Boosting device-independent cryptography with tripartite nonlocality

Federico Grasselli, Gláucia Murta, Hermann Kampermann, and Dagmar Bruß

Institut für Theoretische Physik III, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany

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Abstract

Device-independent (DI) protocols, such as DI conference key agreement (DICKA) and DI randomness expansion (DIRE), certify private randomness by observing nonlocal correlations when two or more parties test a Bell inequality. While most DI protocols are restricted to bipartite Bell tests, harnessing multipartite nonlocal correlations may lead to better performance. Here, we consider tripartite DICKA and DIRE protocols based on testing multipartite Bell inequalities, specifically: the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality, and the Holz and the Parity-CHSH inequalities introduced in the context of DICKA protocols. We evaluate the asymptotic performance of the DICKA (DIRE) protocols in terms of their conference key rate (net randomness generation rate), by deriving lower bounds on the conditional von Neumann entropy of one party's outcome and two parties' outcomes. For the Holz inequality, we prove a tight analytical lower bound on the one-outcome entropy and conjecture a tight lower bound on the two-outcome entropy. We additionally re-derive the analytical one-outcome entropy bound for the MABK inequality with a much simpler method and obtain a numerical lower bound on the two-outcome entropy for the Parity-CHSH inequality. Our simulations show that DICKA and DIRE protocols employing tripartite Bell inequalities can significantly outperform their bipartite counterparts. Moreover, we establish that genuine multipartite entanglement is not a precondition for multipartite DIRE while its necessity for DICKA remains an open question.

In device-independent protocols, the amount of secret randomness that can be certified depends on the nonlocality of the observed correlations, quantified by the violation of a Bell inequality. Most device-independent protocols are based on the violation of bipartite Bell inequalities (e.g. the CHSH inequality). In our work, we show that multipartite nonlocal correlations, testified by the violation of multipartite Bell inequalities, enable the certification of more secret randomness from the outcomes of one or two parties. This is achieved by deriving analytical and numerical bounds on the relevant conditional von Neumann entropies.

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[2] Ekta Panwar, Palash Pandya, and Marcin Wieśniak, "An elegant scheme of self-testing for multipartite Bell inequalities", npj Quantum Information 9 1, 71 (2023).

[3] Peter Brown, Hamza Fawzi, and Omar Fawzi, "Device-independent lower bounds on the conditional von Neumann entropy", arXiv:2106.13692, (2021).

[4] Lewis Wooltorton, Peter Brown, and Roger Colbeck, "Expanding bipartite Bell inequalities for maximum multi-partite randomness", arXiv:2308.07030, (2023).

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