Two quantum Simpson's paradoxes

TL;DR

This paper explores two quantum analogues of Simpson's paradox, demonstrating how quantum measurements can exhibit reversed correlations due to classical mixing or quantum superpositions, with implications for quantum hypothesis testing.

Contribution

It introduces and analyzes two quantum Simpson's paradoxes, one classical-like and one quantum-coherent, showing their independent occurrence and conditions under which they appear.

Findings

Quantum-quantum YS effect more likely than quantum-classical one.

Certain superposition states prevent the quantum-classical YS effect.

The effects have implications for quantum hypothesis testing.

Abstract

The so-called Simpson's "paradox", or Yule-Simpson (YS) effect, occurs in classical statistics when the correlations that are present among different sets of samples are reversed if the sets are combined together, thus ignoring one or more lurking variables. Here we illustrate the occurrence of two analogue effects in quantum measurements. The first, which we term quantum-classical YS effect, may occur with quantum limited measurements and with lurking variables coming from the mixing of states, whereas the second, here referred to as quantum-quantum YS effect, may take place when coherent superpositions of quantum states are allowed. By analyzing quantum measurements on low dimensional systems (qubits and qutrits), we show that the two effects may occur independently, and that the quantum-quantum YS effect is more likely to occur than the corresponding quantum-classical one. We also found that there exist classes of superposition states for which the quantum-classical YS effect cannot occur for any measurement and, at the same time, the quantum-quantum YS effect takes place in a consistent fraction of the possible measurement settings. The occurrence of the effect in the presence of partial coherence is discussed as well as its possible implications for quantum hypothesis testing.