Quantum conditional probabilities

TL;DR

This paper examines the incompatibility between classical and quantum conditional probabilities, revealing that they cannot always be reconciled and that quantum conditional probabilities exhibit inherently nonclassical behavior.

Contribution

It demonstrates that Kolmogorov-Bayes and Born rule conditional probabilities are incompatible, showing limitations of Gleason's theorem and clarifying the nature of quantum conditional probabilities.

Findings

Conditional probabilities are not always compatible with the Born rule.

Gleason's theorem does not extend to quantum conditional probabilities.

Quantum conditional probabilities can lead to nonclassical joint statistics.

Abstract

We investigate the consistency of conditional quantum probabilities. This is whether there is compatibility between the Kolmogorov-Bayes conditional probabilities and the Born rule. We show that they are not compatible in the sense that there are situations where there is no legitimate density matrix that may reproduce the conditional statistics of the other observable via the Born rule. This is to say that the Gleason theorem does not apply to conditional probabilities. Moreover, we show that when this occurs the joint statistics is nonclassical. We show that conditional probabilities are not equivalent to state reduction, so these results do not affect the validity of the L\"{u}ders expression.