Probability sum rules and consistent quantum histories

TL;DR

This paper compares weak and minimal decoherence in quantum histories, showing that weak decoherence is less restrictive but leads to different probabilities, raising questions about their operational meaning.

Contribution

It demonstrates that extending probability formulas from weak to minimal decoherence results in different probabilities, highlighting fundamental differences in quantum history frameworks.

Findings

Weak decoherence is less restrictive than minimal decoherence.

Extending probability formulas changes the calculated probabilities.

Different decoherence structures have implications for quantum history interpretations.

Abstract

An example shows that weak decoherence is more restrictive than the minimal logical decoherence structure that allows probabilities to be used consistently for quantum histories. The probabilities in the sum rules that define minimal decoherence are all calculated by using a projection operator to describe each possibility for the state at each time. Weak decoherence requires more sum rules. They bring in additional variables, that require different measurements and a different way to calculate probabilities, and raise questions of operational meaning. The example shows that extending the linearly positive probability formula from weak to minimal decoherence gives probabilities that are different from those calculated in the usual way using the Born and von Neumann rules and a projection operator at each time.