Bell inequality with an arbitrary number of settings and its applications

Abstract

Based on a geometrical argument introduced byŻukowski, a new multisetting Bell inequality is derived, for the scenario in which many parties make measurements on two-level systems. This generalizes and unifies some previous results. Moreover, a necessary and sufficient condition for the violation of this inequality is presented. It turns out that the class of non-separable states which do not admit local realistic description is extended when compared to the two-setting inequalities. However, supporting the conjecture of Peres, quantum states with positive partial transposes with respect to all subsystems do not violate the inequality. Additionally, we follow a general link between Bell inequalities and communication complexity problems, and present a quantum protocol linked with the inequality, which outperforms the best classical protocol.

Entanglement detection

“…This directly leads to the conjecture of Peres, stating that bipartite states with a positive partial transposition allow a LHV description. While no proof or counterexample has been found, in several special cases it is shown that such states do not violate large classes of Bell inequalities, e.g., for the Mermin-Klyshko inequalities [366] and for a wide range of multi-setting inequalities [367].…”

Entanglement and communication-reducing properties of noisyN-qubit states

“…If one considers all possible settings, restricted to one measurement plane on the Bloch sphere for each observer, the critical value for violation of local realism changes to Υ ∞ lr = 2(2/π) N , see [41], and therefore decreases the Werner gap. This result is a limiting case for inequalities involving M settings per party studied in [42]. These inequalities involve measurement settings (again in a specific plane for each observer) evenly spaced at the Bloch sphere.…”

“…These inequalities involve measurement settings (again in a specific plane for each observer) evenly spaced at the Bloch sphere. One has Υ ∞ lr = lim M→∞ Υ M lr [42], notation is obvious here. One may ask for how many settings the critical entanglement admixture for violation of local realism for finite and continuum number of settings are already very close.…”

“…[41,42]. If one considers all possible settings, restricted to one measurement plane on the Bloch sphere for each observer, the critical value for violation of local realism changes to Υ ∞ lr = 2(2/π) N , see [41], and therefore decreases the Werner gap.…”

Unbounded Violation of Tripartite Bell Inequalities

“…Adapted to our case, the communication complexity problem discussed in [13,15,52] is the following: Each of the three parties (i = 1, 2, 3) obtains initially a random bit string encoding (x i , y i ), where each y i = ±1 is taken from a flat distribution and 3 are the coefficients appearing in the violated Bell inequality. The goal is now that every party first broadcasts a single bit and then attempts to compute the function…”