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arXiv:2105.03837

Exploring Bell nonlocality of quantum networks with stabilizing and logical operators

Li‐Yi Hsu, Ching-Hsu Chen

Published May 9, 2021• 0 citations
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Abstract

In practical quantum networks, a variety of multi-qubit 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 non-locality 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 new way of exploring Bell non-locality 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 non-maximal entangled stabilizer states and a way of achieving the maximal violation are also explored.

DOI
10.48550/arxiv.2105.03837

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In practical quantum networks, a variety of multi-qubit 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 non-locality 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 new way of exploring Bell non-locality 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 non-maximal entangled stabilizer states and a way of achieving the maximal violation are also explored.
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 In K-locality networks, local hidden variables emitted from classical sources are distributed among limited observers. We explore genuine Bell locality in classical networks, where, regarding all local hidden variables as classical objects that can be perfectly cloned and spread throughout the networks, any observer can access all local hidden variables plus shared randomness. In the proposed linear and nonlinear Bell-type inequalities, there are more correlators to reveal genuine Bell locality than those in the K-locality inequalities, and their upper bounds can be specified using the probability normalization of the predetermined probability distribution. On the other hand, the no-cloning theorem limits the broadcast of quantum correlations in quantum networks. To explore genuine Bell nonlocality, the stabilizing operators play an important role in designing the segmented Bell operators and assigning the incompatible measurements for the spatially separated observers. We prove the maximal violations of the proposed Bell-type inequalities tailored for the given qubit distributions in quantum networks.
Abstract In a generic hybrid network, classical, quantum, and no‐signaling sources emit local hidden variables, stabilizer states, and no‐signaling systems, respectively. The maximal correlation strength is investigated as the non‐classical feature in this network. Given the associated fully‐quantum network of a hybrid network, the stabilizing operators of the distributed quantum state are exploited to construct segmented Bell operators and the Bell inequalities tailored to the state. The upper bounds of the maximal correlation strengths in the associated full‐classical, full‐quantum, and fully‐no‐signaling networks are derived as the benchmarks. The study shows that the achievable correlation strength depends on the number of type‐ measurements and that of nonlocal sources. The ‐nonlocality criteria are also introduced, indicating that the achievable maximal correlation strength cannot modeled by the network with at least observers with local hidden variables performing type‐ measurements.
Abstract We initiate the study of the nonlocal correlations in generic asymmetric quantum networks in a star configuration. Therein, the diverse unrelated sources can emit either partially or maximally entangled states, while the observers employ varying numbers of measurement settings. We propose nonlinear Bell inequalities tailored to the distributed entangled states. Specifically, we demonstrate that the algebraic maximal violations of the proposed nonlinear Bell inequalities are physically achievable within the quantum region. To achieve this, we construct the segmented Bell operators through the cut-graft-mix method applied to the Bell operators in the standard Bell tests. Furthermore, we devise the fitting Bell operators using the sum-of-square approach.
In the literature on KKK-locality (K≥2K\geq2K≥2) networks, the local hidden variables are strictly distributed in the specific observers rather than the whole ones. Regarding genuine Bell locality, all local hidden variables, as classical objects that allow for perfect cloning in classical physics, should be cloned and then spread throughout the networks. More correlators are involved in the proposed linear and non-linear Bell-type inequalities, where their upper bounds are specified by the pre-determined output probability distribution. As for the quantum version, the no-clone theorem limits the broadcast of quantum correlations. To explore genuine Bell nonlocality in variant particle distributions in the networks, the Pauli operators stabilizing the two-qubit Bell states or multi-qubit Greenberger--Horne--Zeilinger states (GHZ states) play an essential role in designing the proposed linear and non-linear Bell tests and assigning the local incompatible measurements for the spatially separated observers. We prove the maximal violations of the proposed Bell-type inequalities quantum networks. In the end, how entanglement swapping replaces the joint measurements in the Bell tests is demonstrated.
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.