Nonclassicality thresholds for multiqubit states: Numerical analysis

Abstract

States that strongly violate Bell's inequalities are required in many quantum-informational protocols as, for example, in cryptography, secret sharing, and the reduction of communication complexity. We investigate families of such states with a numerical method which allows us to reveal nonclassicality even without direct knowledge of Bell's inequalities for the given problem. An extensive set of numerical results is presented and discussed.

Solving large-scale optimization problems related to Bell’s Theorem

“…The original approach adopted by the authors of [3] used the implementation of the simplex method [12] from the GLPK library [4] for solving the lower-level LP task, and the downhill simplex method (DSM) [13] from the SciPy package [14] as the higher-level non-linear optimizer. As shown in [7,15] the critical visibility function is continuous but not differentiable so a robust higher-level optimizer was needed.…”

“…Although eliminating redundancy significantly reduces the number of LP constraints and produces more compact problem formulations, it is not sufficient to extend the applicability of the previous approach [3] which was based on the use of the simplex method. The reduced problems still defy both standard LP approaches: the simplex method and the interior point method.…”

“…These problems were described in detail in [2,3] and are, in fact, two-level optimization problems. The higher level problem is a non-convex non-linear optimization task.…”

“…We will focus on LP, as it has been identified as the major computational challenge. Reference [3] reports a range of problems which have been solved, with the small and medium scale LPs being solved using the GLPK [4] simplex implementation. However, even for medium scale problems the CPU times were occasionally prohibitive.…”

“…• we reformulate the problem by removing multiple redundant constraints from its original definition [3];…”