On the existence of quantum representations for two dichotomic measurements

TL;DR

This paper investigates the conditions under which measurement outcome probabilities can be modeled quantum-mechanically, focusing on two dichotomic measurements, and reveals simple relations that distinguish quantum models from general probabilistic theories.

Contribution

It provides a complete characterization of when two dichotomic measurements admit a quantum representation, using operator algebra and moment problem techniques.

Findings

Derived simple relations between quantum-mechanical probabilities

Identified conditions that distinguish quantum models from general probabilistic models

Facilitated potential experimental tests to discriminate quantum mechanics from other theories

Abstract

Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories.