Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0903.3318

Error-Tolerating Bell Inequalities via Graph States

Qing Chen, Sixia Yu, C. H. Oh

Published Mar 19, 2009• 0 citations
arXivHTMLPDF

Abstract

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 $t$-qubit errors involving $3(t+1)$ 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.

DOI
10.48550/arxiv.0903.3318

Related papers

Robustness of Bell violation of graph states to qubit loss

Shahar Silberstein, Rotem Arnon-Friedman

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.

arXiv:2303.07377•Physical Review Research•Oct 30, 2023•1 citation

Two-setting Bell inequalities for graph states

Géza Tóth, Otfried Gühne, Hans J. Briegel

We present Bell inequalities for graph states with a high violation of local realism. In particular, we show that there is a basic Bell inequality for every nontrivial graph state which is violated by the state at least by a factor of 2. This inequality needs the measurement of, at most, two operators for each qubit and involves only some of the qubits. We also show that for some families of graph states composite Bell inequalities can be constructed such that the violation of local realism increases exponentially with the number of qubits. We prove that some of our inequalities are facets of the convex polytope containing the many-body correlations consistent with local hidden variable models. Our Bell inequalities are built from stabilizing operators of graph states.

arXiv:quant-ph/0510007•Physical Review A•Feb 2, 2006•54 citations

Bell Inequalities for Graph States

Otfried Gühne, Géza Tóth, Philipp Hyllus +1 more

We investigate the nonlocal properties of graph states. To this aim, we derive a family of Bell inequalities which require three measurement settings for each party and are maximally violated by graph states. In turn, for each graph state there is an inequality maximally violated only by that state. We show that for certain types of graph states the violation of these inequalities increases exponentially with the number of qubits. We also discuss connections to other entanglement properties such as the positivity of the partial transpose or the geometric measure of entanglement.

arXiv:quant-ph/0410059•Physical Review Letters•Sep 14, 2005•230 citations

Scalable Bell Inequalities for Qubit Graph States and Robust Self-Testing

F. Baccari, Remigiusz Augusiak, Ivan Šupić +2 more

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.

arXiv:1812.10428•Physical Review Letters•Jan 17, 2020•83 citations

Error Correcting Bell Inequalities

Thomas Alfred Walker, Fiona Polack, Samuel L. Braunstein

Quantum error-correcting codes can protect multipartite quantum states from errors on some limited number of their subsystems (usually qubits). We construct a family of Bell inequalities which inherit this property from the underlying code and exhibit the violation of local realism, without any quantum information processing (except for the creation of an entangled state). This family shows no reduction in the size of the violation of local realism for arbitrary errors on a limited number of qubits. Our minimal construction requires preparing an 11-qubit entangled state.

Physical Review Letters•Aug 18, 2008•1 citation

Scalable Bell inequalities for graph states of arbitrary prime local dimension and self-testing

Rafael M. Santos, Debashis Saha, F. Baccari +1 more

Abstract Bell nonlocality—the existence of quantum correlations that cannot be explained by classical means—is certainly one of the most striking features of quantum mechanics. Its range of applications in device-independent protocols is constantly growing. Many relevant quantum features can be inferred from violations of Bell inequalities, including entanglement detection and quantification, and state certification applicable to systems of arbitrary number of particles. A complete characterisation of nonlocal correlations for many-body systems is, however, a computationally intractable problem. Even if one restricts the analysis to specific classes of states, no general method to tailor Bell inequalities to be violated by a given state is known. In this work we provide a general construction of Bell expressions tailored to the graph states of any prime local dimension. These form a broad class of multipartite quantum states that have many applications in quantum information, including quantum error correction. We analytically determine their maximal quantum values, a number of high relevance for device-independent applications of Bell inequalities. Importantly, the number of expectation values to determine in order to test the violation of our inequalities scales only linearly with the system size, which we expect to be the optimal scaling one can hope for in this case. Finally, we show that these inequalities can be used for self-testing of multi-qutrit graph states such as the well-known four-qutrit absolutely maximally entangled state AME(4,3).

arXiv:2212.07133•New Journal of Physics•May 30, 2023•8 citations
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.
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 present Bell inequalities for graph states with a high violation of local realism. In particular, we show that there is a basic Bell inequality for every nontrivial graph state which is violated by the state at least by a factor of 2. This inequality needs the measurement of, at most, two operators for each qubit and involves only some of the qubits. We also show that for some families of graph states composite Bell inequalities can be constructed such that the violation of local realism increases exponentially with the number of qubits. We prove that some of our inequalities are facets of the convex polytope containing the many-body correlations consistent with local hidden variable models. Our Bell inequalities are built from stabilizing operators of graph states.
We investigate the nonlocal properties of graph states. To this aim, we derive a family of Bell inequalities which require three measurement settings for each party and are maximally violated by graph states. In turn, for each graph state there is an inequality maximally violated only by that state. We show that for certain types of graph states the violation of these inequalities increases exponentially with the number of qubits. We also discuss connections to other entanglement properties such as the positivity of the partial transpose or the geometric measure of entanglement.
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.
Quantum error-correcting codes can protect multipartite quantum states from errors on some limited number of their subsystems (usually qubits). We construct a family of Bell inequalities which inherit this property from the underlying code and exhibit the violation of local realism, without any quantum information processing (except for the creation of an entangled state). This family shows no reduction in the size of the violation of local realism for arbitrary errors on a limited number of qubits. Our minimal construction requires preparing an 11-qubit entangled state.
Abstract Bell nonlocality—the existence of quantum correlations that cannot be explained by classical means—is certainly one of the most striking features of quantum mechanics. Its range of applications in device-independent protocols is constantly growing. Many relevant quantum features can be inferred from violations of Bell inequalities, including entanglement detection and quantification, and state certification applicable to systems of arbitrary number of particles. A complete characterisation of nonlocal correlations for many-body systems is, however, a computationally intractable problem. Even if one restricts the analysis to specific classes of states, no general method to tailor Bell inequalities to be violated by a given state is known. In this work we provide a general construction of Bell expressions tailored to the graph states of any prime local dimension. These form a broad class of multipartite quantum states that have many applications in quantum information, including quantum error correction. We analytically determine their maximal quantum values, a number of high relevance for device-independent applications of Bell inequalities. Importantly, the number of expectation values to determine in order to test the violation of our inequalities scales only linearly with the system size, which we expect to be the optimal scaling one can hope for in this case. Finally, we show that these inequalities can be used for self-testing of multi-qutrit graph states such as the well-known four-qutrit absolutely maximally entangled state AME(4,3).