Contextuality in entanglement-assisted one-shot classical communication

Shiv Akshar Yadavalli1 and Ravi Kunjwal2

1Department of Physics, Duke University, Durham, North Carolina, USA 27708
2Centre for Quantum Information and Communication, Ecole polytechnique de Bruxelles, CP 165, Université libre de Bruxelles, 1050 Brussels, Belgium

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Abstract

We consider the problem of entanglement-assisted one-shot classical communication. In the zero-error regime, entanglement can increase the one-shot zero-error capacity of a family of classical channels following the strategy of Cubitt et al., Phys. Rev. Lett. 104, 230503 (2010). This strategy uses the Kochen-Specker theorem which is applicable only to projective measurements. As such, in the regime of noisy states and/or measurements, this strategy cannot increase the capacity. To accommodate generically noisy situations, we examine the one-shot success probability of sending a fixed number of classical messages. We show that preparation contextuality powers the quantum advantage in this task, increasing the one-shot success probability beyond its classical maximum. Our treatment extends beyond Cubitt et al. and includes, for example, the experimentally implemented protocol of Prevedel et al., Phys. Rev. Lett. 106, 110505 (2011). We then show a mapping between this communication task and a corresponding nonlocal game. This mapping generalizes the connection with pseudotelepathy games previously noted in the zero-error case. Finally, after motivating a constraint we term $\textit{context-independent guessing}$, we show that contextuality witnessed by noise-robust noncontextuality inequalities obtained in R. Kunjwal, Quantum 4, 219 (2020), is sufficient for enhancing the one-shot success probability. This provides an operational meaning to these inequalities and the associated hypergraph invariant, the weighted max-predictability, introduced in R. Kunjwal, Quantum 3, 184 (2019). Our results show that the task of entanglement-assisted one-shot classical communication provides a fertile ground to study the interplay of the Kochen-Specker theorem, Spekkens contextuality, and Bell nonlocality.

The fact that quantum theory allows the possibility of quantum advantage over classical resources is powered by its nonclassicality. This nonclassicality can take many forms, e.g., entanglement, incompatibility, contextuality, Bell nonlocality, etc. By studying the task of entanglement-assisted one-shot classical communication, we consider the interplay of three notions of nonclassicality in this paper: 1) Kochen-Specker contextuality, 2) Spekkens contextuality, and 3) Bell nonlocality.

Specifically, we study the following communication problem: Alice (the sender) is connected to Bob (the receiver) via a noisy classical channel. They are allowed access to shared entanglement and can implement local quantum measurements. It is known that for a certain family of classical channels inspired by the Kochen-Specker theorem, the number of messages that can be sent without error over the classical channel (i.e., it one-shot zero-error capacity) can be increased with access to shared entanglement. This zero-error result due to Cubitt et al. [Phys. Rev. Lett. 104, 230503 (2010)] is also intimately related to nonlocal games known as pseudotelepathy games that admit perfect quantum winning strategies.

We study this communication problem in the noisy regime where the Kochen-Specker theorem is inapplicable. In doing so, we show the intimate connection of this problem with noise-robust contextuality in the formulation proposed by Spekkens [Phys. Rev. A 71, 052108 (2005)] and with a family of nonlocal games inspired by the communication problem. Under an assumption that the parties do not trust the probabilities associated with the classical channel, but trust only its possibilistic structure (encoded in the channel hypergraph), we also show that noise-robust contextuality witnessed by a hypergraph invariant is sufficient for quantum advantage in this task. This provides an operational meaning to the contextuality witnesses obtained in R. Kunjwal, Quantum 4, 219 (2020).

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► References

[1] J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1, 195 (1964).
https:/​/​doi.org/​10.1103/​PhysicsPhysiqueFizika.1.195

[2] 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 (1969).
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[3] S. Kochen and E. P. Specker, The Problem of Hidden Variables in Quantum Mechanics, in The logico-algebraic approach to quantum mechanics (Springer, 1975) pp. 293–328.
https:/​/​doi.org/​10.1007/​978-94-010-1795-4_17

[4] R. Renner and S. Wolf, Quantum pseudo-telepathy and the Kochen-Specker theorem, in International Symposium on Information Theory, 2004. ISIT 2004. Proceedings. (IEEE, 2004) pp. 322–322.
https:/​/​doi.org/​10.1109/​ISIT.2004.1365359

[5] G. Brassard, A. Broadbent, and A. Tapp, Quantum pseudo-telepathy, Foundations of Physics 35, 1877 (2005).
https:/​/​doi.org/​10.1007/​s10701-005-7353-4

[6] T. S. Cubitt, D. Leung, W. Matthews, and A. Winter, Improving Zero-Error Classical Communication with Entanglement, Phys. Rev. Lett. 104, 230503 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.230503

[7] M. Howard, J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the `magic' for quantum computation, Nature 510, 351 (2014).
https:/​/​doi.org/​10.1038/​nature13460

[8] J. Barrett and A. Kent, Non-contextuality, finite precision measurement and the Kochen-Specker theorem, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 35, 151 (2004).
https:/​/​doi.org/​10.1016/​j.shpsb.2003.10.003

[9] A. Winter, What does an experimental test of quantum contextuality prove or disprove?, Journal of Physics A: Mathematical and Theoretical 47, 424031 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424031

[10] R. Kunjwal, Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements, Quantum 3, 184 (2019).
https:/​/​doi.org/​10.22331/​q-2019-09-09-184

[11] A. Cabello, What do we learn about quantum theory from Kochen-Specker quantum contextuality?, PIRSA 17070034 (2017).
https:/​/​doi.org/​10.48660/​17070034

[12] G. Chiribella and X. Yuan, Measurement sharpness cuts nonlocality and contextuality in every physical theory, arXiv preprint arXiv:1404.3348 (2014).
https:/​/​doi.org/​10.48550/​arXiv.1404.3348
arXiv:1404.3348

[13] R. W. Spekkens, Contextuality for preparations, transformations, and unsharp measurements, Phys. Rev. A 71, 052108 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.052108

[14] M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W. Spekkens, An experimental test of noncontextuality without unphysical idealizations, Nature Communications 7, 1 (2016).
https:/​/​doi.org/​10.1038/​ncomms11780

[15] M. F. Pusey, L. Del Rio, and B. Meyer, Contextuality without access to a tomographically complete set, arXiv preprint arXiv:1904.08699 (2019).
https:/​/​doi.org/​10.48550/​arXiv.1904.08699
arXiv:1904.08699

[16] M. D. Mazurek, M. F. Pusey, K. J. Resch, and R. W. Spekkens, Experimentally bounding deviations from quantum theory in the landscape of generalized probabilistic theories, PRX Quantum 2, 020302 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.020302

[17] R. Kunjwal and R. W. Spekkens, From the Kochen-Specker Theorem to Noncontextuality Inequalities without Assuming Determinism, Phys. Rev. Lett. 115, 110403 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.110403

[18] R. Kunjwal and R. W. Spekkens, From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities, Phys. Rev. A 97, 052110 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.052110

[19] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Preparation Contextuality Powers Parity-Oblivious Multiplexing, Phys. Rev. Lett. 102, 010401 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.010401

[20] A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora, Optimal bounds for parity-oblivious random access codes, New Journal of Physics 18, 045003 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​4/​045003

[21] D. Schmid and R. W. Spekkens, Contextual Advantage for State Discrimination, Phys. Rev. X 8, 011015 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.011015

[22] D. Saha and A. Chaturvedi, Preparation contextuality as an essential feature underlying quantum communication advantage, Phys. Rev. A 100, 022108 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.022108

[23] D. Saha, P. Horodecki, and M. Pawłowski, State independent contextuality advances one-way communication, New Journal of Physics 21, 093057 (2019).
https:/​/​doi.org/​10.1088/​1367-2630/​ab4149

[24] R. Kunjwal, M. Lostaglio, and M. F. Pusey, Anomalous weak values and contextuality: Robustness, tightness, and imaginary parts, Phys. Rev. A 100, 042116 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.042116

[25] M. Lostaglio and G. Senno, Contextual advantage for state-dependent cloning, Quantum 4, 258 (2020).
https:/​/​doi.org/​10.22331/​q-2020-04-27-258

[26] R. Kunjwal, Contextuality beyond the Kochen-Specker theorem, arXiv preprint arXiv:1612.07250 (2016).
https:/​/​doi.org/​10.48550/​arXiv.1612.07250
arXiv:1612.07250

[27] R. Kunjwal, Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem, Quantum 4, 219 (2020).
https:/​/​doi.org/​10.22331/​q-2020-01-10-219

[28] R. Prevedel, Y. Lu, W. Matthews, R. Kaltenbaek, and K. J. Resch, Entanglement-Enhanced Classical Communication Over a Noisy Classical Channel, Phys. Rev. Lett. 106, 110505 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.110505

[29] B. Hemenway, C. A. Miller, Y. Shi, and M. Wootters, Optimal entanglement-assisted one-shot classical communication, Phys. Rev. A 87, 062301 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.062301

[30] J. Barrett, Information processing in generalized probabilistic theories, Phys. Rev. A 75, 032304 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.032304

[31] A. Acín, T. Fritz, A. Leverrier, and A. B. Sainz, A Combinatorial Approach to Nonlocality and Contextuality, Communications in Mathematical Physics 334, 533 (2015).
https:/​/​doi.org/​10.1007/​s00220-014-2260-1

[32] R. W. Spekkens, The ontological identity of empirical indiscernibles: Leibniz's methodological principle and its significance in the work of Einstein, arXiv preprint arXiv:1909.04628 (2019).
https:/​/​doi.org/​10.48550/​arXiv.1909.04628
arXiv:1909.04628

[33] E. Wolfe, D. Schmid, A. B. Sainz, R. Kunjwal, and R. W. Spekkens, Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes, Quantum 4, 280 (2020).
https:/​/​doi.org/​10.22331/​q-2020-06-08-280

[34] M. F. Pusey, Robust preparation noncontextuality inequalities in the simplest scenario, Phys. Rev. A 98, 022112 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.022112

[35] A. Tavakoli and R. Uola, Measurement incompatibility and steering are necessary and sufficient for operational contextuality, Phys. Rev. Research 2, 013011 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.013011

[36] M. S. Leifer and O. J. E. Maroney, Maximally Epistemic Interpretations of the Quantum State and Contextuality, Phys. Rev. Lett. 110, 120401 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.120401

[37] L. P. Hughston, R. Jozsa, and W. K. Wootters, A complete classification of quantum ensembles having a given density matrix, Physics Letters A 183, 14 (1993).
https:/​/​doi.org/​10.1016/​0375-9601(93)90880-9

[38] M. Banik, S. S. Bhattacharya, S. K. Choudhary, A. Mukherjee, and A. Roy, Ontological models, preparation contextuality and nonlocality, Foundations of Physics 44, 1230 (2014).
https:/​/​doi.org/​10.1007/​s10701-014-9839-4

[39] P. Heywood and M. L. Redhead, Nonlocality and the Kochen-Specker paradox, Foundations of Physics 13, 481 (1983).
https:/​/​doi.org/​10.1007/​BF00729511

[40] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
https:/​/​doi.org/​10.1103/​RevModPhys.86.419

[41] S. Popescu and D. Rohrlich, Quantum nonlocality as an axiom, Foundations of Physics 24, 379 (1994).
https:/​/​doi.org/​10.1007/​BF02058098

[42] A. Peres, Two simple proofs of the Kochen-Specker theorem, Journal of Physics A: Mathematical and General 24, L175 (1991).
https:/​/​doi.org/​10.1088/​0305-4470/​24/​4/​003

[43] A. Peres, Incompatible results of quantum measurements, Physics Letters A 151, 107 (1990).
https:/​/​doi.org/​10.1016/​0375-9601(90)90172-K

[44] N. D. Mermin, Hidden variables and the two theorems of John Bell, Rev. Mod. Phys. 65, 803 (1993).
https:/​/​doi.org/​10.1103/​RevModPhys.65.803

[45] A. Peres, Quantum theory: concepts and methods, Vol. 57 (Springer Science & Business Media, 2006).
https:/​/​doi.org/​10.1007/​0-306-47120-5

[46] A. A. Klyachko, M. A. Can, S. Binicioğlu, and A. S. Shumovsky, Simple Test for Hidden Variables in Spin-1 Systems, Phys. Rev. Lett. 101, 020403 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.020403

[47] S. Uijlen and B. Westerbaan, A Kochen-Specker system has at least 22 vectors, New Generation Computing 34, 3 (2016).
https:/​/​doi.org/​10.1007/​s00354-016-0202-5

[48] F. Arends, A lower bound on the size of the smallest Kochen-Specker vector system, Master's thesis, Oxford University (2009).
http:/​/​www.cs.ox.ac.uk/​people/​joel.ouaknine/​download/​arends09.pdf

[49] R. Kunjwal, C. Heunen, and T. Fritz, Quantum realization of arbitrary joint measurability structures, Phys. Rev. A 89, 052126 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.052126

[50] N. Andrejic and R. Kunjwal, Joint measurability structures realizable with qubit measurements: Incompatibility via marginal surgery, Phys. Rev. Research 2, 043147 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.043147

[51] R. Kunjwal and S. Ghosh, Minimal state-dependent proof of measurement contextuality for a qubit, Phys. Rev. A 89, 042118 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.042118

[52] X. Zhan, E. G. Cavalcanti, J. Li, Z. Bian, Y. Zhang, H. M. Wiseman, and P. Xue, Experimental generalized contextuality with single-photon qubits, Optica 4, 966 (2017).
https:/​/​doi.org/​10.1364/​OPTICA.4.000966

[53] I. Marvian, Inaccessible information in probabilistic models of quantum systems, non-contextuality inequalities and noise thresholds for contextuality, arXiv preprint arXiv:2003.05984 (2020).
https:/​/​doi.org/​10.48550/​arXiv.2003.05984
arXiv:2003.05984

[54] T. S. Cubitt, D. Leung, W. Matthews, and A. Winter, Zero-error channel capacity and simulation assisted by non-local correlations, IEEE Transactions on Information Theory 57, 5509 (2011).
https:/​/​doi.org/​10.1109/​TIT.2011.2159047

[55] C. E. Shannon, A note on a partial ordering for communication channels, Information and control 1, 390 (1958).
https:/​/​doi.org/​10.1016/​S0019-9958(58)90239-0

[56] D. Schmid, T. C. Fraser, R. Kunjwal, A. B. Sainz, E. Wolfe, and R. W. Spekkens, Understanding the interplay of entanglement and nonlocality: motivating and developing a new branch of entanglement theory, arXiv preprint arXiv:2004.09194 (2020).
https:/​/​doi.org/​10.48550/​arXiv.2004.09194
arXiv:2004.09194

[57] L. Hardy, Nonlocality for two particles without inequalities for almost all entangled states, Phys. Rev. Lett. 71, 1665 (1993).
https:/​/​doi.org/​10.1103/​PhysRevLett.71.1665

[58] A. Cabello, J. Estebaranz, and G. García-Alcaine, Bell-Kochen-Specker theorem: A proof with 18 vectors, Physics Letters A 212, 183 (1996).
https:/​/​doi.org/​10.1016/​0375-9601(96)00134-X

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