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:0808.2287 · Physical Review A

Bell inequality for qubits based on the Cauchy-Schwarz inequality

Jing‐Ling Chen, Dong-Ling Deng

Published Jan 30, 2009• 11 citations
arXivHTMLPDF

Abstract

We develop a systematic approach to establish Bell inequalities for qubits based on the Cauchy-Schwarz inequality. We also use the concept of distinct ``roots'' of the Bell function to classify some well-known Bell inequalities for qubits. As applications of the approach, we present three unique and tight Bell inequalities for four and three qubits, respectively.

DOI
10.1103/physreva.79.012115

Related papers

Unified approach to entanglement criteria using the Cauchy-Schwarz and Hölder inequalities

Sabine Wölk, Marcus Huber, Otfried Gühne

We present a unified approach to different recent entanglement criteria. Although they were developed in different ways, we show that they are all applications of a more general principle given by the Cauchy-Schwarz inequality. We explain this general principle and show how to use it to derive not only already known entanglement criteria, but also criteria which were unknown so far. We systematically investigate its potential and limitations to detect bipartite and multipartite entanglement. Furthermore, we describe how to apply our generalized entanglement detection scheme to find the measurement directions to verify the entanglement of a given state experimentally.

arXiv:1405.0986•Physical Review A•Aug 15, 2014•15 citations

Bell inequalities for three particles

Jing‐Ling Chen, Chunfeng Wu, L. C. Kwek +1 more

We present tight Bell inequalities expressed by probabilities for three four- and five-dimensional systems. A tight structure of Bell inequalities for three $d$-dimensional systems (qudits) is proposed. Some interesting Bell inequalities of three qubits reduced from those of three qudits are also studied.

arXiv:quant-ph/0506230•Physical Review A•Sep 8, 2008•19 citations

Bell Inequalities for Three-Particle Systems

Jing‐Ling Chen, Chunfeng Wu, L. C. Kwek +1 more

In the work, we present the structure of the tight Bell inequalities for three-particle systems. Interesting Bell inequalities for three qubits that reducing from those of three qudits are also studied.

•Jun 28, 2005•0 citations

Classifying the simplest Bell inequalities beyond qubits and their applications towards self-testing

Palash Pandya, Shubhayan Sarkar, Remigiusz Augusiak

Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest $(2,2,2)$ scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the $(2,2,3)$ scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.

quant-pharXiv:2602.08469•Feb 10, 2026•0 citations

Generalized Cauchy-Schwarz inequality and Uncertainty Relation

Vishnu M. Bannur

A generalized Cauchy-Schwarz inequality is derived and applied to uncertainty relation in quantum mechanics. We see a modification in the uncertainty relation and minimum uncertainty wave packet.

arXiv:1509.03701•Sep 12, 2015•0 citations

Explicit form of correlation-function three-setting tight Bell inequalities for three qubits

Marcin Wieśniak, Piotr Badziąg, Marek Żukowski

We present a method to derive explicit forms of tight correlation-function Bell inequalities for three systems and dichotomic observables, which involve three settings for each observer. We also give sufficient and necessary conditions for quantum predictions to satisfy the new inequalities.

arXiv:quant-ph/0611083•Physical Review A•Jul 16, 2007•18 citations
We develop a systematic approach to establish Bell inequalities for qubits based on the Cauchy-Schwarz inequality. We also use the concept of distinct ``roots'' of the Bell function to classify some well-known Bell inequalities for qubits. As applications of the approach, we present three unique and tight Bell inequalities for four and three qubits, respectively.
We present a unified approach to different recent entanglement criteria. Although they were developed in different ways, we show that they are all applications of a more general principle given by the Cauchy-Schwarz inequality. We explain this general principle and show how to use it to derive not only already known entanglement criteria, but also criteria which were unknown so far. We systematically investigate its potential and limitations to detect bipartite and multipartite entanglement. Furthermore, we describe how to apply our generalized entanglement detection scheme to find the measurement directions to verify the entanglement of a given state experimentally.
We present tight Bell inequalities expressed by probabilities for three four- and five-dimensional systems. A tight structure of Bell inequalities for three ddd-dimensional systems (qudits) is proposed. Some interesting Bell inequalities of three qubits reduced from those of three qudits are also studied.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.