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

By Antonio Acín, Nicolas Gisin, B. Toner. 2006

156 citations recorded by the source corpus.

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 p W =p|psi–><psi–|+(1–p)[openface 1]/4, we show that there is a local model for projective measurements if and only if p<=1/KG(3), where KG(3) is Grothendieck's constant of order 3. Known bounds on KG(3) prove the existence of this model at least for p<~0.66, quite close to the current region of Bell violation, p~0.71. We generalize this result to arbitrary quantum states.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/747783a3c87c7f0c3a1235f8392e63a52e967949

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Grothendieck's Theorem, past and present (2011, background): “2n 2 1 ⊗ℓ1 ⊗ℓ1 K≥ c √ n, with c>0 independent of n. We give a complete proof of this in the next section. For more information on the connection between Grothendieck’s and Bell’s inequalities, see [78, 1, 34, 18, 66, 59]. 66 20. Trilinear counterexamples Many researchers have looked for a triline”
Unbounded Violation of Tripartite Bell Inequalities (2007, background): “As the violation K is attainable for traceless observables, ρ yields K = pK which is still a violation whenever p � 1/ √ d (see [ 5 ] for a similar reasoning in the bipartite”
Experimental EPR-steering using Bell-local states (2009, background): “case of Bell inequalities [11, 12, 13], increasing the number of measurement settings beyond two—we”
Entanglement detection (2008, background): “6595 [44,316] and it violates a Bell inequality (different from the CHSH one) for p ≥ 0.”