1University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), Boltzmanngasse 5, 1090 Vienna, Austria
2Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
3Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
We analyze the amount of classical communication required to reproduce the statistics of local projective measurements on a general pair of entangled qubits, $|Psi_{AB}rangle=sqrt{p} |00rangle+sqrt{1-p} |11rangle$ (with $1/2leq p leq 1$). We construct a classical protocol that perfectly simulates local projective measurements on all entangled qubit pairs by communicating one classical trit. Additionally, when $frac{2p(1-p)}{2p-1} log{left(frac{p}{1-p}right)}+2(1-p)leq1$, approximately $0.835 leq p leq 1$, we present a classical protocol that requires only a single bit of communication. The latter model even allows a perfect classical simulation with an average communication cost that approaches zero in the limit where the degree of entanglement approaches zero ($p to 1$). This proves that the communication cost for simulating weakly entangled qubit pairs is strictly smaller than for the maximally entangled one.
Featured image: $textbf{a)}$ Alice and Bob perform local projective measurements on an entangled qubit pair $textbf{b)}$ Alice and Bob want to reproduce these quantum correlations with classical resources
[embedded content]
► BibTeX data
► References
[1] J. S. Bell, On the einstein podolsky rosen paradox, Physics Physique Fizika 1, 195–200 (1964).
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
[2] A. K. Ekert, Quantum cryptography based on bell’s theorem, Phys. Rev. Lett. 67, 661–663 (1991).
https://doi.org/10.1103/PhysRevLett.67.661
[3] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Device-Independent Security of Quantum Cryptography against Collective Attacks, Phys. Rev. Lett. 98, 230501 (2007), arXiv:quant-ph/0702152 [quant-ph].
https://doi.org/10.1103/PhysRevLett.98.230501
arXiv:quant-ph/0702152
[4] S. Pironio, A. Acín, S. Massar, A. B. 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 theorem, Nature 464, 1021–1024 (2010), arXiv:0911.3427 [quant-ph].
https://doi.org/10.1038/nature09008
arXiv:0911.3427
[5] U. Vazirani and T. Vidick, Fully device-independent quantum key distribution, Phys. Rev. Lett. 113, 140501 (2014), arXiv:1210.1810 [quant-ph].
https://doi.org/10.1103/PhysRevLett.113.140501
arXiv:1210.1810
[6] I. Šupić and J. Bowles, Self-testing of quantum systems: a review, Quantum 4, 337 (2020), arXiv:1904.10042 [quant-ph].
https://doi.org/10.22331/q-2020-09-30-337
arXiv:1904.10042
[7] T. Maudlin, Bell’s inequality, information transmission, and prism models, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1992, 404–417 (1992).
https://doi.org/10.1086/psaprocbienmeetp.1992.1.192771
[8] G. Brassard, R. Cleve, and A. Tapp, Cost of Exactly Simulating Quantum Entanglement with Classical Communication, Phys. Rev. Lett. 83, 1874–1877 (1999), arXiv:quant-ph/9901035 [quant-ph].
https://doi.org/10.1103/PhysRevLett.83.1874
arXiv:quant-ph/9901035
[9] M. Steiner, Towards quantifying non-local information transfer: finite-bit non-locality, Physics Letters A 270, 239–244 (2000), arXiv:quant-ph/9902014 [quant-ph].
https://doi.org/10.1016/S0375-9601(00)00315-7
arXiv:quant-ph/9902014
[10] N. J. Cerf, N. Gisin, and S. Massar, Classical Teleportation of a Quantum Bit, Phys. Rev. Lett. 84, 2521–2524 (2000), arXiv:quant-ph/9906105 [quant-ph].
https://doi.org/10.1103/PhysRevLett.84.2521
arXiv:quant-ph/9906105
[11] A. K. Pati, Minimum classical bit for remote preparation and measurement of a qubit, Phys. Rev. A 63, 014302 (2000), arXiv:quant-ph/9907022 [quant-ph].
https://doi.org/10.1103/PhysRevA.63.014302
arXiv:quant-ph/9907022
[12] S. Massar, D. Bacon, N. J. Cerf, and R. Cleve, Classical simulation of quantum entanglement without local hidden variables, Phys. Rev. A 63, 052305 (2001), arXiv:quant-ph/0009088 [quant-ph].
https://doi.org/10.1103/PhysRevA.63.052305
arXiv:quant-ph/0009088
[13] B. F. Toner and D. Bacon, Communication Cost of Simulating Bell Correlations, Phys. Rev. Lett. 91, 187904 (2003), arXiv:quant-ph/0304076 [quant-ph].
https://doi.org/10.1103/PhysRevLett.91.187904
arXiv:quant-ph/0304076
[14] D. Bacon and B. F. Toner, Bell Inequalities with Auxiliary Communication, Phys. Rev. Lett. 90, 157904 (2003), arXiv:quant-ph/0208057 [quant-ph].
https://doi.org/10.1103/PhysRevLett.90.157904
arXiv:quant-ph/0208057
[15] J. Degorre, S. Laplante, and J. Roland, Simulating quantum correlations as a distributed sampling problem, Phys. Rev. A 72, 062314 (2005), arXiv:quant-ph/0507120 [quant-ph].
https://doi.org/10.1103/PhysRevA.72.062314
arXiv:quant-ph/0507120
[16] J. Degorre, S. Laplante, and J. Roland, Classical simulation of traceless binary observables on any bipartite quantum state, Phys. Rev. A 75, 012309 (2007), arXiv:quant-ph/0608064 [quant-ph].
https://doi.org/10.1103/PhysRevA.75.012309
arXiv:quant-ph/0608064
[17] O. Regev and B. Toner, Simulating quantum correlations with finite communication, SIAM Journal on Computing 39, 1562–1580 (2010), arXiv:0708.0827 [quant-ph].
https://doi.org/10.1137/080723909
arXiv:0708.0827
[18] C. Branciard and N. Gisin, Quantifying the Nonlocality of Greenberger-Horne-Zeilinger Quantum Correlations by a Bounded Communication Simulation Protocol, Phys. Rev. Lett. 107, 020401 (2011), arXiv:1102.0330 [quant-ph].
https://doi.org/10.1103/PhysRevLett.107.020401
arXiv:1102.0330
[19] C. Branciard, N. Brunner, H. Buhrman, R. Cleve, N. Gisin, S. Portmann, D. Rosset, and M. Szegedy, Classical Simulation of Entanglement Swapping with Bounded Communication, Phys. Rev. Lett. 109, 100401 (2012), arXiv:1203.0445 [quant-ph].
https://doi.org/10.1103/PhysRevLett.109.100401
arXiv:1203.0445
[20] K. Maxwell and E. Chitambar, Bell inequalities with communication assistance, Phys. Rev. A 89, 042108 (2014), arXiv:1405.3211 [quant-ph].
https://doi.org/10.1103/PhysRevA.89.042108
arXiv:1405.3211
[21] G. Brassard, L. Devroye, and C. Gravel, Exact classical simulation of the quantum-mechanical ghz distribution, IEEE Transactions on Information Theory 62, 876–890 (2016), arXiv:1303.5942 [cs.IT].
https://doi.org/10.1109/TIT.2015.2504525
arXiv:1303.5942
[22] G. Brassard, L. Devroye, and C. Gravel, Remote Sampling with Applications to General Entanglement Simulation, Entropy 21, 92 (2019), arXiv:1807.06649 [quant-ph].
https://doi.org/10.3390/e21010092
arXiv:1807.06649
[23] E. Zambrini Cruzeiro and N. Gisin, Bell Inequalities with One Bit of Communication, Entropy 21, 171 (2019), arXiv:1812.05107 [quant-ph].
https://doi.org/10.3390/e21020171
arXiv:1812.05107
[24] M. J. Renner, A. Tavakoli, and M. T. Quintino, Classical Cost of Transmitting a Qubit, Phys. Rev. Lett. 130, 120801 (2023), arXiv:2207.02244 [quant-ph].
https://doi.org/10.1103/PhysRevLett.130.120801
arXiv:2207.02244
[25] N. Brunner, N. Gisin, and V. Scarani, Entanglement and non-locality are different resources, New Journal of Physics 7, 88 (2005), arXiv:quant-ph/0412109 [quant-ph].
https://doi.org/10.1088/1367-2630/7/1/088
arXiv:quant-ph/0412109
[26] N. J. Cerf, N. Gisin, S. Massar, and S. Popescu, Simulating Maximal Quantum Entanglement without Communication, Phys. Rev. Lett. 94, 220403 (2005), arXiv:quant-ph/0410027 [quant-ph].
https://doi.org/10.1103/PhysRevLett.94.220403
arXiv:quant-ph/0410027
[27] P. H. Eberhard, Background level and counter efficiencies required for a loophole-free einstein-podolsky-rosen experiment, Phys. Rev. A 47, R747–R750 (1993).
https://doi.org/10.1103/PhysRevA.47.R747
[28] A. Cabello and J.-Å. Larsson, Minimum Detection Efficiency for a Loophole-Free Atom-Photon Bell Experiment, Phys. Rev. Lett. 98, 220402 (2007), arXiv:quant-ph/0701191 [quant-ph].
https://doi.org/10.1103/PhysRevLett.98.220402
arXiv:quant-ph/0701191
[29] N. Brunner, N. Gisin, V. Scarani, and C. Simon, Detection Loophole in Asymmetric Bell Experiments, Phys. Rev. Lett. 98, 220403 (2007), arXiv:quant-ph/0702130 [quant-ph].
https://doi.org/10.1103/PhysRevLett.98.220403
arXiv:quant-ph/0702130
[30] M. Araújo, M. T. Quintino, D. Cavalcanti, M. F. Santos, A. Cabello, and M. T. Cunha, Tests of Bell inequality with arbitrarily low photodetection efficiency and homodyne measurements, Phys. Rev. A 86, 030101 (2012), arXiv:1112.1719 [quant-ph].
https://doi.org/10.1103/PhysRevA.86.030101
arXiv:1112.1719
[31] S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, Journal of Mathematics and Mechanics 17, 59–87 (1967).
http://www.jstor.org/stable/24902153
[32] N. Gisin and B. Gisin, A local hidden variable model of quantum correlation exploiting the detection loophole, Physics Letters A 260, 323–327 (1999), arXiv:quant-ph/9905018 [quant-ph].
https://doi.org/10.1016/S0375-9601(99)00519-8
arXiv:quant-ph/9905018
[33] N. Gisin, Bell’s inequality holds for all non-product states, Physics Letters A 154, 201–202 (1991).
https://doi.org/10.1016/0375-9601(91)90805-I
[34] A. C. Elitzur, S. Popescu, and D. Rohrlich, Quantum nonlocality for each pair in an ensemble, Physics Letters A 162, 25–28 (1992).
https://doi.org/10.1016/0375-9601(92)90952-I
[35] J. Barrett, A. Kent, and S. Pironio, Maximally Nonlocal and Monogamous Quantum Correlations, Phys. Rev. Lett. 97, 170409 (2006), arXiv:quant-ph/0605182 [quant-ph].
https://doi.org/10.1103/PhysRevLett.97.170409
arXiv:quant-ph/0605182
[36] S. Portmann, C. Branciard, and N. Gisin, Local content of all pure two-qubit states, Phys. Rev. A 86, 012104 (2012), arXiv:1204.2982 [quant-ph].
https://doi.org/10.1103/PhysRevA.86.012104
arXiv:1204.2982
[37] P. Sidajaya, A. Dewen Lim, B. Yu, and V. Scarani, Neural Network Approach to the Simulation of Entangled States with One Bit of Communication, arXiv e-prints (2023), arXiv:2305.19935 [quant-ph].
arXiv:2305.19935
[38] N. Gisin and F. Fröwis, From quantum foundations to applications and back, Philosophical Transactions of the Royal Society of London Series A 376, 20170326 (2018), arXiv:1802.00736 [quant-ph].
https://doi.org/10.1098/rsta.2017.0326
arXiv:1802.00736
[39] G. Brassard, Quantum communication complexity, Foundations of Physics 33, 1593–1616 (2003).
https://doi.org/10.1023/A:1026009100467
[40] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Reviews of Modern Physics 86, 419–478 (2014), arXiv:1303.2849 [quant-ph].
https://doi.org/10.1103/revmodphys.86.419
arXiv:1303.2849
[41] V. Scarani, Local and nonlocal content of bipartite qubit and qutrit correlations, Phys. Rev. A 77, 042112 (2008), arXiv:0712.2307 [quant-ph].
https://doi.org/10.1103/PhysRevA.77.042112
arXiv:0712.2307
[42] C. Branciard, N. Gisin, and V. Scarani, Local content of bipartite qubit correlations, Phys. Rev. A 81, 022103 (2010), arXiv:0909.3839 [quant-ph].
https://doi.org/10.1103/PhysRevA.81.022103
arXiv:0909.3839
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] Peter Sidajaya, Aloysius Dewen Lim, Baichu Yu, and Valerio Scarani, “Neural Network Approach to the Simulation of Entangled States with One Bit of Communication”, arXiv:2305.19935, (2023).
The above citations are from Crossref’s cited-by service (last updated successfully 2023-10-26 14:20:56) and SAO/NASA ADS (last updated successfully 2023-10-26 14:20:57). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
- SEO Powered Content & PR Distribution. Get Amplified Today.
- PlatoData.Network Vertical Generative Ai. Empower Yourself. Access Here.
- PlatoAiStream. Web3 Intelligence. Knowledge Amplified. Access Here.
- PlatoESG. Carbon, CleanTech, Energy, Environment, Solar, Waste Management. Access Here.
- PlatoHealth. Biotech and Clinical Trials Intelligence. Access Here.
- Source: https://quantum-journal.org/papers/q-2023-10-24-1149/
