Separability of quantum states and the violation of Bell-type inequalities

Abstract

In contrast to the wide-spread opinion that any separable quantum state satisfies every classical probabilistic constraint, we present a simple example where a separable quantum state does not satisfy the original Bell inequality although the latter inequality, in its perfect correlation form, is valid for all joint classical measurements.In a very general setting, we discuss inequalities for joint experiments upon a bipartite quantum system in a separable state. We derive quantum analogues of the original Bell inequality and specify the conditions sufficient for a separable state to satisfy the original Bell inequality. We introduce the extended CHSH inequality and prove that, for any separable quantum state, this inequality holds for a variety of linear combinations.

Threshold bounds for noisy bipartite states

“…7 See, for example, in [5] (section 3). 8 For short, we further refer to the perfect correlation of outcomes if the same observable is measured on both "sides" as Bell's perfect correlations.…”

“…In section 2, we shortly list the main properties of DSO states specified in [1,2] and further prove in a general setting that any DSO state satisfies a generalized version of the original CHSH inequality [3] -the extended CHSH inequality, which we introduced in [5].…”

“…A noisy bipartite state satisfying the extended CHSH inequality * erl@erl.msk.ru 1 A classical state satisfies a CHSH-form inequality under any Alice and Bob classical measurements, ideal or randomized, and the perfect correlation form of the original Bell inequality for any three classical observables (see appendix of [5] for a general proof of the latter).…”

“…In section 3, for an arbitrary nonseparable quantum state violating the original CHSH inequality [3], we introduce the amounts of noise sufficient for the resulting noisy state to represent a DSO state and a Bell class DSO state and, therefore, to satisfy the extended CHSH inequality [5] and the perfect correlation form of the original Bell inequality [4] for any quantum observables. A noisy bipartite state satisfying the extended CHSH inequality * erl@erl.msk.ru 1 A classical state satisfies a CHSH-form inequality under any Alice and Bob classical measurements, ideal or randomized, and the perfect correlation form of the original Bell inequality for any three classical observables (see appendix of [5] for a general proof of the latter).…”

Bipartite Bell Inequalities for Hyperentangled States

“…Therefore, separable states (which are convex combinations of product states) satisfy them too. However, some separable states violate the original BI [4]. The explanation is that the original BI is based on assumptions that are not satisfied by these separable states [5].…”