Efficiency of higher dimensional Hilbert spaces for the violation of Bell inequalities

Abstract

We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two dimensional systems, i.e., qubits are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate 'measurements' in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result, that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore, in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.

arXiv comment8 pages, no figures, REVTeX; published version

References in corpus (15)

• Device-independent security of quantum cryptography against collective attacks
• Bell inequalities for arbitrarily high dimensional systems
• From Bell's Theorem to Secure Quantum Key Distribution
• Bounding the set of quantum correlations
• A Relevant Two Qubit Bell Inequality Inequivalent to the CHSH Inequality
• A Bell-type test of energy-time entangled qutrits
• Experimental violation of a spin-1 Bell inequality using maximally-entangled four-photon states
• New optimal tests of quantum nonlocality
• Unbounded violation of tripartite Bell inequalities
• Tailoring photonic entanglement in high-dimensional Hilbert spaces
• Two-Party Bell Inequalities Derived from Combinatorics via Triangular Elimination
• Generalizing Tsirelson's bound on Bell inequalities using a min-max principle
• Some Open Problems in Quantum Information Theory
• Intermediate states in quantum cryptography and Bell inequalities
• Bell inequalities stronger than the CHSH inequality for 3-level isotropic states

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