Grothendieck's constant and local models for noisy entangled quantum states

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Antonio Acín,¹ Nicolas Gisin,² and Benjamin Toner³
¹ICFO—Institut de Ciències Fotóniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain
²GAP-Optique, University of Geneva, 20, Rue de l'École de Médecine, CH-1211 Geneva 4, Switzerland
³Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA

Received 23 November 2005; published 9 June 2006

Werelate the nonlocal properties of noisy entangled states to Grothendieck'sconstant, a mathematical constant appearing in Banach space theory. Fortwo-qubit Werner states rho <sub>p</sub><sup>W</sup> =p| psi ^– psi ^–|+(1–p) [openface 1] /4, we show that there is alocal model for projective measurements if and only if p1/K_G(3),where K_G(3) is Grothendieck's constant of order 3. Known boundson K_G(3) prove the existence of this model at leastfor p0.66, quite close to the current region of Bellviolation, p~0.71. We generalize this result to arbitrary quantum states.

URL:	http://link.aps.org/doi/10.1103/PhysRevA.73.062105
DOI:	10.1103/PhysRevA.73.062105
PACS:	03.65.Ud; 03.67.Dd; 03.67.Mn 03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.) 03.67.Dd Quantum cryptography 03.67.Mn Quantum entanglement production, characterization, and manipulation YEAR: 2006
KEYWORDS:	quantum entanglement, quantum noise, Banach spaces

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