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arXiv:2002.01843 · Physical Review Research

Constructing multipartite Bell inequalities from stabilizers

Qi Zhao, You Zhou

Published Dec 26, 2022• 5 citations
arXivHTMLPDF

Abstract

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 $N$-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 $2N$ correlations, and (2) single pair inequalities improving on all previous robustness self-testing bounds using $N+1$ correlations, which are both efficient and robust for realizations in multipartite systems.

DOI
10.1103/physrevresearch.4.043215

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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.
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Abstract Self-testing was originally introduced as a device-independent method of certification of entangled quantum states and local measurements performed on them. Recently, in (Baccari et al 2020 Phys. Rev. Lett. 125 260507) the notion of state self-testing has been generalized to entangled subspaces and the first self-testing strategies for exemplary genuinely entangled subspaces have been given. The main aim of our work is to pursue this line of research and to address the question how ‘large’ (in terms of dimension) are genuinely entangled subspaces that can be self-tested, concentrating on the multiqubit stabilizer formalism. To this end, we first introduce a framework allowing to efficiently check whether a given stabilizer subspace is genuinely entangled. Building on it, we then determine the maximal dimension of genuinely entangled subspaces that can be constructed within the stabilizer subspaces and provide an exemplary construction of such maximally-dimensional subspaces for any number of qubits. Third, we construct Bell inequalities that are maximally violated by any entangled state from those subspaces and thus also any mixed states supported on them, and we show these inequalities to be useful for self-testing. Interestingly, our Bell inequalities allow for identification of higher-dimensional face structures in the boundaries of the sets of quantum correlations in the simplest multipartite Bell scenarios in which every observer performs two dichotomic measurements.
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Bell nonlocality provides a robust scalable route to the efficient certification of quantum states. Here, we introduce a general framework for constructing Bell inequalities tailored to the Zd\mathbb{Z}_dZd​ toric code for odd prime local dimensions. Selecting a suitable subset of stabilizer operators and mapping them to generalized measurement observables, we compute multipartite Bell expressions whose quantum maxima admit a sum-of-squares decomposition. We show that these inequalities are maximally violated by all states in the ground-state manifold of the Zd\mathbb{Z}_dZd​ toric code, and determine their classical (local) bounds through a combination of combinatorial tiling arguments and explicit optimization. As a concrete application, we analyze the case of d=3d=3d=3 and demonstrate that the maximal violation self-tests the full qutrit toric-code subspace, up to local isometries and complex conjugation. This constitutes, to our knowledge, the first-ever example of self-testing a qutrit subspace. Extending these constructions, we further present schemes to enhance the ratio of classical--quantum bounds and thus improve robustness to experimental imperfections. Our results establish a pathway toward device-independent certification of highly entangled topological quantum matter and provide new tools for validating qudit states in error-correcting codes and quantum simulation platforms.
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.