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

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

2006-06-16

arxiv.org/abs/quant-ph/0606138v1

Phys. Rev. A 73, 062105 (2006)

Physics

Quantum Physics

6 pages, 1 figure

Scientific paper

10.1103/PhysRevA.73.062105

We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states $\rho^W_p=p \proj{\psi^-}+(1-p){\one}/{4}$, we show that there is a local model for projective measurements if and only if $p \le 1/K_G(3)$, 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 $p \lesssim 0.66$, quite close to the current region of Bell violation, $p \sim 0.71$. We generalize this result to arbitrary quantum states.

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