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

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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.

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

[2] Sahil Gopalkrishna Naik, Govind Lal Sidhardh, Samrat Sen, Arup Roy, Ashutosh Rai, and Manik Banik, "Distilling Nonlocality in Quantum Correlations", Physical Review Letters 130 22, 220201 (2023).

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

[4] Martin J. Renner, Armin Tavakoli, and Marco Túlio Quintino, "Classical Cost of Transmitting a Qubit", Physical Review Letters 130 12, 120801 (2023).

[5] Eric Chitambar, Ian George, Brian Doolittle, and Marius Junge, "The Communication Value of a Quantum Channel", IEEE Transactions on Information Theory 69 3, 1660 (2023).

[6] Mir Alimuddin, Ananya Chakraborty, Govind Lal Sidhardh, Ram Krishna Patra, Samrat Sen, Snehasish Roy Chowdhury, Sahil Gopalkrishna Naik, and Manik Banik, "Advantage of Hardy's nonlocal correlation in reverse zero-error channel coding", Physical Review A 108 5, 052430 (2023).

[7] Ram Krishna Patra, Sahil Gopalkrishna Naik, Edwin Peter Lobo, Samrat Sen, Tamal Guha, Some Sankar Bhattacharya, Mir Alimuddin, and Manik Banik, "Classical analogue of quantum superdense coding and communication advantage of a single quantum system", Quantum 8, 1315 (2024).

[8] 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).

[9] Carlos Vieira, Carlos de Gois, Lucas Pollyceno, and Rafael Rabelo, "Interplays between classical and quantum entanglement-assisted communication scenarios", New Journal of Physics 25 11, 113004 (2023).

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

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