Naturally restricted subsets of nonsignaling correlations: typicality and convergence
1Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan
2MTA Atomki Lendület Quantum Correlations Research Group, Institute for Nuclear Research, P.O. Box 51, H-4001 Debrecen, Hungary
3Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
Published: | 2022-07-14, volume 6, page 765 |
Eprint: | arXiv:2107.05646v4 |
Doi: | https://doi.org/10.22331/q-2022-07-14-765 |
Citation: | Quantum 6, 765 (2022). |
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Abstract
It is well-known that in a Bell experiment, the observed correlation between measurement outcomes – as predicted by quantum theory – can be stronger than that allowed by local causality, yet not fully constrained by the principle of relativistic causality. In practice, the characterization of the set $Q$ of quantum correlations is carried out, often, through a converging hierarchy of outer approximations. On the other hand, some subsets of $Q$ arising from additional constraints [e.g., originating from quantum states having positive-partial-transposition (PPT) or being finite-dimensional maximally entangled (MES)] turn out to be also amenable to similar numerical characterizations. How, then, at a quantitative level, are all these naturally restricted subsets of nonsignaling correlations different? Here, we consider several bipartite Bell scenarios and numerically estimate their volume relative to that of the set of nonsignaling correlations. Within the number of cases investigated, we have observed that (1) for a given number of inputs $n_s$ (outputs $n_o$), the relative volume of both the Bell-local set and the quantum set increases (decreases) rapidly with increasing $n_o$ ($n_s$) (2) although the so-called macroscopically local set $Q_1$ may approximate $Q$ well in the two-input scenarios, it can be a very poor approximation of the quantum set when $n_s$$\gt$$n_o$ (3) the almost-quantum set $\tilde{Q}_1$ is an exceptionally-good approximation to the quantum set (4) the difference between $Q$ and the set of correlations originating from MES is most significant when $n_o=2$, whereas (5) the difference between the Bell-local set and the PPT set generally becomes more significant with increasing $n_o$. This last comparison, in particular, allows us to identify Bell scenarios where there is little hope of realizing the Bell violation by PPT states and those that deserve further exploration.
![](https://quantum-journal.org/wp-content/uploads/2022/07/CorrelationsSets.jpg)
Featured image: Illustration of the various naturally restricted subsets of the nonsignaling set $\mathcal{NS}$ of correlations and their inclusion relations. In solid lines and moving inwards, we have, respectively, the boundary of the quantum set of correlations $\mathcal{Q}$ (blue), the convex hull $\mathcal{M}^\text{P}$ (red) of the set of correlations attainable by finite-dimensional maximally entangled states in conjunction with projective measurements, the set of correlations attainable by positive-partial-transposed states $\mathcal{P}$ (green), and the Bell-local polytope $\mathcal{L}$ (sky-blue). The dashed (pink) and dotted (brown) lines that lie between the boundary of $\mathcal{Q}$ and that of $\mathcal{NS}$ mark the border of the lowest-level outer approximation of $\mathcal{Q}$, respectively, due to Navascués-Pironio-Acín ($\mathcal{Q}_1$) and Moroder-Bancal-Liang-Hofmann-Gühne ($\tilde{\mathcal{Q}}_1)$. Accordingly, these are also known as the macroscopically-local set and the almost-quantum set of correlations.
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► References
[1] A. Acín. Statistical distinguishability between unitary operations. Phys. Rev. Lett., 87: 177901, Oct 2001. 10.1103/PhysRevLett.87.177901.
https://doi.org/10.1103/PhysRevLett.87.177901
[2] Antonio Acín. (private communication).
[3] Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani. Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett., 98: 230501, Jun 2007. 10.1103/PhysRevLett.98.230501.
https://doi.org/10.1103/PhysRevLett.98.230501
[4] Rotem Arnon-Friedman and Jean-Daniel Bancal. Device-independent certification of one-shot distillable entanglement. New J. Phys., 21 (3): 033010, 2019. 10.1088/1367-2630/aafef6.
https://doi.org/10.1088/1367-2630/aafef6
[5] Davd Avis. lrs: A revised implementation of the reverse search vertex enumeration algorithm. (unpublished), 1999. URL http://cgm.cs.mcgill.ca/ avis/doc/avis/Av98a.pdf.
http://cgm.cs.mcgill.ca/~avis/doc/avis/Av98a.pdf
[6] Jean-Daniel Bancal, Nicolas Gisin, Yeong-Cherng Liang, and Stefano Pironio. Device-independent witnesses of genuine multipartite entanglement. Phys. Rev. Lett., 106: 250404, Jun 2011. 10.1103/PhysRevLett.106.250404.
https://doi.org/10.1103/PhysRevLett.106.250404
[7] Jean-Daniel Bancal, Nicolas Sangouard, and Pavel Sekatski. Noise-resistant device-independent certification of Bell state measurements. Phys. Rev. Lett., 121: 250506, Dec 2018. 10.1103/PhysRevLett.121.250506.
https://doi.org/10.1103/PhysRevLett.121.250506
[8] Tomer Jack Barnea, Jean-Daniel Bancal, Yeong-Cherng Liang, and Nicolas Gisin. Tripartite quantum state violating the hidden-influence constraints. Phys. Rev. A, 88: 022123, Aug 2013. 10.1103/PhysRevA.88.022123.
https://doi.org/10.1103/PhysRevA.88.022123
[9] Jonathan Barrett. Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality. Phys. Rev. A, 65: 042302, Mar 2002. 10.1103/PhysRevA.65.042302.
https://doi.org/10.1103/PhysRevA.65.042302
[10] Jonathan Barrett, Noah Linden, Serge Massar, Stefano Pironio, Sandu Popescu, and David Roberts. Nonlocal correlations as an information-theoretic resource. Phys. Rev. A, 71: 022101, Feb 2005. 10.1103/PhysRevA.71.022101.
https://doi.org/10.1103/PhysRevA.71.022101
[11] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics, 1: 195–200, Nov 1964. 10.1103/PhysicsPhysiqueFizika.1.195.
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
[12] J. S. Bell. Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy. Cambridge University Press, 2 edition, 2004. 10.1017/CBO9780511815676.
https://doi.org/10.1017/CBO9780511815676
[13] Tim Benham. Uniform distribution over a convex polytope. MATLAB central file exchange, 2014. URL https://www.mathworks.com/matlabcentral/fileexchange/34208-uniform-distribution-over-a-convex-polytope.
https://www.mathworks.com/matlabcentral/fileexchange/34208-uniform-distribution-over-a-convex-polytope
[14] Mario Berta, Omar Fawzi, and Volkher B. Scholz. Quantum bilinear optimization. Siam J. Optim., 26 (3): 1529–1564, 2020/04/04 2016. 10.1137/15M1037731.
https://doi.org/10.1137/15M1037731
[15] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, Cambridge, 1 edition, 2004.
[16] Gilles Brassard, Harry Buhrman, Noah Linden, André Allan Méthot, Alain Tapp, and Falk Unger. Limit on nonlocality in any world in which communication complexity is not trivial. Phys. Rev. Lett., 96: 250401, Jun 2006. 10.1103/PhysRevLett.96.250401.
https://doi.org/10.1103/PhysRevLett.96.250401
[17] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86: 419–478, Apr 2014. 10.1103/RevModPhys.86.419.
https://doi.org/10.1103/RevModPhys.86.419
[18] Benno Büeler, Andreas Enge, and Komei Fukuda. Exact Volume Computation for Polytopes: A Practical Study, pages 131–154. Birkhäuser Basel, Basel, 2000. ISBN 978-3-0348-8438-9. 10.1007/978-3-0348-8438-9_6.
https://doi.org/10.1007/978-3-0348-8438-9_6
[19] Adán Cabello. How much larger quantum correlations are than classical ones. Phys. Rev. A, 72: 012113, Jul 2005. 10.1103/PhysRevA.72.012113.
https://doi.org/10.1103/PhysRevA.72.012113
[20] Shin-Liang Chen, Costantino Budroni, Yeong-Cherng Liang, and Yueh-Nan Chen. Natural framework for device-independent quantification of quantum steerability, measurement incompatibility, and self-testing. Phys. Rev. Lett., 116: 240401, Jun 2016. 10.1103/PhysRevLett.116.240401.
https://doi.org/10.1103/PhysRevLett.116.240401
[21] Shin-Liang Chen, Costantino Budroni, Yeong-Cherng Liang, and Yueh-Nan Chen. Exploring the framework of assemblage moment matrices and its applications in device-independent characterizations. Phys. Rev. A, 98: 042127, Oct 2018a. 10.1103/PhysRevA.98.042127.
https://doi.org/10.1103/PhysRevA.98.042127
[22] Shin-Liang Chen, Costantino Budroni, Yeong-Cherng Liang, and Yueh-Nan Chen. Exploring the framework of assemblage moment matrices and its applications in device-independent characterizations. Phys. Rev. A, 98: 042127, Oct 2018b. 10.1103/PhysRevA.98.042127.
https://doi.org/10.1103/PhysRevA.98.042127
[23] Shin-Liang Chen, Nikolai Miklin, Costantino Budroni, and Yueh-Nan Chen. Device-independent quantification of measurement incompatibility. Phys. Rev. Research, 3: 023143, May 2021. 10.1103/PhysRevResearch.3.023143.
https://doi.org/10.1103/PhysRevResearch.3.023143
[24] Bradley G. Christensen, Yeong-Cherng Liang, Nicolas Brunner, Nicolas Gisin, and Paul G. Kwiat. Exploring the limits of quantum nonlocality with entangled photons. Phys. Rev. X, 5: 041052, Dec 2015. 10.1103/PhysRevX.5.041052.
https://doi.org/10.1103/PhysRevX.5.041052
[25] Andrea Coladangelo and Jalex Stark. An inherently infinite-dimensional quantum correlation. Nat. Commun., 11 (1): 3335, 2020. 10.1038/s41467-020-17077-9.
https://doi.org/10.1038/s41467-020-17077-9
[26] Roger Colbeck. Quantum And Relativistic Protocols For Secure Multi-Party Computation. PhD thesis, University of Cambridge, 2006. URL https://doi.org/10.48550/arXiv.0911.3814.
https://doi.org/10.48550/arXiv.0911.3814
[27] Daniel Collins and Nicolas Gisin. A relevant two qubit Bell inequality inequivalent to the CHSH inequality. J. Phys. A: Math. Theo., 37 (5): 1775, 2004. 10.1088/0305-4470/37/5/021.
https://doi.org/10.1088/0305-4470/37/5/021
[28] Florian John Curchod, Nicolas Gisin, and Yeong-Cherng Liang. Quantifying multipartite nonlocality via the size of the resource. Phys. Rev. A, 91: 012121, Jan 2015. 10.1103/PhysRevA.91.012121.
https://doi.org/10.1103/PhysRevA.91.012121
[29] Andrew C. Doherty, Yeong-Cherng Liang, Ben Toner, and Stephanie Wehner. The quantum moment problem and bounds on entangled multi-prover games. In 23rd Annu. IEEE Conf. on Comput. Comp, 2008, CCC'08, pages 199–210, Los Alamitos, CA, 2008. 10.1109/CCC.2008.26.
https://doi.org/10.1109/CCC.2008.26
[30] Cristhiano Duarte, Samuraí Brito, Barbara Amaral, and Rafael Chaves. Concentration phenomena in the geometry of Bell correlations. Phys. Rev. A, 98: 062114, Dec 2018. 10.1103/PhysRevA.98.062114.
https://doi.org/10.1103/PhysRevA.98.062114
[31] Arthur Fine. Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett., 48: 291–295, Feb 1982. 10.1103/PhysRevLett.48.291.
https://doi.org/10.1103/PhysRevLett.48.291
[32] T. Fritz, A. B. Sainz, R. Augusiak, J. Bohr Brask, R. Chaves, A. Leverrier, and A. Acín. Local orthogonality as a multipartite principle for quantum correlations. Nat. Commun., 4 (1): 2263, 2013. ISSN 2041-1723. 10.1038/ncomms3263.
https://doi.org/10.1038/ncomms3263
[33] Koon Tong Goh, Jędrzej Kaniewski, Elie Wolfe, Tamás Vértesi, Xingyao Wu, Yu Cai, Yeong-Cherng Liang, and Valerio Scarani. Geometry of the set of quantum correlations. Phys. Rev. A, 97: 022104, Feb 2018. 10.1103/PhysRevA.97.022104.
https://doi.org/10.1103/PhysRevA.97.022104
[34] Tomáš Gonda, Ravi Kunjwal, David Schmid, Elie Wolfe, and Ana Belén Sainz. Almost Quantum Correlations are Inconsistent with Specker's Principle. Quantum, 2: 87, August 2018. ISSN 2521-327X. 10.22331/q-2018-08-27-87.
https://doi.org/10.22331/q-2018-08-27-87
[35] Lucien Hardy. Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett., 71: 1665–1668, Sep 1993. 10.1103/PhysRevLett.71.1665.
https://doi.org/10.1103/PhysRevLett.71.1665
[36] Aram W. Harrow, Anand Natarajan, and Xiaodi Wu. Limitations of semidefinite programs for separable states and entangled games. Commun. Math. Phys., 366 (2): 423–468, 2019. ISSN 1432-0916. 10.1007/s00220-019-03382-y.
https://doi.org/10.1007/s00220-019-03382-y
[37] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Mixed-state entanglement and distillation: Is there a ``bound'' entanglement in nature? Phys. Rev. Lett., 80: 5239–5242, Jun 1998. 10.1103/PhysRevLett.80.5239.
https://doi.org/10.1103/PhysRevLett.80.5239
[38] M. Junge and C. Palazuelos. Large violation of bell inequalities with low entanglement. Commun. Math. Phys., 306 (3): 695, 2011. 10.1007/s00220-011-1296-8.
https://doi.org/10.1007/s00220-011-1296-8
[39] Ben Lang, Tamás Vértesi, and Miguel Navascués. Closed sets of correlations: answers from the zoo. J. Phys. A Math. Theor., 47 (42): 424029, oct 2014. 10.1088/1751-8113/47/42/424029.
https://doi.org/10.1088/1751-8113/47/42/424029
[40] Yeong-Cherng Liang, Tamás Vértesi, and Nicolas Brunner. Semi-device-independent bounds on entanglement. Phys. Rev. A, 83: 022108, Feb 2011. 10.1103/PhysRevA.83.022108.
https://doi.org/10.1103/PhysRevA.83.022108
[41] Yeong-Cherng Liang, Denis Rosset, Jean-Daniel Bancal, Gilles Pütz, Tomer Jack Barnea, and Nicolas Gisin. Family of Bell-like inequalities as device-independent witnesses for entanglement depth. Phys. Rev. Lett., 114: 190401, May 2015. 10.1103/PhysRevLett.114.190401.
https://doi.org/10.1103/PhysRevLett.114.190401
[42] Noah Linden, Sandu Popescu, Anthony J. Short, and Andreas Winter. Quantum nonlocality and beyond: Limits from nonlocal computation. Phys. Rev. Lett., 99: 180502, Oct 2007. 10.1103/PhysRevLett.99.180502.
https://doi.org/10.1103/PhysRevLett.99.180502
[43] He Lu, Qi Zhao, Zheng-Da Li, Xu-Fei Yin, Xiao Yuan, Jui-Chen Hung, Luo-Kan Chen, Li Li, Nai-Le Liu, Cheng-Zhi Peng, Yeong-Cherng Liang, Xiongfeng Ma, Yu-Ao Chen, and Jian-Wei Pan. Entanglement structure: Entanglement partitioning in multipartite systems and its experimental detection using optimizable witnesses. Phys. Rev. X, 8: 021072, Jun 2018. 10.1103/PhysRevX.8.021072.
https://doi.org/10.1103/PhysRevX.8.021072
[44] Dominic Mayers and Andrew Yao. Self testing quantum apparatus. Quantum Info. Comput., 4 (4): 273–286, July 2004. ISSN 1533-7146. URL http://dl.acm.org/citation.cfm?id=2011827.2011830.
http://dl.acm.org/citation.cfm?id=2011827.2011830
[45] Tobias Moroder, Jean-Daniel Bancal, Yeong-Cherng Liang, Martin Hofmann, and Otfried Gühne. Device-independent entanglement quantification and related applications. Phys. Rev. Lett., 111: 030501, Jul 2013. 10.1103/PhysRevLett.111.030501.
https://doi.org/10.1103/PhysRevLett.111.030501
[46] Miguel Navascués and Harald Wunderlich. A glance beyond the quantum model. Proc. R. Soc. A, 466: 881, Nov 2009. URL https://doi.org/10.1098/rspa.2009.0453.
https://doi.org/10.1098/rspa.2009.0453
[47] Miguel Navascués, Stefano Pironio, and Antonio Acín. Bounding the set of quantum correlations. Phys. Rev. Lett., 98: 010401, Jan 2007. 10.1103/PhysRevLett.98.010401.
https://doi.org/10.1103/PhysRevLett.98.010401
[48] Miguel Navascués, Stefano Pironio, and Antonio Acín. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys., 10 (7): 073013, 2008. 10.1088/1367-2630/10/7/073013.
https://doi.org/10.1088/1367-2630/10/7/073013
[49] Miguel Navascués, Yelena Guryanova, Matty J. Hoban, and Antonio Acín. Almost quantum correlations. Nat. Commun., 6: 6288, 2015. https://doi.org/10.1038/ncomms7288.
https://doi.org/10.1038/ncomms7288
[50] Marcin Pawlowski, Tomasz Paterek, Dagomir Kaszlikowski, Valerio Scarani, Andreas Winter, and Marek Zukowski. Information causality as a physical principle. Nature, 461 (7267): 1101–1104, 2009. ISSN 1476-4687. 10.1038/nature08400.
https://doi.org/10.1038/nature08400
[51] Asher Peres. Neumark's theorem and quantum inseparability. Found. Phys., 20 (12): 1441–1453, 1990. 10.1007/BF01883517.
https://doi.org/10.1007/BF01883517
[52] Asher Peres. Separability criterion for density matrices. Phys. Rev. Lett., 77: 1413–1415, Aug 1996. 10.1103/PhysRevLett.77.1413.
https://doi.org/10.1103/PhysRevLett.77.1413
[53] Asher Peres. All the Bell inequalities. Found. Phys., 29 (4): 589–614, 1999. 10.1023/A:1018816310000.
https://doi.org/10.1023/A:1018816310000
[54] S. Pironio, A. Acín, S. Massar, A. Boyer de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe. Random numbers certified by Bell's theorems theorem. Nature (London), 464: 1021, April 2010. 10.1038/nature09008.
https://doi.org/10.1038/nature09008
[55] Itamar Pitowsky. Quantum Probability - Quantum Logic. Springer, Berlin, 1989.
[56] Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Found. Phys., 24 (3): 379–385, Mar 1994. ISSN 1572-9516. 10.1007/BF02058098.
https://doi.org/10.1007/BF02058098
[57] Rafael Rabelo, Melvyn Ho, Daniel Cavalcanti, Nicolas Brunner, and Valerio Scarani. Device-independent certification of entangled measurements. Phys. Rev. Lett., 107: 050502, Jul 2011. 10.1103/PhysRevLett.107.050502.
https://doi.org/10.1103/PhysRevLett.107.050502
[58] Valerio Scarani. The device-independent outlook on quantum physics. Acta Physica Slovaca, 62 (4): 347, 2012.
[59] Pavel Sekatski, Jean-Daniel Bancal, Sebastian Wagner, and Nicolas Sangouard. Certifying the building blocks of quantum computers from Bell's theorem. Phys. Rev. Lett., 121: 180505, Nov 2018. 10.1103/PhysRevLett.121.180505.
https://doi.org/10.1103/PhysRevLett.121.180505
[60] Jamie Sikora and Antonios Varvitsiotis. Linear conic formulations for two-party correlations and values of nonlocal games. Math. Program., Ser. A, 162 (1): 431–463, 2017. 10.1007/s10107-016-1049-8.
https://doi.org/10.1007/s10107-016-1049-8
[61] William Slofstra. The set of quantum correlations is not closed. Forum of Mathematics, Pi, 7: e1, 2019. 10.1017/fmp.2018.3.
https://doi.org/10.1017/fmp.2018.3
[62] William Slofstra. Tsirelson's problem and an embedding theorem for groups arising from non-local games. J. Amer. Math. Soc., 33: 1–56, 2020. 10.1090/jams/929.
https://doi.org/10.1090/jams/929
[63] James Vallins, Ana Belén Sainz, and Yeong-Cherng Liang. Almost-quantum correlations and their refinements in a tripartite Bell scenario. Phys. Rev. A, 95: 022111, Feb 2017. 10.1103/PhysRevA.95.022111.
https://doi.org/10.1103/PhysRevA.95.022111
[64] Tamás Vértesi and Nicolas Brunner. Quantum nonlocality does not imply entanglement distillability. Phys. Rev. Lett., 108: 030403, Jan 2012. 10.1103/PhysRevLett.108.030403.
https://doi.org/10.1103/PhysRevLett.108.030403
[65] Tamas Vertesi and Nicolas Brunner. Disproving the Peres conjecture by showing Bell nonlocality from bound entanglement. Nat. Commun., 5: 5297, 05 2014. 10.1038/ncomms6297.
https://doi.org/10.1038/ncomms6297
[66] Thomas Vidick and Stephanie Wehner. More nonlocality with less entanglement. Phys. Rev. A, 83: 052310, May 2011. 10.1103/PhysRevA.83.052310.
https://doi.org/10.1103/PhysRevA.83.052310
[67] Ivan Šupić and Joseph Bowles. Self-testing of quantum systems: a review. Quantum, 4: 337, Sep 2020. ISSN 2521-327X. 10.22331/q-2020-09-30-337.
https://doi.org/10.22331/q-2020-09-30-337
[68] Sebastian Wagner, Jean-Daniel Bancal, Nicolas Sangouard, and Pavel Sekatski. Device-independent characterization of quantum instruments. Quantum, 4: 243, March 2020. ISSN 2521-327X. 10.22331/q-2020-03-19-243.
https://doi.org/10.22331/q-2020-03-19-243
[69] R. F. Werner and M. M. Wolf. Bell's inequalities for states with positive partial transpose. Phys. Rev. A, 61: 062102, May 2000. 10.1103/PhysRevA.61.062102.
https://doi.org/10.1103/PhysRevA.61.062102
[70] R. F. Werner and M. M. Wolf. All-multipartite Bell-correlation inequalities for two dichotomic observables per site. Phys. Rev. A, 64: 032112, Aug 2001. 10.1103/PhysRevA.64.032112.
https://doi.org/10.1103/PhysRevA.64.032112
[71] Reinhard F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A, 40: 4277–4281, Oct 1989. 10.1103/PhysRevA.40.4277.
https://doi.org/10.1103/PhysRevA.40.4277
[72] Edwin B. Wilson. Probable inference, the law of succession, and statistical inference. J. Amer. Statist. Assoc, 22 (158): 209–212, 06 1927. 10.1080/01621459.1927.10502953.
https://doi.org/10.1080/01621459.1927.10502953
[73] H M Wiseman. The two Bell's theorems of John Bell. J. Phys. A Math. Theor., 47 (42): 424001, 2014. 10.1088/1751-8113/47/42/424001.
https://doi.org/10.1088/1751-8113/47/42/424001
[74] Peter Wittek. Algorithm 950: Ncpol2sdpa—sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables. ACM Trans. Math. Softw., 41 (3), jun 2015. ISSN 0098-3500. 10.1145/2699464.
https://doi.org/10.1145/2699464
[75] Elie Wolfe and S. F. Yelin. Quantum bounds for inequalities involving marginal expectation values. Phys. Rev. A, 86: 012123, Jul 2012. 10.1103/PhysRevA.86.012123.
https://doi.org/10.1103/PhysRevA.86.012123
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