# 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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### Cited by

[1] Victor Gitton and Mischa P. Woods, "On the system loophole of generalized noncontextuality", arXiv:2209.04469.

[2] Lorenzo Catani, Matthew Leifer, David Schmid, and Robert W. Spekkens, "Why interference phenomena do not capture the essence of quantum theory", arXiv:2111.13727.

[3] John H. Selby, Elie Wolfe, David Schmid, and Ana Belén Sainz, "An open-source linear program for testing nonclassicality", arXiv:2204.11905.

[4] David Schmid, Haoxing Du, John H. Selby, and Matthew F. Pusey, "Uniqueness of Noncontextual Models for Stabilizer Subtheories", Physical Review Letters 129 12, 120403 (2022).

[5] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens, "Contextuality without incompatibility", arXiv:2106.09045.

[6] Armin Tavakoli, Emmanuel Zambrini Cruzeiro, Roope Uola, and Alastair A. Abbott, "Bounding and Simulating Contextual Correlations in Quantum Theory", PRX Quantum 2 2, 020334 (2021).

[7] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens, "Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality", arXiv:2112.04521.

[8] Lorenzo Catani and Matthew Leifer, "A mathematical framework for operational fine tunings", arXiv:2003.10050.

[9] Victoria J Wright and Ravi Kunjwal, "Contextuality in composite systems: the role of entanglement in the Kochen-Specker theorem", arXiv:2109.13594.

[10] Anubhav Chaturvedi, Máté Farkas, and Victoria J Wright, "Characterising and bounding the set of quantum behaviours in contextuality scenarios", arXiv:2010.05853.

[11] Lorenzo Catani, Ricardo Faleiro, Pierre-Emmanuel Emeriau, Shane Mansfield, and Anna Pappa, "Connecting XOR and XOR* games", arXiv:2210.00397.

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