Neural Network Approach to the Simulation of Entangled States with One Bit of Communication

Peter Sidajaya1, Aloysius Dewen Lim2, Baichu Yu1,3,4, and Valerio Scarani1,2

1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542
3Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Nanshan District, Shenzhen, 518055, China
4International Quantum Academy (SIQA), Shenzhen 518048, China

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Bell's theorem states that Local Hidden Variables (LHVs) cannot fully explain the statistics of measurements on some entangled quantum states. It is natural to ask how much supplementary classical communication would be needed to simulate them. We study two long-standing open questions in this field with neural network simulations and other tools. First, we present evidence that all projective measurements on partially entangled pure two-qubit states require only one bit of communication. We quantify the statistical distance between the exact quantum behaviour and the product of the trained network, or of a semianalytical model inspired by it. Second, while it is known on general grounds (and obvious) that one bit of communication cannot eventually reproduce all bipartite quantum correlation, explicit examples have proved evasive. Our search failed to find one for several bipartite Bell scenarios with up to 5 inputs and 4 outputs, highlighting the power of one bit of communication in reproducing quantum correlations.

Bell's theorem tells us that local hidden variable (LHV) cannot describe the statistics of quantum theory. However, entanglement and Bell inequality are not exactly the most intuitive of concepts. Thus, we turn to the question of: "If we were to stick with LHV and supplement it with communication, how much communication would we need?" Answering this would give us a more intuitive understanding of the power of quantum entanglement.

We first investigate the problem for two-qubit states, where for some states, it is still not known whether one-bit of classical communication is enough. We used a neural network to generate numerical protocols for simulating such states and analysed the resulting protocol. In the second half, we tried to see whether we can find an explicit quantum behaviour that is unsimulatable with one-bit of communication.

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[1] Albert Einstein, Boris Podolsky, and Nathan Rosen. Can quantum-mechanical description of physical reality be considered complete? Physical review, 47 (10): 777, 1935. 10.1103/​PhysRev.47.777.

[2] John S Bell. On the einstein podolsky rosen paradox. Physics Physique Fizika, 1 (3): 195, 1964. 10.1103/​PhysicsPhysiqueFizika.1.195.

[3] Valerio Scarani. Bell nonlocality. Oxford University Press, 2019. ISBN 978-0198788416. 10.1093/​oso/​9780198788416.001.0001.

[4] Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Foundations of Physics, 24 (3): 379–385, 1994. 10.1007/​bf02058098.

[5] Nicolas Brunner, Nicolas Gisin, and Valerio Scarani. Entanglement and non-locality are different resources. New Journal of Physics, 7 (1): 88, 2005. 10.1088/​1367-2630/​7/​1/​088.

[6] Nicolas Brunner, Nicolas Gisin, Sandu Popescu, and Valerio Scarani. Simulation of partial entanglement with nonsignaling resources. Phys. Rev. A, 78: 052111, 2008. 10.1103/​PhysRevA.78.052111.

[7] Gilles Brassard, Richard Cleve, and Alain Tapp. Cost of exactly simulating quantum entanglement with classical communication. Physical Review Letters, 83 (9): 1874, 1999. 10.1103/​PhysRevLett.83.1874.

[8] Michael Steiner. Towards quantifying non-local information transfer: finite-bit non-locality. Physics Letters A, 270 (5): 239–244, 2000. 10.1016/​s0375-9601(00)00315-7.

[9] János A. Csirik. Cost of exactly simulating a bell pair using classical communication. Phys. Rev. A, 66: 014302, 2002. 10.1103/​PhysRevA.66.014302.

[10] B. F. Toner and D. Bacon. Communication cost of simulating bell correlations. Phys. Rev. Lett., 91: 187904, 2003. 10.1103/​PhysRevLett.91.187904.

[11] Julien Degorre, Sophie Laplante, and Jérémie Roland. Simulating quantum correlations as a distributed sampling problem. Physical Review A, 72 (6): 062314, 2005. 10.1103/​PhysRevA.72.062314.

[12] Martin J Renner, Armin Tavakoli, and Marco Túlio Quintino. Classical cost of transmitting a qubit. Physical Review Letters, 130 (12): 120801, 2023. 10.1103/​PhysRevLett.130.120801.

[13] Martin J Renner and Marco Túlio Quintino. The minimal communication cost for simulating entangled qubits. arXiv preprint arXiv:2207.12457, 2022. 10.48550/​arXiv.2207.12457.

[14] D. Bacon and B. F. Toner. Bell inequalities with auxiliary communication. Phys. Rev. Lett., 90: 157904, 2003. 10.1103/​PhysRevLett.90.157904.

[15] Katherine Maxwell and Eric Chitambar. Bell inequalities with communication assistance. Phys. Rev. A, 89: 042108, 2014. 10.1103/​PhysRevA.89.042108.

[16] Emmanuel Zambrini Cruzeiro and Nicolas Gisin. Bell inequalities with one bit of communication. Entropy, 21 (2): 171, 2019. 10.3390/​e21020171.

[17] Serge Massar, Dave Bacon, Nicolas J Cerf, and Richard Cleve. Classical simulation of quantum entanglement without local hidden variables. Physical Review A, 63 (5): 052305, 2001. 10.1103/​PhysRevA.63.052305.

[18] Julien Degorre, Sophie Laplante, and Jérémie Roland. Classical simulation of traceless binary observables on any bipartite quantum state. Phys. Rev. A, 75: 012309, 2007. 10.1103/​PhysRevA.75.012309.

[19] Gilles Brassard, Luc Devroye, and Claude Gravel. Remote sampling with applications to general entanglement simulation. Entropy, 21 (1): 92, 2019. 10.3390/​e21010092.

[20] T. Vértesi and E. Bene. Lower bound on the communication cost of simulating bipartite quantum correlations. Phys. Rev. A, 80: 062316, 2009. 10.1103/​PhysRevA.80.062316.

[21] Dong-Ling Deng. Machine learning detection of bell nonlocality in quantum many-body systems. Phys. Rev. Lett., 120: 240402, 2018. 10.1103/​PhysRevLett.120.240402.

[22] Yue-Chi Ma and Man-Hong Yung. Transforming bell’s inequalities into state classifiers with machine learning. npj Quantum Information, 4 (1): 34, 2018. 10.1038/​s41534-018-0081-3.

[23] Askery Canabarro, Samuraí Brito, and Rafael Chaves. Machine learning nonlocal correlations. Phys. Rev. Lett., 122: 200401, 2019. 10.1103/​PhysRevLett.122.200401.

[24] Tamás Kriváchy, Yu Cai, Daniel Cavalcanti, Arash Tavakoli, Nicolas Gisin, and Nicolas Brunner. A neural network oracle for quantum nonlocality problems in networks. npj Quantum Information, 6 (1): 1–7, 2020. 10.1038/​s41534-020-00305-x.

[25] Cillian Harney, Mauro Paternostro, and Stefano Pirandola. Mixed state entanglement classification using artificial neural networks. New Journal of Physics, 23 (6): 063033, 2021. 10.1088/​1367-2630/​ac0388.

[26] Antoine Girardin, Nicolas Brunner, and Tamás Kriváchy. Building separable approximations for quantum states via neural networks. Physical Review Research, 4 (2): 023238, 2022. 10.1103/​PhysRevResearch.4.023238.

[27] Solomon Kullback. Information Theory and Statistics. Dover Publications, 1968. ISBN 0-8446-5625-9.

[28] 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, 2008. 10.1088/​1367-2630/​10/​7/​073013.

[29] Nathaniel Johnston, Alessandro Cosentino, and Vincent Russo. Qetlab: Qetlab v0.9. 10.5281/​zenodo.44637.

[30] 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, 2005. 10.1103/​PhysRevA.71.022101.

[31] Nick S. Jones and Lluís Masanes. Interconversion of nonlocal correlations. Phys. Rev. A, 72: 052312, 2005. 10.1103/​PhysRevA.72.052312.

[32] István Márton, Erika Bene, Péter Diviánszky, and Tamás Vértesi. Beating one bit of communication with and without quantum pseudo-telepathy. arXiv preprint arXiv:2308.10771, 2023. 10.48550/​arXiv.2308.10771.

Cited by

[1] Armin Tavakoli, "The classical price tag of entangled qubits", Quantum Views 7, 76 (2023).

[2] Martin J. Renner and Marco Túlio Quintino, "The minimal communication cost for simulating entangled qubits", Quantum 7, 1149 (2023).

[3] István Márton, Erika Bene, Péter Diviánszky, and Tamás Vértesi, "Beating one bit of communication with and without quantum pseudo-telepathy", arXiv:2308.10771, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-15 06:52:03) and SAO/NASA ADS (last updated successfully 2024-04-15 06:52:04). The list may be incomplete as not all publishers provide suitable and complete citation data.

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