Law of Total Probability in Quantum Theory and Its Application in Wigner's Friend Scenario

TL;DR

This paper extends the concept of conditional probability in quantum theory to POVM measurements, providing a more general formulation of the law of total probability and revealing loopholes in quantum no-go theorems related to the Wigner's Friend scenario.

Contribution

It introduces a new rule for two-time conditional probability with incompatible POVMs and identifies conditions where the law of total probability holds in quantum theory.

Findings

The law of total probability does not generally hold in quantum theory.

Logical loopholes in quantum no-go theorems are due to assumptions about the law's validity.

The paper clarifies conditions under which the law of total probability is valid in quantum contexts.

Abstract

It is well-known that the law of total probability does not hold in general in quantum theory. However, the recent arguments on some of the fundamental assumptions in quantum theory based on the extended Wigner's Friend scenario show a need to clarify how the law of total probability should be formulated in quantum theory and under what conditions it still holds. In this work, the definition of conditional probability in quantum theory is extended to POVM measurements. Rule to assign two-time conditional probability is proposed for incompatible POVM operators, which leads to a more general and precise formulation of the law of total probability. Sufficient conditions under which the law of total probability holds are identified. Applying the theory developed here to analyze several quantum no-go theorems related to the extended Wigner's friend scenario reveals logical loopholes in these no-go theorems. The loopholes exist as a consequence of taking for granted the validity of the law of total probability without verifying the sufficient conditions. Consequently, the contradictions in these no-go theorems only reconfirm the invalidity of the law of total probability in quantum theory, rather than invalidating the physical statements that the no-go theorems attempt to refute.