Probabilistic logic of quantum observations

TL;DR

This paper introduces a probabilistic logic framework for reasoning about quantum measurement outcomes, incorporating epistemic aspects of observable compatibility, with a formal axiomatization based on real closed fields.

Contribution

It presents a novel probabilistic propositional logic with an epistemic component for quantum observations, including a sound and weakly complete axiomatization.

Findings

Logic is a conservative extension of classical propositional logic.

Axiomatization relies on decidable first-order theory of real closed fields.

Framework effectively models quantum measurement reasoning.

Abstract

A probabilistic propositional logic, endowed with an epistemic component for asserting (non-)compatibility of diagonizable and bounded observables, is presented and illustrated for reasoning about the random results of projective measurements made on a given quantum state. Simultaneous measurements are assumed to imply that the underlying observables are compatible. A sound and weakly complete axiomatization is provided relying on the decidable first-order theory of real closed ordered fields. The proposed logic is proved to be a conservative extension of classical propositional logic.