Bounds on Quantum Correlations in Bell Inequality Experiments

Abstract

Bell inequality violation is one of the most widely known manifestations of entanglement in quantum mechanics; indicating that experiments on physically separated quantum mechanical systems cannot be given a local realistic description. However, despite the importance of Bell inequalities, it is not known in general how to determine whether a given entangled state will violate a Bell inequality. This is because one can choose to make many different measurements on a quantum system to test any given Bell inequality and the optimization over measurements is a high-dimensional variational problem. In order to better understand this problem we present algorithms that provide, for a given quantum state, both a lower bound and an upper bound on the maximal expectation value of a Bell operator. Both bounds apply techniques from convex optimization and the methodology for creating upper bounds allows them to be systematically improved. In many cases these bounds determine measurements that would demonstrate violation of the Bell inequality or provide a bound that rules out the possibility of a violation. Examples are given to illustrate how these algorithms can be used to conclude definitively if some quantum states violate a given Bell inequality.

13 pages, 1 table, 2 figures. Updated version as published in PRA

References in corpus (32)

• Separability Criterion for Density Matrices
• Separability of Mixed States: Necessary and Sufficient Conditions
• Separability criterion and inseparable mixed states with positive partial transposition
• Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?
• No Signalling and Quantum Key Distribution
• From Bell's Theorem to Secure Quantum Key Distribution
• Bounding the set of quantum correlations
• The Bloch Vector for N-Level Systems
• A complete family of separability criteria
• A Relevant Two Qubit Bell Inequality Inequivalent to the CHSH Inequality
• Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality
• Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation
• Teleportation, Bell's Inequalities and Inseparability
• Violation of Bell's inequality: criterion for quantum communication complexity advantage
• Quantum Communication between N partners and Bell's inequalities
• Grothendieck's constant and local models for noisy entangled quantum states
• Quantum communication complexity protocol with two entangled qutrits
• Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities
• Quantum Entanglement and Communication Complexity
• Complete hierarchies of efficient approximations to problems in entanglement theory
• Local Hidden Variable Theories for Quantum States
• Simulating quantum correlations as a distributed sampling problem
• Bell's inequalities for states with positive partial transpose
• On the Relationship between Convex Bodies Related to Correlation Experiments with Dichotomic Observables
• Causality and Cirel'son bounds
• Generalizing Tsirelson's bound on Bell inequalities using a min-max principle
• Better Bell Inequality Violation by Collective Measurements
• Modeling the Singlet State with Local Variables
• Bell inequalities stronger than the CHSH inequality for 3-level isotropic states
• Proposed experiment to test the bounds of quantum correlations
• Testing the bounds on quantum probabilities
• Experimental evidence for bounds on quantum correlations

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