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.
Yaoyun Shi received his Bachelor's degree from Beijing University in 1997, and his PhD from Princeton University in 2001, both in computer science. He was a postdoc at Institute of Quantum Information at Caltech and is currently an Associate Professor at University of Michigan, Ann Arbor. His research focuses on the theory of quantum information processing.