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arXiv:2508.01071

Robustly self-testing all maximally entangled states in every finite dimension

Uta Isabella Meyer, Ivan Šupić, Frédéric Grosshans, Damian Markham

Published May 19, 2026• 0 citations
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Abstract

Abstract We establish a device-independent, noise-tolerant certification of maximally entangled states in every finite dimension $d$. The core ingredient is a $d$-input, $d$-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 $d\geq 3$ 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 $\epsilon$-near-optimal strategy is, up to local isometries, within trace distance $\delta=\mathcal{O}(\sqrt{\epsilon})$ 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 $d$, thereby yielding a fully rigorous, robust, and genuinely qudit self-test of maximally entangled states in all finite dimensions.

DOI
10.21203/rs.3.rs-9359447/v1

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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.
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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 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.
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.
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.
Abstract Device-independent (DI) certification schemes are based on minimal assumptions about the quantum system under study, which makes the most desirable among certification schemes. However, they are often the most challenging to implement. In order to reduce the implementation cost one can consider semi-DI (SDI) schemes such as those based on quantum steering. Here we provide a construction of a family of steering inequalities which are tailored to large classes of d-outcomes projective measurements being a certain linear combination of the Heisenberg–Weyl operators on the untrusted side and a fixed set of known measurements on the trusted side. We then prove that the maximal quantum violation of those inequalities can be used for certification of those measurements and the maximally entangled state of two qudits. Importantly, in our self-testing proof, we do not assume the shared state to be pure, nor do we assume the measurements to be projective. Before concluding, we also show how robust to noise our self-testing statement is. We believe that our construction broadens the scope of SDI certification, paving the way for more general but still less costly quantum certification protocols.