Conditional probability and interferences in generalized measurements with or without definite causal order

TL;DR

This paper explores how interference effects influence probability assignments in generalized quantum measurements, especially when causal order is indefinite, revealing non-uniqueness and classical-quantum probability distinctions.

Contribution

It analyzes the impact of indefinite causal order on measurement probabilities, extending previous results to cases involving composition of measurements and interference effects.

Findings

Probability cannot be uniquely assigned in indefinite causal order scenarios.

Interference effects lead to two distinct probability expressions: Born rule and classical probability.

A causal inequality is derived for indefinite causal order, illustrating fundamental limits.

Abstract

In the context of generalized measurement theory, the Gleason-Busch theorem assures the unique form of the associated probability function. Recently, in Flatt et al. Phys. Rev. A 96, 062125 (2017), the case of subsequent measurements has been treated, with the derivation of the L\"uders rule and its generalization (Kraus update rule). Here we investigate the special case of subsequent measurements where an intermediate measurement is a composition of two measurements (a or b) and the case where the causal order is not defined (a and b or b and a). In both cases interference effects can arise. We show that the associated probability cannot be written univocally, and the distributive property on its arguments cannot be taken for granted. The two probability expressions correspond to the Born rule and the classical probability; they are related to the intrinsic possibility of obtaining definite results for the intermediate measurement. For indefinite causal order, a causal inequality is also deduced. The frontier between the two cases is investigated in the framework of generalized measurements with a toy model, a Mach-Zehnder interferometer with a movable beam splitter.