Origin of which-way information and generalization of the Born rule

TL;DR

The paper introduces the quantum typicality rule, a new postulate that correlates particle positions over time, generalizing the Born rule and enabling a trajectory-based formulation of quantum mechanics.

Contribution

It explicitly states and formalizes the quantum typicality rule, linking it to the Born rule and proposing a new trajectory-based quantum framework.

Findings

Quantum typicality rule correlates particle positions at different times.

The rule is equivalent to the Born rule for statistical experiments.

Supports a trajectory-based formulation of quantum mechanics.

Abstract

The possibility to recover the which-way information, for example in the two slit experiment, is based on a natural but implicit assumption about the position of a particle {\it before} a position measurement is performed on it. This assumption cannot be deduced from the standard postulates of quantum mechanics. In the present paper this assumption is made explicit and formally postulated as a new rule, the {\it quantum typicality rule}. This rule correlates the positions of the particles at two different times, thus defining their trajectories. Unexpectedly, this rule is also equivalent to the Born rule with regard to the explanation of the results of statistical experiments. For this reason it can be considered a generalization of the Born rule. The existence of the quantum typicality rule strongly suggests the possibility of a new trajectory-based formulation of quantum mechanics. According to this new formulation, a closed quantum system is represented as a {\it quantum process}, which corresponds to a canonical stochastic process in which the probability measure is replaced by the wave function and the usual frequentist interpretation of probability is replaced by the quantum typicality rule.