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arXiv:2505.19917 · Physical Review A

Robust self-testing and certified randomness based on an arbitrary-input Bell inequality

Rajdeep Paul, Sneha Munshi, A. K. Pan

Published Feb 27, 2026• 0 citations
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Abstract

Self-testing is the strongest certification procedure that uniquely characterizes the physical system based on the observed statistics, without any knowledge of the inner workings of the devices. The optimal quantum violation of a Bell inequality enables such a device-independent (DI) self-testing of the source and the measurement devices. In this work, we demonstrate the DI self-testing based on the arbitrary-input chained Bell inequality. We devise a systematic and elegant sum-of-squares (SOS) technique enabling dimension-independent optimization of the quantum violation. Our approach enables the derivation of the state along with the relationship between the local observables directly from the optimization condition. One significant aspect is the robustness of such self-testing in real experimental situations involving noise and imperfection, leading to deviation from the optimal quantum violation. We provide an analytical technique for robust self-testing in the presence of noise. As an application of our scheme, we demonstrate the generation of two bit DI randomness and analyze the robustness of such randomness. Our optimization method is both simple and elegant, making it suitable for deriving the optimal quantum violation of various arbitrary-input Bell inequalities.

DOI
10.1103/z4tp-mc5z

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Self-testing is the strongest certification procedure that uniquely characterizes the physical system based on the observed statistics, without any knowledge of the inner workings of the devices. The optimal quantum violation of a Bell inequality enables such a device-independent (DI) self-testing of the source and the measurement devices. In this work, we demonstrate the DI self-testing based on the arbitrary-input chained Bell inequality. We devise a systematic and elegant sum-of-squares (SOS) technique enabling dimension-independent optimization of the quantum violation. Our approach enables the derivation of the state along with the relationship between the local observables directly from the optimization condition. One significant aspect is the robustness of such self-testing in real experimental situations involving noise and imperfection, leading to deviation from the optimal quantum violation. We provide an analytical technique for robust self-testing in the presence of noise. As an application of our scheme, we demonstrate the generation of two bit DI randomness and analyze the robustness of such randomness. Our optimization method is both simple and elegant, making it suitable for deriving the optimal quantum violation of various arbitrary-input Bell inequalities.
Self-testing refers to the strongest device-independent (DI) certification method that validates the nature of a quantum system and devices solely based on the observed statistics. We demonstrate the self-testing of state and measurements based on the Gisin Bell inequality (GBI) featuring arbitrary inputs for both parties. We introduce a systematic and elegant sum-of-squares (SOS) approach that enables the dimension-independent derivation of the optimal quantum violation of GBI. We derive the state and the interrelation between the local observables directly from the optimization condition. Since the practical experimental scenario involves inevitable noise and imperfection, we present a comprehensive strategy for robust self-testing.
Bell nonlocality provides the foundation for device-independent (DI) certification of quantum devices. We introduce a Bell inequality capable of identifying genuine multipartite nonlocality (GMNL) in an arbitrary m-partite scenario with an arbitrary odd number of measurements per party. Since the multi-setting nature of this inequality precludes the use of Jordan's Lemma, we construct an analytical sum-of-squares (SOS) decomposition to obtain the optimal quantum violation without assuming any bound on the Hilbert space dimension. This, in turn, enables self-testing of the shared entangled state and the corresponding measurement observables, up to local isometries, whose existence we confirm using a swap-based certification scheme. In addition, we show that our framework enables the extraction of maximal global DI randomness (m bits) at the optimal quantum violation, thereby exceeding previous limitations in the GMNL regime. Finally, we demonstrate that the architecture of our inequality yields improved robustness to noise as the number of measurement settings grows, ensuring experimental feasibility.
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 Semi-device-independent certification of an unsharp instrument has recently been demonstrated (2019 New J. Phys. 21 083034) based on the sequential sharing of quantum advantages in a prepare-measure communication game by assuming the system to be qubit. In this work, we provide device-independent (DI) self-testing of the unsharp instrument through the quantum violation of two Bell inequalities where the devices are uncharacterized and the dimension of the system remains unspecified. We introduce an elegant sum-of-squares approach to derive the dimension-independent optimal quantum violation of Bell inequalities which plays a crucial role. Note that the standard Bell test cannot self-test the post-measurement states and consequently cannot self-test unsharp instrument. The sequential Bell test possess the potential to self-test an unsharp instrument. We demonstrate that there exists a trade-off between the maximum sequential quantum violations of the Clauser–Horne–Shimony–Holt inequality, and they form an optimal pair that enables the DI self-testing of the entangled state, the observables, and the unsharpness parameter. Further, we extend our study to the case of elegant Bell inequality and we argue that it has two classical bounds—the local bound and the non-trivial preparation non-contextual bound, lower than the local bound. Based on the sharing of preparation contextuality by three independent sequential observers, we demonstrate the DI self-testing of two unsharpness parameters. Since an actual experimental scenario involves losses and imperfection, we demonstrate robustness of our certification to noise.
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 Quantum entanglement is the key resource for quantum information processing. Device-independent certification of entangled states is a long standing open question, which arouses the concept of self-testing. The central aim of self-testing is to certify the state and measurements of quantum systems without any knowledge of their inner workings, even when the used devices cannot be trusted. Specifically, utilizing Bell’s theorem, one can infer the appearance of certain entangled state when the maximum violation is observed, e.g., to self-test singlet state using CHSH inequality. In this work, by constructing a versatile entanglement source, we experimentally demonstrate a generalized self-testing proposal for various bipartite entangled states up to four dimensions. We show that the high-quality generated states can approach the maximum violations of the utilized Bell inequalities, and thus, their Schmidt coefficients can be precisely inferred by self-testing them into respective target states with near-unity fidelities. Our results indicate the superior completeness and robustness of this method and promote self-testing as a practical tool for developing quantum techniques.