Grothendieck's constant and local models for noisy entangled quantum states
Abstract
We relate the nonlocal properties of noisy entangled states to Grothendieck’s constant, a mathematical constant appearing in Banach space theory. For twoqubit Werner states ρ_{p}^{W}=p∣ψ^{}⟩⟨ψ^{}∣+(1p)1/4 , we show that there is a local model for projective measurements if and only if p⩽1/K_{G}(3) , where K_{G}(3) is Grothendieck’s constant of order 3. Known bounds on K_{G}(3) prove the existence of this model at least for p≲0.66 , quite close to the current region of Bell violation, ptilde 0.71 . We generalize this result to arbitrary quantum states.
 Publication:

Physical Review A
 Pub Date:
 June 2006
 DOI:
 10.1103/PhysRevA.73.062105
 arXiv:
 arXiv:quantph/0606138
 Bibcode:
 2006PhRvA..73f2105A
 Keywords:

 03.65.Ud;
 03.67.Dd;
 03.67.Mn;
 Entanglement and quantum nonlocality;
 Quantum cryptography;
 Entanglement production characterization and manipulation;
 Quantum Physics
 EPrint:
 6 pages, 1 figure