. Wikipedia. Schmidt decomposition. https://en.wikipedia.org/wiki/Schmidt_decomposition.

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

  83. 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.

  84. 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.

  85. 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.

  86. 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.

  87. 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.

  88. 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.

  89. 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.

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

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

  92. Quantiki. Entanglement of formation. https://www.quantiki.org/wiki/entanglement-formation.

  93. 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.

  94. 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.

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

  96. Wikipedia. Orthogonality. https://en.wikipedia.org/wiki/Orthogonality.

  97. Wikipedia. Mutually unbiased bases. hhttps://en.wikipedia.org/wiki/Mutually_unbiased_bases.

  98. 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.

  99. Wikipedia. Separable state. https://en.wikipedia.org/wiki/Separable_state.

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

  101. 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.

  102. 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.

  103. Wikipedia. Negativity. https://en.wikipedia.org/wiki/Negativity_(quantum_mechanics).

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

  105. Wikipedia. Von neumann entropy. https://en.wikipedia.org/wiki/Von_Neumann_entropy.

  106. Wikipedia. Bra-ket notation. https://en.wikipedia.org/wiki/Bra.

  107. Wikipedia. Bb84. https://en.wikipedia.org/wiki/BB84.

  108. Wikipedia. Bell state. https://en.wikipedia.org/wiki/Bell_state.

  109. Wikipedia. Brauer algebra. https://en.wikipedia.org/wiki/Brauer_algebra.

  110. 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.

  111. 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.

  112. 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.

  113. 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.

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

  115. 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.

  116. 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.

  117. Dariusz Chruściński and Andrzej Kossakowski. On the symmetry of the seminal horodecki state. Physics Letters A, 375(3):434–436, Jan 2011. URL: http://dx.doi.org/10.1016/j.physleta.2010.11.069, doi:10.1016/j.physleta.2010.11.069.

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

  119. Wikipedia. Quantum entanglement. https://en.wikipedia.org/wiki/Quantum_entanglement.

  120. Scott Aaronson. Lecture 6: mixed states. https://www.scottaaronson.com/qclec/6.pdf.

  121. 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.

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

  123. 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.

  124. 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, June 2015. URL: http://dx.doi.org/10.1109/TIT.2015.2417755, doi:10.1109/tit.2015.2417755.

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

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

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

Books

  1. John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. doi:10.1017/9781316848142.