Bell inequalities and the separability criterion

1999/11/30 by Barbara M. Terhal · 19 citations
Computer Science · Mathematics · Physics and Astronomy · #Bell state #Bell test experiments #Bell's theorem #Bipartite graph #CHSH inequality #Computer science #Discrete mathematics #Graph #Hidden variable theory #Inequality #Local hidden variable theory #Mathematical analysis #Mathematics #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum entanglement #Quantum mechanics #Set (abstract data type) #Statistical physics #Variable (mathematics) #Witness #quant-ph

paper · pdf · doi:10.1016/s0375-9601(00)00401-1

published as Physics Letters A Vol. 271, 319 (2000) · 16 pages Revtex, typo in equation (14) corrected

openalex publication_date 2000/07/01 · arxiv created 2000/11/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We analyze and compare the mathematical formulations of the criterion for separability for bipartite density matrices and the Bell inequalities. We show that a violation of a Bell inequality can formally be expressed as a witness for entanglement. We also show how the criterion for separability and a description of the state by a local hidden variable theory, become equivalent when we restrict the set of local hidden variable theories to the domain of quantum mechanics. This analysis sheds light on the essential difference between the two criteria and may help us in understanding whether there exist entangled states for which the statistics of the outcomes of all possible local measurements can be described by a local hidden variable theory.

Citations

• Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels
• Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement
• Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model
• Quantum vs. classical communication and computation
• On the Einstein Podolsky Rosen paradox
• Separability of mixed states: necessary and sufficient conditions
• Quantum vs. Classical Communication and Computation
• Convex Analysis

Cited by

• Retrodiction of measurement outcomes on a single quantum system reveals entanglement with its environment
• Efficient Bipartite Entanglement Detection Scheme with a Quantum Adversarial Solver
• Scalable method to estimate experimentally the entanglement of multipartite systems
• Iterations of nonlinear entanglement witnesses
• Experimental Detection of Entanglement with Optimal-Witness Families
• Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps
• Entanglement witnesses: construction, analysis and classification
• A class of symmetric Bell diagonal entanglement witnesses—a geometric perspective
• Controllable generation of mixed two-photon states
• Multipartite Bound Entangled States that Violate Bell’s Inequality
• Tripartite thermal correlations in an inhomogeneous spin–star system
• Quantum separability and entanglement detection via entanglement-witness search and global optimization
• Entanglement and nonclassicality: A mutual impression
• How to construct indecomposable entanglement witnesses
• Construction of exposed indecomposable positive linear maps between matrix algebras
• Symmetry-resolved entanglement detection using partial transpose moments
• Entanglement sudden death as an indicator of fidelity in a four-qubit cluster state
• Quantum Information
• Observable estimation of entanglement for arbitrary finite-dimensional mixed states