# Probabilistic inequalities and measurements in bipartite systems. (arXiv:1903.06591v1 [quant-ph])

Various inequalities (Boole inequality, Chung-Erd\"os inequality, Frechet

inequality) for Kolmogorov (classical) probabilities are considered. Quantum

counterparts of these inequalities are introduced, which have an extra `quantum

correction' term, and which hold for all quantum states. When certain

sufficient conditions are satisfied, the quantum correction term is zero, and

the classical version of these inequalities holds for all states. But in

general, the classical version of these inequalities is violated by some of the

quantum states. For example in bipartite systems, classical Boole inequalities

hold for all rank one (factorizable) states, and are violated by some rank two

(entangled) states. A logical approach to CHSH inequalities (which are related

to the Frechet inequalities), is studied in this context.It is shown that CHSH

inequalities hold for all rank one (factorizable) states, and are violated by

some rank two (entangled) states. The reduction of the rank of a pure state by

a quantum measurement with both orthogonal and coherent projectors, is studied.

Bounds for the average rank reduction are given.