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

IEEE Access

Mathematically Proving Bell Nonlocality Motivated by the GHZ Argument

Qiaowei Zhang, Zhihua Guo, Huaixin Cao

Published Jan 1, 2021• 4 citations
Publisher

Abstract

Bell nonlocality of quantum states is an important resource in quantum information and then has various applications. It is usually detected by the violation of some Bell’s inequalities and the all-versus-nothing test. In the present paper, we aim to establish some mathematical methods for proving Bell nonlocality without inequalities, inspired by the work [Phys. Rev. Lett., 89, 080402 (2002)] regarding the GHZ paradox. For self-containedness, we recall the mathematical definition of Bell nonlocality proposed in [Sci. China-Phys. Mech. Astron. 62, 030311 (2019)] and then give some basic properties on it. Then we derive some necessary conditions for a multipartite state to be Bell local and obtain some sufficient conditions for a state to be Bell nonlocal in terms of “expectations” of local observables without invoking Bell inequalities. Unlike the standard approach to nonlocality detection based on violation of Bell inequalities, the obtained criteria are formulated in terms of certain relations for expectation values of local observables that are constructed from the well-known GHZ paradoxes.

DOI
10.1109/access.2021.3063744

Related papers

Detecting Bell nonlocality based on weak Hardy-like paradoxes and Hardy-Bell inequalities

HAN KanYuan, GUO ZhiHua, Cao Huai-xin +1 more

Bell nonlocality is a special quantum nonlocality and has emerged as an important resource in quantum information processing tasks. Typically, there are two types of strategies for testing the Bell nonlocality: the Bell inequality method and the “all-versus-nothing proof", the later includes methods such as the GHZ argument and Hardy paradox. The objective of this work is to provide new methods for detecting Bell nonlocality by understanding Hardy-like paradoxes (HLPs) and Hardy-Bell inequalities (HBIs) as well as Hardy inequality. First, the weak HLP (WHLP) is established, and an HBI is obtained for tripartite correlation tensors (CTs) with two inputs and two outcomes. Based on an existing Hardy inequality for an n-qubit pure state, a Hardy inequality is proven for n-partite Bell local CTs. Second, the WHLP and HBI are proven for tripartite states in terms of conditional probabilities. Our WHLP is theoretically and practically easier to construct and more efficient in checking Bell nonlocality compared to the usual HLP.

Zhongguo kexue. Wulixue Lixue Tianwenxue•Nov 20, 2023•0 citations

Quantum non-locality—it ainʼt necessarily so...

Marek Żukowski, Časlav Brukner

Bell's theorem is 50 years old. Still there is a controversy about its implications. Much of it has its roots in confusion regarding the premises from which the theorem can be derived. Some claim that a derivation of Bell's inequalities requires just locality assumption, and nothing more. Violations of the inequalities are then interpreted as ``nonlocality'' or ``quantum nonlocality''. We show that such claims are unfounded and that every derivation of Bell's inequalities requires a premise -- in addition to locality and freedom of choice -- which is either assumed tacitly, or unconsciously, or is embedded in a single compound condition (like Bell's ``local causality''). The premise is equivalent to the assumption of existence of additional variables which do not appear in the quantum formalism (in form of determinism, or joint probability for outcomes of all conceivable measurements, or ``additional causes`'', or ``hidden variables'', ``complete description of the state'' or counterfactual definiteness, etc.). A certain irony is that perhaps the main message of violation of Bell's inequalities is that our notion of locality should be based on an operationally well-defined no-signalling condition, rather than on local causality.

arXiv:1501.04618•Journal of Physics A Mathematical and Theoretical•Oct 9, 2014•100 citations

Detecting nonlocality in many-body quantum states

Jordi Tura, Remigiusz Augusiak, Ana Belén Sainz +3 more

Intensive studies of entanglement properties have proven essential for our understanding of quantum many-body systems. In contrast, much less is known about the role of quantum nonlocality in these systems because the available multipartite Bell inequalities involve correlations among many particles, which are difficult to access experimentally. We constructed multipartite Bell inequalities that involve only two-body correlations and show how they reveal the nonlocality in many-body systems relevant for nuclear and atomic physics. Our inequalities are violated by any number of parties and can be tested by measuring total spin components, opening the way to the experimental detection of many-body nonlocality, for instance with atomic ensembles.

arXiv:1306.6860•Science•Jun 12, 2014•209 citations

Test of Genuine Multipartite Nonlocality

Ya-Li Mao, Zheng-Da Li, Sixia Yu +1 more

While Bell nonlocality of a bipartite system is counterintuitive, multipartite nonlocality in our many-body world turns out to be even more so. Recent theoretical study reveals in a theory-agnostic manner that genuine multipartite nonlocal correlations cannot be explained by any causal theory involving fewer-partite nonclassical resources and global shared randomness. Here, we provide a Bell-type inequality as a test for genuine multipartite nonlocality in network by exploiting a matrix representation of the causal structure of a multipartite system. We further present experimental demonstrations that both four-photon GHZ state and generalized four-photon GHZ state significantly violate the inequality, i.e., the observed four-partite correlations resist explanations involving three-way nonlocal resources subject to local operations and common shared randomness, hence confirming that nature is boundless multipartite nonlocal.

arXiv:2201.12753•Physical Review Letters•Oct 4, 2022•27 citations

Deterministic all-versus-nothing proofs of Bell nonlocality based on non-stabilizer states

Weidong Tang

Abstract The all-versus-nothing proof of Bell nonlocality is a prominent demonstration of Bell’s theorem without inequalities. There are two kinds of such proofs: the deterministic all-versus-nothing proof and the probabilistic all-versus-nothing proof, which have received extensive research attention. Traditionally, all previous deterministic all-versus-nothing proofs are constructed based on stabilizer states. However, this work presents new deterministic proofs derived from non-stabilizer states, thereby breaking away from this conventional approach. These novel results not only significantly broaden the range of demonstrations of Bell nonlocality without inequalities but also offer valuable resources for certain quantum information processing applications.

arXiv:2201.01886•New Journal of Physics•Jun 1, 2023•1 citation

Volume of violation of Bell-type inequalities as a measure of nonlocality

Artur Barasiński, Mateusz Nowotarski

Motivated by the recent proposal for the quantification of the Bell nonlocality [Phys. Rev. A 92, 030101(R) (2015)], we provide an extensive analysis of this concept, namely, the volume of violation. This new measure of nonlocality has been proposed to deliver an evidence that the anomaly between maximally entangled states and states that maximally violate a Bell inequality is caused by the method, which has been applied to quantify nonlocality and the anomaly disappears when the volume of violation in used. We prove that such conclusion is not true for all bipartite quantum systems with dimension $d\ensuremath{\bigotimes}d$. In fact, if one assumes the same condition as in Phys. Rev. A 92, 030101(R) (2015), it is limited to $d\ensuremath{\le}7$. Furthermore, we discuss several type of local measurements and their influence on the quantification of nonlocality by mean of volume of violation. In particular, we propose a set of local observers that significantly enhance the volume of violation, which may have important meaning in a real-world application. Finally, we present a comparison between the volume of violation and the maximal violation of a Bell inequality.

Physical Review A•Aug 24, 2018•15 citations
Bell nonlocality of quantum states is an important resource in quantum information and then has various applications. It is usually detected by the violation of some Bell’s inequalities and the all-versus-nothing test. In the present paper, we aim to establish some mathematical methods for proving Bell nonlocality without inequalities, inspired by the work [Phys. Rev. Lett., 89, 080402 (2002)] regarding the GHZ paradox. For self-containedness, we recall the mathematical definition of Bell nonlocality proposed in [Sci. China-Phys. Mech. Astron. 62, 030311 (2019)] and then give some basic properties on it. Then we derive some necessary conditions for a multipartite state to be Bell local and obtain some sufficient conditions for a state to be Bell nonlocal in terms of “expectations” of local observables without invoking Bell inequalities. Unlike the standard approach to nonlocality detection based on violation of Bell inequalities, the obtained criteria are formulated in terms of certain relations for expectation values of local observables that are constructed from the well-known GHZ paradoxes.
Bell nonlocality is a special quantum nonlocality and has emerged as an important resource in quantum information processing tasks. Typically, there are two types of strategies for testing the Bell nonlocality: the Bell inequality method and the “all-versus-nothing proof", the later includes methods such as the GHZ argument and Hardy paradox. The objective of this work is to provide new methods for detecting Bell nonlocality by understanding Hardy-like paradoxes (HLPs) and Hardy-Bell inequalities (HBIs) as well as Hardy inequality. First, the weak HLP (WHLP) is established, and an HBI is obtained for tripartite correlation tensors (CTs) with two inputs and two outcomes. Based on an existing Hardy inequality for an n-qubit pure state, a Hardy inequality is proven for n-partite Bell local CTs. Second, the WHLP and HBI are proven for tripartite states in terms of conditional probabilities. Our WHLP is theoretically and practically easier to construct and more efficient in checking Bell nonlocality compared to the usual HLP.
Bell's theorem is 50 years old. Still there is a controversy about its implications. Much of it has its roots in confusion regarding the premises from which the theorem can be derived. Some claim that a derivation of Bell's inequalities requires just locality assumption, and nothing more. Violations of the inequalities are then interpreted as ``nonlocality'' or ``quantum nonlocality''. We show that such claims are unfounded and that every derivation of Bell's inequalities requires a premise -- in addition to locality and freedom of choice -- which is either assumed tacitly, or unconsciously, or is embedded in a single compound condition (like Bell's ``local causality''). The premise is equivalent to the assumption of existence of additional variables which do not appear in the quantum formalism (in form of determinism, or joint probability for outcomes of all conceivable measurements, or ``additional causes`'', or ``hidden variables'', ``complete description of the state'' or counterfactual definiteness, etc.). A certain irony is that perhaps the main message of violation of Bell's inequalities is that our notion of locality should be based on an operationally well-defined no-signalling condition, rather than on local causality.
Intensive studies of entanglement properties have proven essential for our understanding of quantum many-body systems. In contrast, much less is known about the role of quantum nonlocality in these systems because the available multipartite Bell inequalities involve correlations among many particles, which are difficult to access experimentally. We constructed multipartite Bell inequalities that involve only two-body correlations and show how they reveal the nonlocality in many-body systems relevant for nuclear and atomic physics. Our inequalities are violated by any number of parties and can be tested by measuring total spin components, opening the way to the experimental detection of many-body nonlocality, for instance with atomic ensembles.
While Bell nonlocality of a bipartite system is counterintuitive, multipartite nonlocality in our many-body world turns out to be even more so. Recent theoretical study reveals in a theory-agnostic manner that genuine multipartite nonlocal correlations cannot be explained by any causal theory involving fewer-partite nonclassical resources and global shared randomness. Here, we provide a Bell-type inequality as a test for genuine multipartite nonlocality in network by exploiting a matrix representation of the causal structure of a multipartite system. We further present experimental demonstrations that both four-photon GHZ state and generalized four-photon GHZ state significantly violate the inequality, i.e., the observed four-partite correlations resist explanations involving three-way nonlocal resources subject to local operations and common shared randomness, hence confirming that nature is boundless multipartite nonlocal.
Abstract The all-versus-nothing proof of Bell nonlocality is a prominent demonstration of Bell’s theorem without inequalities. There are two kinds of such proofs: the deterministic all-versus-nothing proof and the probabilistic all-versus-nothing proof, which have received extensive research attention. Traditionally, all previous deterministic all-versus-nothing proofs are constructed based on stabilizer states. However, this work presents new deterministic proofs derived from non-stabilizer states, thereby breaking away from this conventional approach. These novel results not only significantly broaden the range of demonstrations of Bell nonlocality without inequalities but also offer valuable resources for certain quantum information processing applications.
Motivated by the recent proposal for the quantification of the Bell nonlocality [Phys. Rev. A 92, 030101(R) (2015)], we provide an extensive analysis of this concept, namely, the volume of violation. This new measure of nonlocality has been proposed to deliver an evidence that the anomaly between maximally entangled states and states that maximally violate a Bell inequality is caused by the method, which has been applied to quantify nonlocality and the anomaly disappears when the volume of violation in used. We prove that such conclusion is not true for all bipartite quantum systems with dimension $d\ensuremath{\bigotimes}d$. In fact, if one assumes the same condition as in Phys. Rev. A 92, 030101(R) (2015), it is limited to $d\ensuremath{\le}7$. Furthermore, we discuss several type of local measurements and their influence on the quantification of nonlocality by mean of volume of violation. In particular, we propose a set of local observers that significantly enhance the volume of violation, which may have important meaning in a real-world application. Finally, we present a comparison between the volume of violation and the maximal violation of a Bell inequality.