Covariance Bell inequalities

Abstract

We introduce Bell inequalities based on covariance, one of the most common measures of correlation. Explicit examples are discussed, and violations in quantum theory are demonstrated. A crucial feature of these covariance Bell inequalities is their nonlinearity; this has nontrivial consequences for the derivation of their local bound, which is not reached by deterministic local correlations. For our simplest inequality, we derive analytically tight bounds for both local and quantum correlations. An interesting application of covariance Bell inequalities is that they can act as "shared randomness witnesses": specifically, the value of the Bell expression gives device-independent lower bounds on both the dimension and the entropy of the shared random variable in a local model.

15 pages, 3 figures

References in corpus (36)

• Quantum entanglement
• Bell nonlocality
• Device-independent security of quantum cryptography against collective attacks
• Steering, Entanglement, Nonlocality, and the EPR Paradox
• Random Numbers Certified by Bell's Theorem
• Bounding the set of quantum correlations
• Bound entangled Gaussian states
• A Relevant Two Qubit Bell Inequality Inequivalent to the CHSH Inequality
• Testing the Hilbert space dimension
• Device-independent tests of classical and quantum dimensions
• Characterizing the nonlocal correlations of particles that never interacted
• Bilocal versus non-bilocal correlations in entanglement swapping experiments
• Symmetries of the Bell correlation inequalities
• Distillability Criterion for all Bipartite Gaussian States
• Separability Criterion for all bipartite Gaussian States
• Covariance matrices and the separability problem
• Maximal violation of the I3322 inequality using infinite dimensional quantum systems
• New optimal tests of quantum nonlocality
• Nonlocal correlations in the star-network configuration
• Nonlinear Bell inequalities tailored for quantum networks
• Polynomial Bell inequalities
• Entropic Bell Inequalities
• Einstein-Podolsky-Rosen steering and the steering ellipsoid
• Bell inequalities for continuous-variable correlations
• An entropic approach to local realism and noncontextuality
• Classifying 50 years of Bell inequalities
• Partial list of bipartite Bell inequalities with four binary settings
• Efficiency of higher dimensional Hilbert spaces for the violation of Bell inequalities
• Device-Independent Tests of Entropy
• Complementarity and correlations
• Minimum Dimension of a Hilbert Space Needed to Generate a Quantum Correlation
• Proposal for a cumulant-based Bell test for mesoscopic junctions
• Local hidden variable models for entangled quantum states using finite shared randomness
• Shared randomness and device-independent dimension witnessing
• Concavity of the quantum body for any given dimension
• Testing locality and noncontextuality with the lowest moments

Cited by in corpus (14)

• Pearson Correlation Coefficient as a measure for Certifying and Quantifying High Dimensional Entanglement
• Optimization over trace polynomials
• Semidefinite tests for latent causal structures
• Multiphoton Bell-type inequality: a tool to unearth nonlocality of continuous variable quantum optical systems
• Multiplicative Bell Inequalities
• State polynomials: positivity, optimization and nonlinear Bell inequalities
• Geometrical self-testing of partially entangled two-qubit states
• Nonlinear Bell inequality for macroscopic measurements
• Correlation Minor Norms, Entanglement Detection and Discord
• Improved estimation for energy dissipation in biochemical oscillations
• Extended Bell inequality and maximum violation
• Constructing local models for general measurements on bosonic Gaussian states
• On a certain inequality for the sum of norms and reverse uncertainty relations
• Sets of Marginals and Pearson-Correlation-based CHSH Inequalities for a Two-Qubit System