# On entanglement assistance to a noiseless classical channel

Péter E. Frenkel1,2 and Mihály Weiner3,4

1Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest, 1117 Hungary
2Rényi Institute, Budapest, Reáltanoda u. 13-15, 1053 Hungary
3Budapest University of Technology and Economics (BME), Department of Analysis, H-1111 Budapest Műegyetem rkp. 3–9 Hungary
4MTA-BME Lendület Quantum Information Theory Research Group

### Abstract

For a classical channel, neither the Shannon ca-pa-city, nor the sum of conditional probabilities corresponding to the cases of successful transmission can be increased by the use of shared entanglement, or, more generally, a non-signaling resource. Yet, perhaps somewhat counterintuitively, entanglement assistance can help and actually elevate the chances of success even in a one-way communicational task that is to be completed by a single-shot use of a noiseless classical channel.

To quantify the help that a non-signaling resource provides to a noiseless classical channel, one might ask how many extra letters should be added to the alphabet of the channel in order to perform equally well $without$ the specified non-signaling resource. As was observed by Cubitt, Leung, Matthews, and Winter, there is no upper bound on the number of extra letters required for substituting the assistance of a general non-signaling resource to a noiseless one-bit classical channel. In contrast, here we prove that if this resource is a bipartite quantum system in a maximally entangled state, then an extra classical bit always suffices as a replacement.

You can find the authors’ online classes related to this publication here.

### ► References

[1] A. Acín, T. Durt, N. Gisin and J. I. Latorre: Quantum nonlocality in two three-level systems. Phys. Rev. A 65 (2002), 052325. https:/​/​doi.org/​10.1103/​physreva.65.052325.
https:/​/​doi.org/​10.1103/​physreva.65.052325

[2] C. H. Bennett, P. W. Shor, J. A. Smolin and A. V. Thapliyal: Entanglement-Assisted Classical Capacity of Noisy Quantum Channels Phys. Rev. Lett. 83, (1999) 3081. https:/​/​doi.org/​10.1103/​PhysRevLett.83.3081.
https:/​/​doi.org/​10.1103/​PhysRevLett.83.3081

[3] C. H. Bennett and S. J. Wiesner: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69 (1992), pg. 2881–2884. https:/​/​doi.org/​10.1103/​PhysRevLett.69.2881.
https:/​/​doi.org/​10.1103/​PhysRevLett.69.2881

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

[5] T. S. Cubitt, D. Leung, W. Matthews and A. Winter: Zero-Error Channel Capacity and Simulation Assisted by Non-Local Correlations. IEEE Trans. Inf. Theory 57 (2011), pg. 5509–5523. https:/​/​doi.org/​10.1109/​TIT.2011.2159047.
https:/​/​doi.org/​10.1109/​TIT.2011.2159047

[6] M. Dall'Arno, S. Brandsen, A. Tosini, F. Buscemi and V. Vedral: No-hypersignaling principle. Phys. Rev. Lett. 119 (2017), 020401. https:/​/​doi.org/​10.1103/​PhysRevLett.119.020401.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.020401

[7] P. E. Frenkel, Classical simulations of communication channels, arXiv: 2101.10985.

[8] P. E. Frenkel and M. Weiner: Classical information storage in an $n$-level quantum system. Commun. Math. Phys. 340 (2015), pg. 563–574. https:/​/​doi.org/​10.1007/​s00220-015-2463-0.
https:/​/​doi.org/​10.1007/​s00220-015-2463-0

[9] S. Hao, H. Shi, W. Li, Q. Zhuang and Z. Zhang: Entanglement-Assisted Communication Surpassing the Ultimate Classical Capacity. Phys. Rev. Lett. 126, (2021) 250501. https:/​/​doi.org/​10.1103/​PhysRevLett.126.250501.
https:/​/​doi.org/​10.1103/​PhysRevLett.126.250501

[10] F. Leditzky, M. A. Alhejji, J. Levin and G. Smith: Playing games with multiple access channels. Nat. Commun. 11, (2020) 1497. https:/​/​doi.org/​10.1038/​s41467-020-15240-w.
https:/​/​doi.org/​10.1038/​s41467-020-15240-w

[11] L. Lovász and M. D. Plummer: Matching Theory. North-Holland, 1986.

[12] L. Mančinska and D. E. Roberson: Quantum homomorphisms. J. Comb. Theory Ser. B. 118 (2016), pg. 228–267. https:/​/​doi.org/​10.4230/​LIPIcs.TQC.2014.212.
https:/​/​doi.org/​10.4230/​LIPIcs.TQC.2014.212

[13] K. Matsumoto and G. Kimura: Information storing yields a point-asymmetry of state space in general probabilistic theories. arXiv:1802.01162.
arXiv:1802.01162

[14] M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter and M. Zukowski: Information causality as a physical principle. Nature 461 (2009), pg. 1101–1104. https:/​/​doi.org/​10.1038/​nature08400.
https:/​/​doi.org/​10.1038/​nature08400

[15] S. Popescu and D. Rohrlich: Quantum nonlocality as an axiom. Found. Phys. 24, pg. (1994) 379–385. https:/​/​doi.org/​10.1007/​BF02058098.
https:/​/​doi.org/​10.1007/​BF02058098

### Cited by

[1] Michele Dall&apos;Arno, "The signaling dimension of physical systems", Quantum Views 6, 66 (2022).

[2] Jef Pauwels, Stefano Pironio, Emmanuel Zambrini Cruzeiro, and Armin Tavakoli, "Adaptive Advantage in Entanglement-Assisted Communications", Physical Review Letters 129 12, 120504 (2022).

[3] Jef Pauwels, Armin Tavakoli, Erik Woodhead, and Stefano Pironio, "Entanglement in prepare-and-measure scenarios: many questions, a few answers", New Journal of Physics 24 6, 063015 (2022).

[4] Ram Krishna Patra, Sahil Gopalkrishna Naik, Edwin Peter Lobo, Samrat Sen, Tamal Guha, Some Sankar Bhattacharya, Mir Alimuddin, and Manik Banik, "Classical superdense coding and communication advantage of a single quantum", arXiv:2202.06796.

[5] Martin J. Renner, Armin Tavakoli, and Marco Túlio Quintino, "The classical cost of transmitting a qubit", arXiv:2207.02244.

[6] Armin Tavakoli, Jef Pauwels, Erik Woodhead, and Stefano Pironio, "Correlations in Entanglement-Assisted Prepare-and-Measure Scenarios", PRX Quantum 2 4, 040357 (2021).

[7] Vaisakh M, Ram krishna Patra, Mukta Janpandit, Samrat Sen, Manik Banik, and Anubhav Chaturvedi, "Mutually unbiased balanced functions and generalized random access codes", Physical Review A 104 1, 012420 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2022-10-04 21:14:35) and SAO/NASA ADS (last updated successfully 2022-10-04 21:14:36). The list may be incomplete as not all publishers provide suitable and complete citation data.