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arXiv:2209.12828 · Quantum

Boosting device-independent cryptography with tripartite nonlocality

Federico Grasselli, Gláucia Murta, Hermann Kampermann, Dagmar Bruß

Published Apr 13, 2023• 16 citations
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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.

DOI
10.22331/q-2023-04-13-980

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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.
Multiparty quantum cryptography based on distributed entanglement will find its natural application in the upcoming quantum networks.The security of many multipartite device-independent (DI) protocols, such as DI conference-key agreement, relies on bounding the von Neumann entropy of the parties' outcomes conditioned on the eavesdropper's information, given the violation of a multipartite Bell inequality.We consider three parties testing the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and certify the privacy of their outcomes by bounding the conditional entropy of a single party's outcome and the joint conditional entropy of two parties' outcomes.From the former bound, we show that genuine multipartite entanglement is necessary to certify the privacy of a party's outcome, while the latter significantly improves previous results.We obtain the entropy bounds thanks to two general results of independent interest.The first one drastically simplifies the quantum setup of an N -partite Bell scenario.The second one provides an upper bound on the violation of the MABK inequality by an arbitrary N -qubit state, as a function of the state's parameters.
Bell nonlocality provides a device-independent (DI) way to certify quantum randomness, based on which true random numbers can be extracted from the observed correlations without detail characterizations on devices for quantum state preparation and measurement. However, the efficiency of current strategies for DI randomness certification is still heavily constrained when it comes to non-maximal Bell values, especially for multiple parties. Here, we present a family of multipartite Bell inequalities that allows to certify optimal quantum randomness and self-test GHZ (Greenberger-Horne-Zeilinger) states, which are inspired from the stabilizer group of the GHZ state. Due to the simple representation of stabilizer group for GHZ states, this family of Bell inequalities is of simple structure and can be easily expanded to more parties. Compared with the Mermin-type inequalities, this family of Bell inequality is more efficient in certifying quantum randomness when non-maximal Bell values achieved. Meanwhile, the general analytical upper bound for the Holevo quantity is presented, and achieves better performance compared with the MABK (Mermin-Ardehali-Belinskii-Klyshko) inequality, Parity-CHSH (Clauser-Horne-Shimony-Holt) inequality and Holz inequality at N=3N=3N=3, which is of particular interests for experimental researches on DI quantum cryptography in quantum networks.
Conference key agreement aims to establish shared, private randomness among many separated parties in a network. Device-independent conference key agreement (DICKA) is a variant in which the source and the measurement devices used by each party need not be trusted. So far, DICKA protocols largely fall into two categories: those that rely on violating a joint Bell inequality using genuinely multipartite entangled states and those that concatenate many bipartite protocols. The question of whether a hybrid protocol exists, where a multipartite Bell inequality can be violated using only bipartite entanglement, was asked by Grasselli et al. [Quantum 7, 980 (2023)2521-327X10.22331/q-2023-04-13-980]. We answer this question affirmatively, by constructing an asymptotically secure DICKA protocol achieving the same rate as the concatenation of bipartite device-independent quantum key distribution, yet relying on a single joint Bell violation. Our results prompt further discussion on the benefits of multipartite entanglement for DICKA over its bipartite alternative, and we give an overview of different arguments for near-term devices.
Conference key agreement allows multiple remote parties to establish a shared secret key with information-theoretical security. In device-independent conference key agreement, security can be guaranteed with minimal assumptions on the devices used, provided that a violation of a Bell inequality is observed. However, implementations are extremely challenging because high detection efficiency is required to observe loophole-free Bell violations. Here, we enhance the robustness of device-independent conference key agreement by introducing a new family of multipartite Bell inequalities called the asymmetric parity-Clauser-Horne-Shimony-Holt (CHSH) inequalities. We derive a tight analytical lower bound on the conditional von Neumann entropy of the outcomes of one of the parties in a protocol based on this inequality, including noisy preprocessing. Using this bound, we analyze robustness to detection inefficiencies as well as local and global depolarizing noise. We show that the combination of the asymmetric parity-CHSH inequality and noisy preprocessing can significantly improve the robustness to imperfections.
Device-independent conference key agreement (DI-CKA) enables N>2N > 2N>2 distant parties to establish a common secret key without detailed characterization of the source or measurement devices. Many existing protocols rely on full-body Bell correlators, whose measurement can be demanding for a large number of parties. Here, we show that two-body correlators alone suffice for DI-CKA using balanced Dicke states for every even N≥4N \geq 4N≥4. We construct two-body Bell inequalities and derive a tight analytical lower bound on the conditional von Neumann entropy by comparing their Bell operators with those of asymmetric Clauser-Horne-Shimony-Holt inequalities. In the ideal setting, certified key rates can approach 1−h[(N−2)/(4(N−1))]1-h[(N-2)/(4(N-1))]1−h[(N−2)/(4(N−1))], where hhh is the binary entropy function. This rate remains bounded away from zero as N→∞N \rightarrow \inftyN→∞, tending to 1−h(1/4)≈0.1891-h(1/4) \approx 0.1891−h(1/4)≈0.189 bits per key-generation round. We show that the honest implementation of our protocol is invariant under collective phase noise. Moreover, a complementary full-body Bell construction certifies positive conference-key rates from correlations arbitrarily close to the local set.
Nonlocal tests on multi-partite quantum correlations form the basis of protocols that certify randomness in a device-independent (DI) way. Such correlations admit a rich structure, making the task of choosing an appropriate test difficult. For example, extremal Bell inequalities are tight witnesses of nonlocality, but achieving their maximum violation places constraints on the underlying quantum system, which can reduce the rate of randomness generation. As a result there is often a trade-off between maximum randomness and the amount of violation of a given Bell inequality. Here, we explore this trade-off for more than two parties. More precisely, we study the maximum amount of randomness that can be certified by correlations with a particular violation of the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality. For any even number of parties, we find that maximum randomness cannot occur beyond a threshold quantum violation, which increases with the number of parties, and we give a conjectured form of the maximum randomness in terms of the MABK value. We also show that maximum randomness can be obtained for any MABK violation for odd numbers of parties. To obtain our results, we derive new families of Bell inequalities certifying maximum randomness from a technique for randomness certification, which we call "expanding Bell inequalities''. Our technique allows a bipartite Bell expression to be used as a seed, and transformed into a multi-partite Bell inequality tailored for randomness certification, showing how intuition learned in the bipartite case can find use in more complex scenarios.