Combinatorics and Quantum Nonlocality

Abstract

We use techniques for lower bounds on communication to derive necessary conditions (in terms of detector efficiency or amount of super-luminal communication) for being able to reproduce the quantum correlations occurring in EPR-type experiments with classical local hidden-variable theories. As an application, we consider n parties sharing a GHZ-type state and show that the amount of superluminal classical communication required to reproduce the correlations is at least n(log 2 n − 3) bits and the maximum detector efficiency η * for which the resulting correlations can still be reproduced by a local hidden-variable theory is upper bounded by η * ≤ 8/n and thus decreases with n.

Multisetting Bell-type inequalities for detecting genuine multipartite entanglement

“…These Bell coefficients correspond to the so-called extended parity game considered in Ref. [14]. An equivalent definition, more similar to the definition of the Bell inequality (5) treated in section II is that M αβγ = cos[π(α + β + γ)/m], whenever the absolute value of this expression is one, and M αβγ = 0 otherwise.…”

“…This generalization is discussed in section II, whereas another family of Bell inequalities, based on the extended parity game [14], is discussed in section III. Notably, this game exhibits pseudo-telepathy [15], and the corresponding Bell inequality has the same performance (for m a power of 2) as our Bell inequality of section II.…”

Multipartite nonlocal quantum correlations resistant to imperfections

“…The case k = 2 and arbitrary n was studied by Mermin [19] and recently revisited by Brassard et al [20,21]. In Buhrman et al [12] and our earlier research [15] the case where the number of settings k is a power of two was considered. In [12] it was shown that the amount c of classical communication which the parties must broadcast in order to reproduce exactly the correlations from Definition 10 is c = O(n log n) when k = O(n).…”

“…Our result constitutes to our knowledge the first example in which the degree to which the quantum correlations are nonlocal increases with the size of the entangled system in the presence of noise and as such constitute a significant advance in our understanding of quantum communication complexity and of quantum nonlocality. The present work builds upon the earlier results of [12] and [15]. As in these references we rely heavily on techniques and ideas from the field of communication complexity.…”

“…In [12] it was shown that the amount c of classical communication which the parties must broadcast in order to reproduce exactly the correlations from Definition 10 is c = O(n log n) when k = O(n). And in [15] it was shown that the maximum detector efficiency η * for which a local classical model can reproduce exactly the correlations from Definition 10 decreases as 1/n. Furthermore the classical strategy described above shows that in the absence of noise these results are essentially optimal.…”