Local commutativity versus Bell inequality violation for entangled states and versus non-violation for separable states

Journal Article · Mon Oct 15 04:00:00 EDT 2007 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.76.042105· OSTI ID:21020647

Seevinck, Michael; Uffink, Jos ^[1]

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1. Institute of History and Foundations of Science, Utrecht University, P.O Box 80.000, 3508 TA Utrecht (Netherlands)

By introducing a quantitative 'degree of commutativity' in terms of the angle between spin observables we present two tight quantitative trade-off relations in the case of two qubits. First, for entangled states, between the degree of commutativity of local observables and the maximal amount of violation of the Bell inequality: if both local angles increase from zero to {pi}/2 (i.e., the degree of local commutativity decreases), the maximum violation of the Bell inequality increases. Secondly, a converse trade-off relation holds for separable states: if both local angles approach {pi}/2 the maximal value obtainable for the correlations in the Bell inequality decreases and thus the non-violation increases. As expected, the extremes of these relations are found in the case of anticommuting local observables where, respectively, the bounds of 2{radical}(2) and {radical}(2) hold for the expectation value of the Bell operator. The trade-off relations show that noncommmutativity gives 'a more than classical result' for entangled states, whereas 'a less than classical result' is obtained for separable states. The experimental relevance of the trade-off relation for separable states is that it provides an experimental test for two qubit entanglement. Its advantages are twofold: in comparison to violations of Bell inequalities it is a stronger criterion and in comparison to entanglement witnesses it needs to make less strong assumptions about the observables implemented in the experiment.

• OSTI ID: 21020647
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 4 Vol. 76; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BELL THEOREM
COMPARATIVE EVALUATIONS
CORRELATIONS
EXPECTATION VALUE
QUANTUM ENTANGLEMENT
QUANTUM MECHANICS
QUBITS
SPIN