# Probabilistic inequalities and measurements in bipartite systems

## TL;DR

This paper explores classical and quantum probabilistic inequalities in bipartite systems, highlighting how quantum states can violate classical bounds and analyzing the effects of measurements on state rank.

## Contribution

It introduces quantum counterparts to classical inequalities with quantum corrections and studies their validity across different quantum states.

## Key findings

- Classical inequalities hold for factorizable states but are violated by entangled states.
- Quantum corrections vanish under certain conditions, restoring classical inequalities.
- Measurements can reduce the rank of quantum states, with bounds on average rank reduction.

## Abstract

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.