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npj Quantum Information

Device-independent certification of Majorana fermions and a new self-testing equivalence

Patryk Michalski, Arturo Konderak, Wojciech Bruzda, Remigiusz Augusiak

Published Oct 24, 2026• 0 citations
Publisher

Abstract

Abstract Bell inequalities provide a fundamental tool for probing nonlocal correlations, yet their quantum bound, that is, the maximal value attainable by quantum strategies, is rarely accessible analytically. In this work, we introduce a general construction of Bell inequalities for which this bound can be computed exactly. The construction is based on a class of rectangular matrices with pairwise orthogonal rows and normalized columns, which we call row-orthogonal, column-normalized (ROCN) matrices. Our framework generalizes both the Clauser–Horne–Shimony–Holt inequality and Gisin’s elegant Bell inequality, yielding Bell expressions maximally violated by arbitrary families of pairwise anticommuting Clifford observables together with the corresponding maximally entangled state. Under suitable assumptions, these inequalities also enable the device-independent certification of Majorana fermions, understood as multiqubit realizations of Clifford algebra generators. Importantly, we identify an additional equivalence that must be incorporated into self-testing, beyond the standard equivalences generated by local isometries, the addition of auxiliary degrees of freedom, and complex conjugation. This equivalence arises from applying partial transposition jointly to the shared state and the measurements, which, whenever it is well-defined, preserves all observed correlations.

DOI
10.1038/s41534-026-01383-z

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Abstract Bell inequalities provide a fundamental tool for probing nonlocal correlations, yet their quantum bound, that is, the maximal value attainable by quantum strategies, is rarely accessible analytically. In this work, we introduce a general construction of Bell inequalities for which this bound can be computed exactly. The construction is based on a class of rectangular matrices with pairwise orthogonal rows and normalized columns, which we call row-orthogonal, column-normalized (ROCN) matrices. Our framework generalizes both the Clauser–Horne–Shimony–Holt inequality and Gisin’s elegant Bell inequality, yielding Bell expressions maximally violated by arbitrary families of pairwise anticommuting Clifford observables together with the corresponding maximally entangled state. Under suitable assumptions, these inequalities also enable the device-independent certification of Majorana fermions, understood as multiqubit realizations of Clifford algebra generators. Importantly, we identify an additional equivalence that must be incorporated into self-testing, beyond the standard equivalences generated by local isometries, the addition of auxiliary degrees of freedom, and complex conjugation. This equivalence arises from applying partial transposition jointly to the shared state and the measurements, which, whenever it is well-defined, preserves all observed correlations.
Bell inequalities provide a fundamental tool for probing nonlocal correlations, yet their quantum bound, that is, the maximal value attainable through quantum strategies, is rarely accessible analytically. In this work, we introduce a general construction of Bell inequalities for which this bound can be computed exactly. Our framework generalizes both the Clauser-Horne-Shimony-Holt and Gisin's elegant inequalities, yielding Bell expressions maximally violated by any number of pairwise anticommuting Clifford observables together with the corresponding maximally entangled state. Under suitable assumptions, our inequalities also enable the device-independent certification of Majorana fermions, understood as multiqubit realizations of Clifford algebra generators. Importantly, we identify an additional equivalence that must be incorporated into the definition of self-testing beyond invariance under local isometries and transposition. This equivalence arises from partial transposition applied to the shared state and to the measurements, which in specific cases leaves all observed correlations unchanged.
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The next frontier in device-independent quantum information lies in the certification of scalable and parallel quantum resources, which underpin advanced quantum technologies. We put forth a simultaneous self-testing framework for maximally entangled two-qudit state of local dimension m∗=2⌊n/2⌋m_*=2^{\lfloor n/2 \rfloor}m∗​=2⌊n/2⌋ (equivalently ⌊n/2⌋\lfloor n/2 \rfloor⌊n/2⌋ copies of maximally entangled two-qubit pairs), together with nnn numbers of anti-commuting observables on one side. To this end, we employ an nnn-settings Bell inequality comprising two space-like separated observers, Alice and Bob, having 2n−12^{n-1}2n−1 and nnn number of measurement settings, respectively. We derive the local ontic bound of this inequality and, crucially, employ the Sum-of-Squares decomposition to determine the optimal quantum bound without presupposing the dimension of the state or observables. We then establish that any physical realisation achieving the maximal quantum violation must, up to local isometries and complex conjugation, correspond to a reference strategy consisting of a maximally entangled state of local dimension of at least 2⌊n/2⌋2^{\lfloor n/2 \rfloor}2⌊n/2⌋ and local observables forming an irreducible representation of the Clifford algebra. This construction thereby demonstrates that the minimal dimension compatible with nnn mutually anticommuting observables is naturally self-tested by the maximal violation of the proposed Bell functional. Finally, we analyse the robustness of the protocol by establishing quantitative bounds relating deviations in the observed Bell value to the fidelity between the realised and the ideal strategies. Our results thus provide a scalable, dimension-independent route for the certification of high-dimensional entanglement and Clifford measurements in a fully device-independent framework.
Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is defined in the simplest scenario involving two dichotomic measurements and whose all key properties are well understood. There have been many attempts to generalise the CHSH Bell inequality to higher-dimensional quantum systems, however, for most of them the maximal quantum violation---the key quantity for most device-independent applications---remains unknown. On the other hand, the constructions for which the maximal quantum violation can be computed, do not preserve the natural property of the CHSH inequality, namely, that the maximal quantum violation is achieved by the maximally entangled state and measurements corresponding to mutually unbiased bases. In this work we propose a novel family of Bell inequalities which exhibit precisely these properties, and whose maximal quantum violation can be computed analytically. In the simplest scenario it recovers the CHSH Bell inequality. These inequalities involve d measurements settings, each having d outcomes for an arbitrary prime number d ≥ 3 . We then show that in the three-outcome case our Bell inequality can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which motivates the definition of a new weak form of self-testing. The ability to certify high-dimensional MUBs makes these inequalities attractive from the device-independent cryptography point of view.
Self-testing constitutes one of the most powerful forms of device certification, enabling a complete and device-independent characterization of a quantum apparatus solely from the observed correlations. In recent work by the authors [23], a general framework was introduced for constructing Bell inequalities that self-test entire families of Clifford generators. In this manuscript, we develop an alternative and complementary self-testing criterion based on symmetric spanning sets. This formulation provides an explicit and constructive route to designing self-testing Bell inequalities in arbitrary dimensions.
An experiment in which the Clauser-Horne-Shimony-Holt inequality is maximally violated is self-testing (i.e., it certifies in a device-independent way both the state and the measurements). We prove that an experiment maximally violating Gisin's elegant Bell inequality is not similarly self-testing. The reason can be traced back to the problem of distinguishing an operator from its complex conjugate. We provide a complete and explicit characterization of all scenarios in which the elegant Bell inequality is maximally violated. This enables us to see exactly how the problem plays out.