Quantum nonlocality based on the violation of Bell-like inequalities constitutes a fundamental feature of quantum physics and drives the development of device-independent (DI) quantum information technologies. Existing studies of Bell nonlocality have mainly focused on physical qubit systems, where the observed nonlocal correlations are directly encoded in physical degrees of freedom. The decoherence sensitivity of Bell nonlocality largely limits the performance and security of its DI applications. Here, we investigate the robust logical Bell nonlocality based on quantum error correction codes. We construct the general logical Bell inequality in the stabilizer coding subspace and prove its violation indicates the global nonlocal feature of the logical system. Then, we indicate that the logical Bell nonlocality is robust against decoherence. Comparing with the physical qubit system, the fidelity thresholds for the logical Bell inequality violation based on the [[3,1,1]] and [[7,1,1]] repetition codes under the bit-flip error model can be reduced from 82.8% to 73.10% and 66.35%, increasing DI QKD's bit-flip noise threshold from 10.64% to 14.42% and 23.36%, respectively. Such stabilizer-based framework can be also used to characterize the multipartite logical Bell nonlocality in principle. Finally, a logical Bell test implementation circuit based on the [[3,1,1]] repetition code is presented. This work provides a feasible avenue for unlocking robust Bell nonlocality in scalable logical quantum systems and facilitates its applications in future scalable quantum network.
Quantum nonlocality based on the violation of Bell-like inequalities constitutes a fundamental feature of quantum physics and drives the development of device-independent (DI) quantum information technologies. Existing studies of Bell nonlocality have mainly focused on physical qubit systems, where the observed nonlocal correlations are directly encoded in physical degrees of freedom. The decoherence sensitivity of Bell nonlocality largely limits the performance and security of its DI applications. Here, we investigate the robust logical Bell nonlocality based on quantum error correction codes. We construct the general logical Bell inequality in the stabilizer coding subspace and prove its violation indicates the global nonlocal feature of the logical system. Then, we indicate that the logical Bell nonlocality is robust against decoherence. Comparing with the physical qubit system, the fidelity thresholds for the logical Bell inequality violation based on the [[3,1,1]] and [[7,1,1]] repetition codes under the bit-flip error model can be reduced from 82.8% to 73.10% and 66.35%, increasing DI QKD's bit-flip noise threshold from 10.64% to 14.42% and 23.36%, respectively. Such stabilizer-based framework can be also used to characterize the multipartite logical Bell nonlocality in principle. Finally, a logical Bell test implementation circuit based on the [[3,1,1]] repetition code is presented. This work provides a feasible avenue for unlocking robust Bell nonlocality in scalable logical quantum systems and facilitates its applications in future scalable quantum network.
Abstract We develop explicit device-independent self-testing constructions for logical subspaces of quantum error-correcting codes. By constructing multipartite Bell inequalities tailored to stabilizer code structures, we show that maximal quantum violations uniquely certify that the underlying shared state resides within the logical subspace of a given code, up to local isometries. These constructions extend earlier stabilizer-subspace self-testing results beyond the five-qubit code to several representative stabilizer codes, including the six-qubit code, the Steane code, and Shor’s nine-qubit code. The tilted sum-of-squares variants further impose algebraic constraints on the relative weights of the logical sectors. Together, these results provide explicit examples of device-independent self-testing for representative stabilizer-code logical subspaces and suggest possible directions for verification and benchmarking of encoded quantum information.
Abstract Bell inequalities have long been essential for the device-independent detection of entanglement. This study introduces subcorrelation Bell inequalities as a robust framework for detecting entanglement and probing quantum non-locality. We derived a family of such inequalities, categorized according to their dependence on measurement settings, and numerically identified 19 distinct quantum states that achieve the maximal violation of these inequalities. Importantly, our focus on maximal violation is not for its own sake but because, in this framework, maximal violation directly corresponds to maximal robustness of nonlocal correlations against particle loss. This connection reveals a broad class of multipartite entangled states that remain resilient under loss, making them highly relevant for practical implementations of quantum information protocols. We further characterized their entanglement properties using concurrence and the geometric measure, confirming their utility for critical areas such as quantum communication, cryptography, and device-independent protocols. The framework established herein sets the stage for future investigations, including the generalization of these inequalities to higher-dimensional systems and a more comprehensive characterization of their robustness against various forms of noise, thereby expanding the potential for non-local states to drive progress in quantum technologies.
In practical quantum networks, a variety of multiqubit stabilized states emitted from independent sources are distributed among the agents, and the correlations across the entire network can be derived from each agent's local measurements on the shared composite quantum systems.To reveal the Bell nonlocality in such cases as a quantum feature, minimal knowledge of the emitted stabilizer state is required.Here, we demonstrate that knowing the stabilizing and logical operators indeed provides a way of exploring Bell nonlocality in quantum networks.For the qubit distribution in quantum networks, the associated nonlinear Bell inequalities are derived.On the other hand, to violate these inequalities, one can design local incompatible observables using minimal knowledge of the emitted states.The tilted nonlinear Bell inequalities tailored for specific nonmaximal entangled stabilizer states and a way of achieving the maximal violation are also explored.
Abstract Quantum computers are powerful tools for certain computational problems, but their fragile quantum states are easily disrupted by environmental noise, causing errors that degrade computational accuracy. Quantum error correction (QEC) addresses this challenge by encoding one logical qubit across multiple physical qubits, so that individual errors can be identified and reversed through redundancy. We used Qiskit Aer, a free open-source quantum circuit simulator, to test the repetition code, one of the simplest QEC methods, under three types of noise: bit-flip, phase-flip, and depolarizing. We compared an unprotected single-qubit baseline (n = 1) to 3-qubit (n = 3) and 5-qubit (n = 5) repetition codes, sweeping the physical error rate p from 0 to 0.15 with 4,000 simulation shots per data point. We hypothesized that larger codes would achieve higher logical success probabilities at all tested noise levels, with n = 5 outperforming n = 3, which would outperform the unprotected baseline. Our results confirmed this scaling trend across all three noise types. At p = 0.10 under bit-flip noise, the baseline achieved 0.713 logical success, while n = 3 reached 0.791 and n = 5 reached 0.843. At low noise (p = 0.02), the n = 5 code approached 0.994 logical success. Simulation results closely matched exact theoretical predictions derived from the binomial failure probability formula, validating our experimental implementation. These findings confirm that even the simplest QEC methods provide meaningful error reduction and that larger codes offer greater protection, demonstrating a foundational principle of quantum error correction using only free simulation tools.
Abstract Quantum computers are powerful tools for certain computational problems, but their fragile quantum states are easily disrupted by environmental noise, causing errors that degrade computational accuracy. Quantum error correction (QEC) addresses this challenge by encoding one logical qubit across multiple physical qubits, so that individual errors can be identified and reversed through redundancy. We used Qiskit Aer, a free open-source quantum circuit simulator, to test the repetition code, one of the simplest QEC methods, under three types of noise: bit-flip, phase-flip, and depolarizing. We compared an unprotected single-qubit baseline (n = 1) to 3-qubit (n = 3) and 5-qubit (n = 5) repetition codes, sweeping the physical error rate p from 0 to 0.15 with 4,000 simulation shots per data point. We hypothesized that larger codes would achieve higher logical success probabilities at all tested noise levels, with n = 5 outperforming n = 3, which would outperform the unprotected baseline. Our results confirmed this scaling trend across all three noise types. At p = 0.10 under bit-flip noise, the baseline achieved 0.713 logical success, while n = 3 reached 0.791 and n = 5 reached 0.843. At low noise (p = 0.02), the n = 5 code approached 0.994 logical success. Simulation results closely matched exact theoretical predictions derived from the binomial failure probability formula, validating our experimental implementation. These findings confirm that even the simplest QEC methods provide meaningful error reduction and that larger codes offer greater protection, demonstrating a foundational principle of quantum error correction using only free simulation tools.
Fault tolerance is essential for scalable quantum technologies and is enabled by quantum error-correction codes. Bell-state measurements (BSMs) are a fundamental building block for modern quantum technologies such as measurement-based quantum computation and fusion-based quantum computation, as well as quantum networks. Therefore, performing BSMs on error-corrected qubits is a necessary step for achieving fault tolerance in these applications. In this work, we realise a logical BSM using linear optics, based on a two-qubit repetition code, an instance of a quantum parity code that allows detection of bit-flip errors, and experimentally achieve a mean success probability of (70.8 +/- 0.4)%. While standard linear-optical BSMs are fundamentally limited to a maximum success probability of 50%, this increased success probability enables higher secure key rates in quantum communication and facilitates the generation of large graph states for quantum computation. Since fault-tolerant schemes require error-correction codes regardless, this improvement comes at no additional resource overhead. Our results demonstrate that error-correction codes can be used to surpass the linear-optics limit of BSMs, which is an important step towards practical, fault-tolerant, and scalable photonic quantum technologies.