Quantum Simpsons Paradox and High Order Bell-Tsirelson Inequalities

TL;DR

This paper explores a quantum analogue of Simpson's Paradox, demonstrating that quantum probabilities can exhibit reversal effects impossible in classical statistics, and identifies inequalities that distinguish quantum from classical scenarios.

Contribution

It introduces a quantum version of Simpson's Paradox, formulates a simple inequality violated by quantum probabilities, and characterizes the maximum quantum violation.

Findings

Quantum probabilities can reverse classical statistical trends.

A simple inequality distinguishes quantum from classical probabilities.

Maximum quantum violation of the inequality is fully characterized.

Abstract

The well-known Simpson's Paradox, or Yule-Simpson Effect, in statistics is often illustrated by the following thought experiment: A drug may be found in a trial to increase the survival rate for both men and women, but decrease the rate for all the subjects as a whole. This paradoxical reversal effect has been found in numerous datasets across many disciplines, and is now included in most introductory statistics textbooks. In the language of the drug trial, the effect is impossible, however, if both treatment groups' survival rates are higher than both control groups'. Here we show that for quantum probabilities, such a reversal remains possible. In particular, a "quantum drug", so to speak, could be life-saving for both men and women yet deadly for the whole population. We further identify a simple inequality on conditional probabilities that must hold classically but is violated by our quantum scenarios, and completely characterize the maximum quantum violation. As polynomial inequalities on entries of the density operator, our inequalities are of degree 6.