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arXiv:2303.07377 · Physical Review Research

Robustness of Bell violation of graph states to qubit loss

Shahar Silberstein, Rotem Arnon-Friedman

Published Oct 30, 2023• 1 citation
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Abstract

Graph states are special entangled states advantageous for many quantum technologies, including quantum error correction, multiparty quantum communication, and measurement-based quantum computation. Yet, their fidelity is often disrupted by various errors, most notably qubit loss. In general, given an entangled state, Bell inequalities can be used to certify whether quantum entanglement remains despite errors. Here, we study the robustness of graph states to loss in terms of their Bell violation. Considering the recently proposed linearly scalable Bell operators by Baccari et al. [Phys. Rev. Lett. 124, 020402 (2020)], we use the stabilizer formalism to derive a formula for the extent by which the Bell violation of a given graph state is decreased with qubit loss. Our analysis allows one to determine which graph topologies are tolerable to qubit loss as well as pinpointing the Achilles' heel of each graph, namely the sets of qubits whose loss jeopardizes the Bell violation. Our results serve as an analytical tool for optimizing experiments and protocols involving graph states in realistically lossy systems. An experimental demonstration of a Bell violation in our noise-tolerant graphs is within reach using state of the art technology.

DOI
10.1103/physrevresearch.5.043099

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Graph states are special entangled states advantageous for many quantum technologies, including quantum error correction, multiparty quantum communication, and measurement-based quantum computation. Yet, their fidelity is often disrupted by various errors, most notably qubit loss. In general, given an entangled state, Bell inequalities can be used to certify whether quantum entanglement remains despite errors. Here, we study the robustness of graph states to loss in terms of their Bell violation. Considering the recently proposed linearly scalable Bell operators by Baccari et al. [Phys. Rev. Lett. 124, 020402 (2020)], we use the stabilizer formalism to derive a formula for the extent by which the Bell violation of a given graph state is decreased with qubit loss. Our analysis allows one to determine which graph topologies are tolerable to qubit loss as well as pinpointing the Achilles' heel of each graph, namely the sets of qubits whose loss jeopardizes the Bell violation. Our results serve as an analytical tool for optimizing experiments and protocols involving graph states in realistically lossy systems. An experimental demonstration of a Bell violation in our noise-tolerant graphs is within reach using state of the art technology.
We investigate the Bell inequalities derived from the graph states with violations detectable even with the presence of noises, which generalizes the idea of error-correcting Bell inequalities [Phys. Rev. Lett. 101, 080501 (2008)]. Firstly we construct a family of valid Bell inequalities tolerating arbitrary ttt-qubit errors involving 3(t+1)3(t+1)3(t+1) qubits, e.g., 6 qubits suffice to tolerate single qubit errors. Secondly we construct also a single-error-tolerating Bell inequality with a violation that increases exponentially with the number of qubits. Exhaustive computer search for optimal error-tolerating Bell inequalities based on graph states on no more than 10 qubits shows that our constructions are optimal for single- and double-error tolerance.
Bell inequalities constitute a key tool in quantum information theory: they not only allow one to reveal nonlocality in composite quantum systems, but, more importantly, they can be used to certify relevant properties thereof. We provide a general construction of Bell inequalities that are maximally violated by the multiqubit graph states and can be used for their robust self-testing. Apart from their theoretical relevance, our inequalities offer two main advantages from an experimental viewpoint: (i) they present a significant reduction of the experimental effort needed to violate them, as the number of correlations they contain scales only linearly with the number of observers; (ii) numerical results indicate that the self-testing statements for graph states derived from our inequalities tolerate noise levels that are met by present experimental data. We also discuss possible generalizations of our approach to entangled states whose stabilizers are not tensor products of Pauli matrices. Our work introduces a promising approach for the certification of complex many-body quantum states.
Testing and verifying imperfect multi-qubit quantum devices are important as such noisy quantum devices are widely available today. Bell inequalities are known to be useful for testing and verifying the quality of the quantum devices from their nonlocal quantum states and local measurements. There have been many experiments demonstrating the violations of Bell inequalities, but they are limited in the number of qubits and the types of quantum states. We report violations of Bell inequalities on IBM Quantum devices based on the scalable and robust inequalities maximally violated by graph states as proposed by Baccari et al.. The violations are obtained from the quantum states of path graphs up to 57 and 21 qubits on a 65-qubit and two 27-qubit IBM Quantum devices, respectively, and from those of star graphs up to 11 qubits with quantum readout error mitigation (QREM). We are able to show violations of the inequalities on various graph states by constructing low-depth quantum circuits and by applying the QREM technique. We also point out that quantum circuits for star graph states of sizeNNNcan be realized with circuits of depthO(N)O(\sqrt {N})O(N​)on subdivided honeycomb lattices which are the topology of the 65-qubit IBM Quantum device. Our experiments show encouraging results on the ability of existing quantum devices to prepare entangled quantum states and provide experimental evidence on the benefit of scalable Bell inequalities for testing them.
Cluster states and graph states in general offer a useful model of the stabilizer formalism and a path toward the development of measurement-based quantum computation. Their defining structure – the stabilizer group – encodes all possible correlations that can be observed during measurement. The measurement outcomes which are consistent with the stabilizer structure make error correction possible. Here, we leverage both properties to design feasible families of states that can be used as robust building blocks of quantum computation. This procedure reduces the effect of experimentally relevant noise models on the extraction of smaller entangled states from the larger noisy graph state. In particular, we study the extraction of Bell pairs from linearly extended graph states – this has the immediate consequence for state teleportation across the graph. We show that robust entanglement can be extracted by proper design of the linear graph with only a minimal overhead of the physical qubits. This scenario is relevant to systems in which the entanglement can be created between neighboring sites. The results shown in this work provide a mathematical framework for noise reduction in measurement-based quantum computation. With proper connectivity structures, the effect of noise can be minimized for a large class of realistic noise processes.
Graph states-one of the most representative families of multipartite entangled states-are important resources for multiparty quantum communication, quantum error correction, and quantum computation. Device-independent certification of highly entangled graph states plays a prominent role in quantum information processing tasks. Here we have experimentally demonstrated device-independent certification for multipartite graph states by adopting the robust self-testing scheme based on scalable Bell inequalities. Specifically, the prepared multi-qubit Greenberger-Horne-Zeilinger (GHZ) states and linear cluster states achieve a high degree of Bell violation, which are beyond the nontrivial bounds of the robust self-testing scheme. Furthermore, our work paves the way to the device-independent certification of complex multipartite quantum states.
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.