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arXiv:2202.06908 · npj Quantum Information

An elegant scheme of self-testing for multipartite Bell inequalities

Ekta Panwar, Palash Pandya, Marcin Wieśniak

Published Jul 17, 2023• 18 citations
arXivHTMLPDF

Abstract

Abstract Self-testing is the most accurate form of certification of quantum devices. While self-testing in bipartite Bell scenarios has been thoroughly studied, self-testing in the more complex multipartite Bell scenarios remains largely unexplored. We present a simple and broadly applicable self-testing scheme for N -partite correlation Bell inequalities with two binary outcome observables per party. To showcase the versatility of our proof technique, we obtain self-testing statements for the MABK and WWWŻB family of linear Bell inequalities and Uffink’s family of quadratic Bell inequalities. In particular, we show that the N -partite MABK and Uffink’s quadratic Bell inequalities self-test the GHZ state and anti-commuting observables for each party. While the former uniquely specifies the state, the latter allows for an arbitrary relative phase. To demonstrate the operational relevance of the relative phase, we introduce Uffink’s complex-valued N partite Bell expression, whose extremal values self-test the GHZ states and uniquely specify the relative phase.

DOI
10.1038/s41534-023-00735-3

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Abstract Self-testing is the most accurate form of certification of quantum devices. While self-testing in bipartite Bell scenarios has been thoroughly studied, self-testing in the more complex multipartite Bell scenarios remains largely unexplored. We present a simple and broadly applicable self-testing scheme for N -partite correlation Bell inequalities with two binary outcome observables per party. To showcase the versatility of our proof technique, we obtain self-testing statements for the MABK and WWWŻB family of linear Bell inequalities and Uffink’s family of quadratic Bell inequalities. In particular, we show that the N -partite MABK and Uffink’s quadratic Bell inequalities self-test the GHZ state and anti-commuting observables for each party. While the former uniquely specifies the state, the latter allows for an arbitrary relative phase. To demonstrate the operational relevance of the relative phase, we introduce Uffink’s complex-valued N partite Bell expression, whose extremal values self-test the GHZ states and uniquely specify the relative phase.
Robust self-testing of the m−m-m−partite maximally entangled state and observables
As quantum technologies continue to advance rapidly, the device-independent testing of the functioning of a quantum device has become increasingly important. Self-testing, a correlation based protocol, enables such certification of a promised quantum state as well as measurements performed on it without requiring knowledge of the device's internal workings. This approach typically relies on achieving the optimal quantum violation of a suitable Bell inequality. Self-testing has been extensively investigated in the context of bipartite Bell experiments. However, its extension to multipartite scenarios remains largely unexplored, owing to the intricate nature of multipartite quantum correlations. In this work, we propose a simple and efficient self-testing protocol that certifies the state and observables based on the optimal quantum violation of the Svetlichny inequality involving an arbitrary number of parties, each with two inputs. Our method leverages an elegant sum-of-squares approach to derive the optimal quantum value of the Svetlichny functional, devoid of assuming the dimension of the quantum system. This enables the self-testing of the m−m-m−partite maximally entangled state and local anti-commuting observables for each party. Moreover, we develop a swap circuit isometry to assess the proximity of reference states and measurements to their ideal counterparts in the presence of noise and imperfections in real experiments, thereby demonstrating the robustness of our self-testing protocol. Finally, we illustrate how our self-testing protocol facilitates the generation of certified genuine randomness from correlations that enable the optimal violation of the Svetlichny inequality.
Self-testing is a device-independent technique that allows for verification of unknown quantum systems only when observed correlations achieve some quantum supremum. In this paper, we present self-testing schemes for a class of multipartite Bell inequalities with a specific structure, which we refer to as the unit. Our research is based on the graph-theoretic approach, where correlations are characterized via Lov\'asz theta body in the framework of exclusivity graphs. The derived statements are subsequently applied to prove the self-testability of NNN-partite Mermin-Ardehali-Belinski\ifmmode \check{i}\else \v{i}\fi{}-Klyshko (MABK) family of Bell inequalities. In particular, we take special cases to demonstrate that the maximal violation of the multipartite MABK inequality is a self-test for the corresponding multipartite Greenberger-Horne-Zeilinger (GHZ) state. Moreover, we propose a device-independent verification algorithm for distributed blind quantum computation in the noisy intermediate-scale quantum era combined with multipartite GHZ state self-testing, where multiple quantum servers are involved and work in collaboration to reduce the burden on a single server.
Self-testing is a device-independent examination of quantum devices based on correlations of observed statistics. Motivated by elegant progresses on self-testing strategies for measurements [Phys. Rev. Lett. 121, 250507 (2018)] and for states [New J. Phys. 20, 083041 (2018)], we develop a general self-testing procedure for multipartite generalized GHZ-state measurements. The key step is self-testing all measurement eigenstates for a general N-qubit multipartite GHZ-state measurement. Following our procedure, one only needs to perform local measurements on N-2 chosen parties of the N-partite eigenstate and maintain to achieve the maximal violation of tilted Clauser-Horne-Shimony-Holt (CHSH) Bell inequality for remaining two parties. Moreover, this approach is physically operational from an experimental point of view. It turns out that the existing result for three-qubit GHZ-state measurement is recovered as a special case. Meanwhile, we develop the self-testing method to be robust against certain white noise.
Bell inequality with a self-testing property has played an important role in quantum information processing with both fundamental and practical significance. However, it is generally challenging to find Bell inequalities used for self-testing multipartite states, and actually, there are not many known candidates. In this work we propose a systematic framework to construct Bell inequalities from stabilizers which are maximally violated by general stabilizer states, with two observables for each local party. We show that the constructed Bell inequalities can self-test any stabilizer state if and only if these stabilizers can uniquely determine the state in a device-dependent manner. This bridges the gap between device-independent and device-dependent verification methods. Our framework can provide plenty of Bell inequalities for self-testing NNN-party stabilizer states. Among them, we give two families of Bell inequalities with different advantages: (1) a family of Bell inequalities with a constant ratio of quantum and classical bounds using 2N2N2N correlations, and (2) single pair inequalities improving on all previous robustness self-testing bounds using N+1N+1N+1 correlations, which are both efficient and robust for realizations in multipartite systems.
Device-independent certification of multipartite entangled states plays a central role in a wide range of practical applications, including quantum networks, conference key agreement, and verifiable distributed quantum computation. A particularly important class of multipartite entangled states is the class of Greenberger-Horne-Zeilinger (GHZ) states. Many Bell operators have been proposed to self-test GHZ states. However, in practical scenarios, due to imperfections and the finite collection of statistics, the observed statistics do not satisfy the ideal self-testing relations. Hence, it becomes essential to investigate and compare the robustness of the different self-testing protocols. In this work, we investigate the robustness of self-testing schemes constructed from Bell operators due to Svetlichny and Mermin--Ardehali--Belinskii--Klyshko (MABK), using the analytic operator-inequality framework developed by Kaniewski [\href{https://doi.org/10.1103/PhysRevLett.117.070402}{Phys. Rev. Lett. 117, 070402 (2016)}]. We derive lower bounds on the extractable fidelity as a function of the observed value of these Bell operators. Although these protocols self-test the same underlying state, they exhibit markedly different levels of robustness. By comparing the resulting fidelity bounds, we demonstrate that the self-testing scheme based on the Svetlichny's Bell operator is the more robust among the two. Our results thus identify the Svetlichny operator based self-testing protocol as the most favorable candidate for device-independent certification of GHZ states in realistic, noisy experimental scenarios.
We consider the problem of certifying binary observables based on a Bell inequality violation alone, a task known as self-testing of measurements. We introduce a family of commutation-based measures, which encode all the distinct arrangements of two projective observables on a qubit. These quantities by construction take into account the usual limitations of self-testing and since they are ``weighted'' by the (reduced) state, they automatically deal with rank-deficient reduced density matrices. We show that these measures can be estimated from the observed Bell violation in several scenarios and the proofs rely only on standard linear algebra. The trade-offs turn out to be tight, and in particular, they give nontrivial statements for arbitrarily small violations. On the other extreme, observing the maximal violation allows us to deduce precisely the form of the observables, which immediately leads to a complete rigidity statement. In particular, we show that for all $n\ensuremath{\ge}3$ the nnn-partite Mermin-Ardehali-Belinskii-Klyshko inequality self-tests the nnn-partite Greenberger-Horne-Zeilinger state and maximally incompatible qubit measurements on every party. Our results imply that any pair of projective observables on a qubit can be certified in a truly robust manner. Finally, we show that commutation-based measures give a convenient way of expressing relations among more than two observables.