Two-Qubit Pure Entanglement as Optimal Social Welfare Resource in Bayesian Game

Manik Banik1, Some Sankar Bhattacharya2, Nirman Ganguly3, Tamal Guha4, Amit Mukherjee5, Ashutosh Rai6,7,8, and Arup Roy1

1S.N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700098, India.
2Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong.
3Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad campus, Telengana 500078,India.
4Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India.
5Optics and Quantum Information Group, The Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai 600113, India.
6School of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea.
7International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil.
8Centre for Quantum Computer Science, University of Latvia, Raina Bulv. 19, Riga, LV-1586, Latvia.

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Entanglement is of paramount importance in quantum information theory. Its supremacy over classical correlations has been demonstrated in a numerous information theoretic protocols. Here we study possible adequacy of quantum entanglement in Bayesian game theory, particularly in social welfare solution (SWS), a strategy which the players follow to maximize sum of their payoffs. Given a multi-partite quantum state as an advice, players can come up with several correlated strategies by performing local measurements on their parts of the quantum state. A quantum strategy is called quantum-SWS if it is advantageous over a classical equilibrium (CE) strategy in the sense that none of the players has to sacrifice their CE-payoff rather some have incentive and at the same time it maximizes sum of all players' payoffs over all possible quantum advantageous strategies. Quantum state yielding such a quantum-SWS is called a quantum social welfare advice (SWA). We show that any two-qubit pure entangled state, even if it is arbitrarily close to a product state, can serve as quantum-SWA in some Bayesian game. Our result, thus, gives cognizance to the fact that every two-qubit pure entanglement is the best resource for some operational task.

Quantum entanglement: a social welfare advice in Bayesian game.

Game theory - the study of human conflict and cooperation within a competitive situation – has found applications in a wide range of studies starting from economics, political sciences, biological phenomena, logic, computer science and psychology. The seminal Nash theorem assures the existence of strategies, the so called Nash equilibrium, adopted by the players from which none of the players has any incentive to shift those. In a Bayesian scenario, where each player has some private information unknown to the other players, Robert Aumann proved that the proper notion of equilibrium is not the ordinary Nash equilibrium but a more general – correlated equilibrium. In Kenneth Arrow’s social choice theory it becomes apparent that instead of pursuing solely their own payoffs, considering additional social goals can benefit a person.

Quantum mechanics – the most precise theory of microscopic world - predicts a vast range of astonishing and often strikingly counterintuitive phenomena, such as quantum entanglement. Entanglement captures the physical phenomenon involving composite quantum systems where the quantum states of the subsystems cannot be described independently even when the subsystems are separated by a large distance. Apart from several underlying foundational puzzles of quantum theory, such as, Einstein–Podolsky–Rosen paradox, Schrödinger Steering phenomenon, and Bell Nonlocality, quantum entanglement has also been established as a useful resource for a number of computational and communication tasks.

In the present work, the authors study possible adequacy of quantum entanglement in Bayesian game theory, particularly in social welfare solution (SWS), a strategy which the players follow to maximize the sum of their payoffs. A quantum strategy is called quantum-SWS if it provides an advantage over a classical equilibrium (CE) strategy in the sense that none of the players has to sacrifice their CE-payoffs. Instead some of them have incentive and at the same time it maximizes the sum of all players' payoffs among all possible quantum advantageous strategies. A quantum state yielding such a quantum-SWS is called a quantum social welfare advice (SWA). It has been shown that any pure entangled state of a two-qubit system (the most elementary composite quantum system) can serve as quantum-SWA in some Bayesian game.

In the time, when quantum technologies are on the verge of evolving into their second commercial generation, and researchers are trying to find applications of quantum resources in different new tasks, the present work gives cognizance to the fact that every two-qubit pure entanglement is the optimal resource – the ‘gold coin’ - for some operational task.

► BibTeX data

► References

[1] R. Gibbons. Game Theory for Applied Economists. Princeton University Press, Princeton, NJ, 1992. URL https:/​/​​titles/​4993.html.

[2] P. Ordeshook. Game Theory and Political Theory: An Introduction. Cambridge University Press, 1986.

[3] A. M. Colman. Game Theory and its Applications: In the Social and Biological Sciences. Routledge, Taylor & Francis group, 1995.

[4] M. J. Osborne. An Introduction to Game Theory. Oxford University Press, New York, 2003.

[5] J. Von Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Princeton University Press, 1944. URL https:/​/​​titles/​7802.html.

[6] John F. Nash. Equilibrium points in n-person games. PNAS, 36: 48, 1950. URL https:/​/​​10.1073/​pnas.36.1.48.

[7] John F. Nash. Non-cooperative games. The Annals of Mathematics, 54: 286295, 1951. URL https:/​/​​10.2307/​1969529.

[8] John C. Harsanyi. Games with Incomplete Information Played by “Bayesian” Players, I-III Part I. The Basic Model. Management Science, 14 (3): 159–182, 1967. 10.1287/​mnsc.14.3.159. URL https:/​/​​10.1287/​mnsc.14.3.159.

[9] John C. Harsanyi. Games with Incomplete Information Played by “Bayesian” Players Part II. Bayesian Equilibrium Points. Management Science, 14 (5): 320–334, 1968a. 10.1287/​mnsc.14.5.320. URL https:/​/​​10.1287/​mnsc.14.5.320.

[10] John C. Harsanyi. Games with Incomplete Information Played by ‘Bayesian’ Players, Part III. The Basic Probability Distribution of the Game. Management Science, 14 (7): 486–502, 1968b. 10.1287/​mnsc.14.7.486. URL https:/​/​​10.1287/​mnsc.14.7.486.

[11] R. J. Aumann. Subjectivity and correlation in randomized strategies. Journal of mathematical economics, 1: 67, 1974. URL https:/​/​​10.1016/​0304-4068(74)90037-8.

[12] M. Rabin. Incorporating Fairness Into Game Theory. UC Berkeley: Department of Economics, UCB, 1991. URL https:/​/​​uc/​item/​3s87d1tm.

[13] K. Binmore. Game Theory and the Social Contract, Vol. 2: Just Playing (Economic Learning and Social Evolution). MIT Press, Cambridge, MA, 1998.

[14] J. G. March. Bounded rationality, ambiguity, and the engineering of choice. The Bell Journal of Economics, 9: 587, 1978. URL https:/​/​​10.2307/​3003600.

[15] W. B. Arthur. Inductive reasoning and bounded rationality. The American Economic Review, 48: 406, 1994. URL https:/​/​​stable/​2117868?seq=1#page_scan_tab_contents.

[16] Vincenzo Auletta, Diodato Ferraioli, Ashutosh Rai, Giannicola Scarpa, and Andreas Winter. Belief-Invariant and Quantum Equilibria in Games of Incomplete Information. arXiv:1605.07896, 2016. URL https:/​/​​abs/​1605.07896.

[17] R. J. Aumann. Correlated equilibrium as an expression of bayesian rationality. Econometrica, 55: 1, 1987. URL https:/​/​​10.2307/​1911154.

[18] Christos H. Papadimitriou and Tim Roughgarden. Computing correlated equilibria in multi-player games. J. ACM, 55 (3): 14:1–14:29, August 2008. ISSN 0004-5411. 10.1145/​1379759.1379762. URL https:/​/​​10.1145/​1379759.1379762.

[19] A. Blaquiere. Necessary and sufficiency conditions for optimal strategies in impulsive control. In Differential Games and Control Theory III, 1979.

[20] A. Blaquiere. Wave mechanics as a two-player game. In Dynamical Systems and Microphysics, pages 33–69. Springer Vienna, 1980. 10.1007/​978-3-7091-4330-8_2. URL https:/​/​​10.1007/​978-3-7091-4330-8_2.

[21] David A. Meyer. Quantum strategies. Phys. Rev. Lett., 82: 1052–1055, Feb 1999. 10.1103/​PhysRevLett.82.1052. URL https:/​/​​10.1103/​PhysRevLett.82.1052.

[22] Jens Eisert, Martin Wilkens, and Maciej Lewenstein. Quantum games and quantum strategies. Physical Review Letters, 83 (15): 3077–3080, October 1999. 10.1103/​physrevlett.83.3077. URL https:/​/​​10.1103/​physrevlett.83.3077.

[23] A. P. Flitney and D. Abbott. An Introduction to Quantum Game Theory. Fluctuation and Noise Letters, 02 (04): R175–R187, December 2002. 10.1142/​s0219477502000981. URL https:/​/​​10.1142/​s0219477502000981.

[24] Hong Guo, Juheng Zhang, and Gary J. Koehler. A survey of quantum games. Decision Support Systems, 46 (1): 318–332, December 2008. 10.1016/​j.dss.2008.07.001. URL https:/​/​​10.1016/​j.dss.2008.07.001.

[25] Faisal Shah Khan, Neal Solmeyer, Radhakrishnan Balu, and Travis S. Humble. Quantum games: a review of the history, current state, and interpretation. Quantum Information Processing, 17 (11), October 2018. 10.1007/​s11128-018-2082-8. URL https:/​/​​10.1007/​s11128-018-2082-8.

[26] N. Brunner and N. Linden. Connection between Bell nonlocality and Bayesian game theory. Nat. Comm., 4: 2057, 2013. URL https:/​/​​10.1038/​ncomms3057.

[27] Taksu Cheon and Azhar Iqbal. Bayesian nash equilibria and bell inequalities. Journal of the Physical Society of Japan, 77 (2): 024801, February 2008. 10.1143/​jpsj.77.024801. URL https:/​/​​10.1143/​jpsj.77.024801.

[28] Azhar Iqbal, James M. Chappell, and Derek Abbott. Social optimality in quantum bayesian games. Physica A: Statistical Mechanics and its Applications, 436: 798–805, October 2015. 10.1016/​j.physa.2015.05.020. URL https:/​/​​10.1016/​j.physa.2015.05.020.

[29] Ashutosh Rai and Goutam Paul. Strong quantum solutions in conflicting-interest bayesian games. Physical Review A, 96 (4), October 2017. 10.1103/​physreva.96.042340. URL https:/​/​​10.1103/​physreva.96.042340.

[30] Faisal Shah Khan and Travis S. Humble. Nash embedding and equilibrium in pure quantum states. In Quantum Technology and Optimization Problems, pages 51–62. Springer International Publishing, 2019, (arXiv:1801.02053). 10.1007/​978-3-030-14082-3_5. URL https:/​/​​10.1007/​978-3-030-14082-3_5.

[31] A. Pappa, N. Kumar, T. Lawson, M. Santha, S. Zhang, E. Diamanti, and I. Kerenidis. Nonlocality and Conflicting Interest Games. Phys. Rev. Lett., 114: 020401, Jan 2015. 10.1103/​PhysRevLett.114.020401. URL https:/​/​​10.1103/​PhysRevLett.114.020401.

[32] Arup Roy, Amit Mukherjee, Tamal Guha, Sibasish Ghosh, Some Sankar Bhattacharya, and Manik Banik. Nonlocal correlations: Fair and unfair strategies in Bayesian games. Phys. Rev. A, 94: 032120, Sep 2016. 10.1103/​PhysRevA.94.032120. URL https:/​/​​10.1103/​PhysRevA.94.032120.

[33] N. Gisin. Bell's inequality holds for all non-product states. Physics Letters A, 154 (5-6): 201–202, April 1991. 10.1016/​0375-9601(91)90805-i. URL https:/​/​​10.1016/​0375-9601(91)90805-i.

[34] Francesco Buscemi. All entangled quantum states are nonlocal. Physical Review Letters, 108 (20), May 2012. 10.1103/​physrevlett.108.200401. URL https:/​/​​10.1103/​physrevlett.108.200401.

[35] Lluís Masanes. All bipartite entangled states are useful for information processing. Physical Review Letters, 96 (15), April 2006a. 10.1103/​physrevlett.96.150501. URL https:/​/​​10.1103/​physrevlett.96.150501.

[36] Marco Piani and John Watrous. All entangled states are useful for channel discrimination. Physical Review Letters, 102 (25), June 2009. 10.1103/​physrevlett.102.250501. URL https:/​/​​10.1103/​physrevlett.102.250501.

[37] Daniel Cavalcanti, Paul Skrzypczyk, and Ivan Šupić. All entangled states can demonstrate nonclassical teleportation. Physical Review Letters, 119 (11), September 2017. 10.1103/​physrevlett.119.110501. URL https:/​/​​10.1103/​physrevlett.119.110501.

[38] 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. URL https:/​/​​10.1103/​PhysRevA.40.4277.

[39] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. Quantum entanglement. Rev. Mod. Phys., 81: 865–942, Jun 2009. 10.1103/​RevModPhys.81.865. URL https:/​/​​10.1103/​RevModPhys.81.865.

[40] J. S. Bell. On the Einstein Podolsky Rosen Paradox. Physics, 1 (3): 195, 1964. URL https:/​/​​10.1103/​PhysicsPhysiqueFizika.1.195.

[41] J. S. Bell. On the Problem of Hidden Variables in Quantum Mechanics. Rev. Mod. Phys., 38: 447, 1966. URL https:/​/​​10.1103/​RevModPhys.38.447.

[42] 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. URL https:/​/​​10.1103/​RevModPhys.86.419.

[43] edited by, K. Arrow, A. Sen, and K. Suzumura. Handbook of Social Choice and Welfare–Vol.I. 2002. URL https:/​/​​handbook/​handbook-of-social-choice-and-welfare/​vol/​1/​suppl/​C.

[44] edited by, K. Arrow, A. Sen, and K. Suzumura. Handbook of Social Choice and Welfare–Vol.II. 2011. URL https:/​/​​handbook/​handbook-of-social-choice-and-welfare/​vol/​2/​suppl/​C.

[45] 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. URL https:/​/​​10.1103/​PhysRevA.71.022101.

[46] Stefano Pironio. Lifting bell inequalities. Journal of Mathematical Physics, 46 (6): 062112, 2005. 10.1063/​1.1928727. URL https:/​/​​10.1063/​1.1928727.

[47] Nicolas Brunner, Valerio Scarani, and Nicolas Gisin. Bell-type inequalities for nonlocal resources. Journal of Mathematical Physics, 47 (11): 112101, 2006. 10.1063/​1.2352857. URL https:/​/​​10.1063/​1.2352857.

[48] Frédéric Dupuis, Nicolas Gisin, Avinatan Hasidim, André Allan Méthot, and Haran Pilpel. No nonlocal box is universal. Journal of Mathematical Physics, 48 (8): 082107, 2007. URL https:/​/​​10.1063/​1.2767538.

[49] J. M. Méndez and Jesús Urías. On the no-signaling approach to quantum nonlocality. Journal of Mathematical Physics, 56 (3): 032101, 2015. 10.1063/​1.4914336. URL https:/​/​​10.1063/​1.4914336.

[50] Antonio Acín, Serge Massar, and Stefano Pironio. Randomness versus Nonlocality and Entanglement. Phys. Rev. Lett., 108: 100402, Mar 2012. 10.1103/​PhysRevLett.108.100402. URL https:/​/​​10.1103/​PhysRevLett.108.100402.

[51] Tzyh Haur Yang and Miguel Navascués. Robust self-testing of unknown quantum systems into any entangled two-qubit states. Phys. Rev. A, 87: 050102, May 2013. 10.1103/​PhysRevA.87.050102. URL https:/​/​​10.1103/​PhysRevA.87.050102.

[52] Cédric Bamps and Stefano Pironio. Sum-of-squares decompositions for a family of clauser-horne-shimony-holt-like inequalities and their application to self-testing. Phys. Rev. A, 91: 052111, May 2015. 10.1103/​PhysRevA.91.052111. URL https:/​/​​10.1103/​PhysRevA.91.052111.

[53] Artur K. Ekert. Quantum cryptography based on Bell's theorem. Phys. Rev. Lett., 67: 661–663, Aug 1991. 10.1103/​PhysRevLett.67.661. URL https:/​/​​10.1103/​PhysRevLett.67.661.

[54] C.H. Bennett and S.J. Wiesner. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett., 69: 2881, 1992. 10.1103/​PhysRevLett.69.2881. URL https:/​/​​10.1103/​PhysRevLett.69.2881.

[55] Charles H. Bennett, Gilles Brassard, and N. David Mermin. Quantum cryptography without Bell's theorem. Phys. Rev. Lett., 68: 557–559, Feb 1992. 10.1103/​PhysRevLett.68.557. URL https:/​/​​10.1103/​PhysRevLett.68.557.

[56] C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett., 70: 1895, 1993. 10.1103/​PhysRevLett.70.1895. URL https:/​/​​10.1103/​PhysRevLett.70.1895.

[57] A. Einstein, B. Podolsky, and N. Rosen. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev., 47: 777–780, May 1935. 10.1103/​PhysRev.47.777. URL https:/​/​​10.1103/​PhysRev.47.777.

[58] E. Schrödinger. Discussion of Probability Relations between Separated Systems. Proc. Cambridge Philos. Soc., 31: 553, 1935. URL https:/​/​​10.1017/​S0305004100013554.

[59] E. Schrödinger. Probability relations between separated systems. Proc. Cambridge Philos. Soc., 32: 446, 1936. URL https:/​/​​10.1017/​S0305004100019137.

[60] H.M. Wiseman, S.J. Jones, and A.C. Doherty. Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox. Phys. Rev. Lett., 98: 140402, 2007. 10.1103/​PhysRevLett.98.140402. URL https:/​/​​10.1103/​PhysRevLett.98.140402.

[61] Lluís Masanes. Asymptotic violation of bell inequalities and distillability. Phys. Rev. Lett., 97: 050503, Aug 2006b. 10.1103/​PhysRevLett.97.050503. URL https:/​/​​10.1103/​PhysRevLett.97.050503.

[62] Shizuo Kakutani. A generalization of brouwer's fixed point theorem. Duke Mathematical Journal, 8 (3): 457–459, September 1941. 10.1215/​s0012-7094-41-00838-4. URL https:/​/​​10.1215/​s0012-7094-41-00838-4.

[63] I. L. Glicksberg. A further generalization of the kakutani fixed theorem, with application to nash equilibrium points. Proceedings of the American Mathematical Society, 3 (1): 170–170, January 1952. 10.1090/​s0002-9939-1952-0046638-5. URL https:/​/​​10.1090/​s0002-9939-1952-0046638-5.

[64] John Nash. The imbedding problem for riemannian manifolds. The Annals of Mathematics, 63 (1): 20, January 1956. 10.2307/​1969989. URL https:/​/​​10.2307/​1969989.

[65] H. Reichenbach. The Direction of Time. University of Los Angeles Press, Berkeley, 1956.

[66] Eric G Cavalcanti and Raymond Lal. On modifications of reichenbach's principle of common cause in light of bell's theorem. J. Phys. A: Math. Theor., 47: 424018, 2014. URL https:/​/​​10.1088/​1751-8113/​47/​42/​424018.

[67] Valerio Scarani, Nicolas Gisin, Nicolas Brunner, Lluis Masanes, Sergi Pino, and Antonio Acín. Secrecy extraction from no-signaling correlations. Phys. Rev. A, 74: 042339, Oct 2006. 10.1103/​PhysRevA.74.042339. URL https:/​/​​10.1103/​PhysRevA.74.042339.

[68] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Proposed Experiment to Test Local Hidden-Variable Theories. Phys. Rev. Lett., 23: 880–884, Oct 1969. 10.1103/​PhysRevLett.23.880. URL https:/​/​​10.1103/​PhysRevLett.23.880.

Cited by

[1] Sutapa Saha, Some Sankar Bhattacharya, Tamal Guha, Saronath Halder, and Manik Banik, "Advantage of Quantum Theory over Nonclassical Models of Communication", Annalen der Physik 532 12, 2000334 (2020).

[2] Subhendu B. Ghosh, Snehasish Roy Chowdhury, Ranendu Adhikary, Arup Roy, and Tamal Guha, "Minimum detection efficiencies for loophole-free genuine nonlocality tests", Physical Review A 109 5, 052202 (2024).

[3] Ram Krishna Patra, Sahil Gopalkrishna Naik, Edwin Peter Lobo, Samrat Sen, Tamal Guha, Some Sankar Bhattacharya, Mir Alimuddin, and Manik Banik, "Classical analogue of quantum superdense coding and communication advantage of a single quantum system", Quantum 8, 1315 (2024).

[4] Zhonghua Ma, Markus Rambach, Kaumudibikash Goswami, Some Sankar Bhattacharya, Manik Banik, and Jacquiline Romero, "Randomness-Free Test of Nonclassicality: A Proof of Concept", Physical Review Letters 131 13, 130201 (2023).

[5] Subhendu B. Ghosh, Tathagata Gupta, Ardra A. V., Anandamay Das Bhowmik, Sutapa Saha, Tamal Guha, and Amit Mukherjee, "Activating strong nonlocality from local sets: An elimination paradigm", Physical Review A 106 1, L010202 (2022).

[6] Subhendu B. Ghosh, Snehasish Roy Chowdhury, Guruprasad Kar, Arup Roy, Tamal Guha, and Manik Banik, "Quantum Nonlocality: Multicopy Resource Interconvertibility and Their Asymptotic Inequivalence", Physical Review Letters 132 25, 250205 (2024).

[7] Ekta Panwar, Palash Pandya, and Marcin Wieśniak, "An elegant scheme of self-testing for multipartite Bell inequalities", npj Quantum Information 9 1, 71 (2023).

[8] Berry Groisman, Michael Mc Gettrick, Mehdi Mhalla, and Marcin Pawlowski, "How Quantum Information Can Improve Social Welfare", IEEE Journal on Selected Areas in Information Theory 1 2, 445 (2020).

[9] Tamal Guha, Mir Alimuddin, Sumit Rout, Amit Mukherjee, Some Sankar Bhattacharya, and Manik Banik, "Quantum Advantage for Shared Randomness Generation", Quantum 5, 569 (2021).

[10] Sagnik Dutta, Amit Mukherjee, and Manik Banik, "Operational characterization of multipartite nonlocal correlations", Physical Review A 102 5, 052218 (2020).

[11] George Moreno, Ranieri Nery, Alberto Palhares, and Rafael Chaves, "Multistage games and Bell scenarios with communication", Physical Review A 102 4, 042412 (2020).

[12] Ileana Badea, Carmen Mocanu, Florin Nichita, and Ovidiu Păsărescu, "Applications of Non-Standard analysis in Topoi to Mathematical Neurosciences and Artificial Intelligence: Infons, Energons, Receptons (I)", Mathematics 9 17, 2048 (2021).

[13] Arnaud Z. Dragicevic, "The Price Identity of Replicator(–Mutator) Dynamics on Graphs with Quantum Strategies in a Public Goods Game", Dynamic Games and Applications (2024).

[14] Mir Alimuddin, Tamal Guha, and Preeti Parashar, "Bound on ergotropic gap for bipartite separable states", Physical Review A 99 5, 052320 (2019).

[15] Ashutosh Rai and Goutam Paul, "Strong quantum solutions in conflicting-interest Bayesian games", Physical Review A 96 4, 042340 (2017).

[16] Tamal Guha, Mir Alimuddin, and Preeti Parashar, "Allowed and forbidden bipartite correlations from thermal states", Physical Review E 100 1, 012147 (2019).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 12:25:49) and SAO/NASA ADS (last updated successfully 2024-06-22 12:25:50). The list may be incomplete as not all publishers provide suitable and complete citation data.