N-term pairwise-correlation inequalities, steering, and joint measurability.

Abstract

Chained inequalities involving pairwise correlations of qubit observables in the equatorial plane are constructed based on the positivity of a sequence of moment matrices. When a jointly measurable set of positive-operator-valued measures (POVMs) is employed in the first measurement of every pair of sequential measurements, the chained pairwise correlations do not violate the classical bound imposed by the moment matrix positivity. We find that incompatibility of the set of POVMs employed in first measurements is only necessary, but not sufficient, in general, for the violation of the inequality. On the other hand, there exists a one-to-one equivalence between the degree of incompatibility (which quantifies the joint measurability) of the equatorial qubit POVMs and the optimal violation of a nonlocal steering inequality, proposed by Jones and Wiseman [S. J. Jones and H. M. Wiseman, Phys. Rev. A 84, 012110 (2011)]. To this end, we construct a local analog of this steering inequality in a single-qubit system and show that its violation is a mere reflection of measurement incompatibility of equatorial qubit POVMs, employed in first measurements in the sequential unsharp-sharp scheme.