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arXiv:2603.15018 · Quantum Science and Technology

Scalable Self-Testing of Mutually Anticommuting Observables and Maximally Entangled Two-Qudits

Souradeep Sasmal, Ritesh K. Singh, Prabuddha Roy, A. K. Pan

quant-phSubmitted Mar 16, 2026• Announced Mar 17, 2026• 0 citations
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Abstract

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 $m_*=2^{\lfloor n/2 \rfloor}$ (equivalently $\lfloor n/2 \rfloor$ copies of maximally entangled two-qubit pairs), together with $n$ numbers of anti-commuting observables on one side. To this end, we employ an $n$-settings Bell inequality comprising two space-like separated observers, Alice and Bob, having $2^{n-1}$ and $n$ 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 $2^{\lfloor n/2 \rfloor}$ and local observables forming an irreducible representation of the Clifford algebra. This construction thereby demonstrates that the minimal dimension compatible with $n$ 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.

DOI
10.1088/2058-9565/ae9800

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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.
Abstract We establish a device-independent, noise-tolerant certification of maximally entangled states in every finite dimension ddd. The core ingredient is a ddd-input, ddd-outcome Bell experiment that recovers the genuinely qudit Heisenberg–Weyl structure. Yielding the Tsirelson bound in closed form, the associated Bell operator has an exact sum-of-positive-operators decomposition for d≥3d\geq 3d≥3 prime, from which we reconstruct the Heisenberg–Weyl commutation relations on the support of the state. Representation-theoretically, once the twisted-commutation relations are established, the finite Stone–von Neumann theorem identifies the realization with the canonical Heisenberg–Weyl model; in addition, we provide an explicit SWAP-type local isometry that yields the robustness constants. This gives a genuinely qudit certification: rather than reducing to qubit subtests, the higher-dimensional Heisenberg-Weyl algebra itself is recovered. We show that any ϵ\epsilonϵ-near-optimal strategy is, up to local isometries, within trace distance δ=O(ϵ)\delta=\mathcal{O}(\sqrt{\epsilon})δ=O(ϵ​) of the ideal maximally entangled state (with dimension-explicit constants), and the implemented measurements are correspondingly close to the target observables. Via a tensor-factor argument, the prime-dimensional result extends the self-testing protocol to every composite dimension ddd, thereby yielding a fully rigorous, robust, and genuinely qudit self-test of maximally entangled states in all finite dimensions.
While quantum networks have been extensively studied as a natural extension of the standard Bell scenario with a richer correlation structure, general constructions of nonlinear Bell inequalities with self-testing properties are still largely lacking. In this work, we present a general method for constructing such inequalities in the simplest network scenario, in which two independent sources distribute bipartite quantum states to three spatially separated observers. These inequalities allow for arbitrary numbers of binary measurements and are maximally violated by maximally entangled states of the corresponding local dimensions together with sets of pairwise anticommuting Clifford observables. Importantly, their maximal quantum values can be determined analytically, which makes them particularly promising for device-independent applications. In particular, we prove that these Bell inequalities can be used to device-independently certify the underlying quantum network, including both the quantum states produced by the sources and the observables measured by all parties. To the best of our knowledge, this is the first self-testing result for quantum networks that relies solely on the maximal violation of a nonlocality witness.
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.
Abstract Bell nonlocality as a resource for device-independent certification schemes has been studied extensively in recent years. The strongest form of device-independent certification is referred to as self-testing, which given a device, certifies the promised quantum state as well as quantum measurements performed on it without any knowledge of the internal workings of the device. In spite of various results on self-testing protocols, it remains a highly nontrivial problem to propose a certification scheme of qudit–qudit entangled states based on violation of a single d -outcome Bell inequality. Here we address this problem and propose a self-testing protocol for the maximally entangled state of any local dimension using the minimum number of measurements possible, i.e., two per subsystem. Our self-testing result can be used to establish unbounded randomness expansion, log 2d{{{\mathrm{log}}}\,}_{2}dlog2​d log 2 d perfect random bits, while it requires only one random bit to encode the measurement choice.