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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Cited by

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

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

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

The above citations are from Crossref's cited-by service (last updated successfully 2023-12-07 11:14:59) and SAO/NASA ADS (last updated successfully 2023-12-07 11:15:00). The list may be incomplete as not all publishers provide suitable and complete citation data.

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