Preparation Uncertainty Implies Measurement Uncertainty in a Class of Generalized Probabilistic Theories

TL;DR

This paper demonstrates that in a broad class of generalized probabilistic theories, preparation uncertainty necessarily implies measurement uncertainty, revealing a universal relationship similar to that in quantum theory.

Contribution

It extends the known quantum uncertainty relations to a class of GPTs characterized by transitivity and self-duality, showing their universal applicability.

Findings

Preparation uncertainty implies measurement uncertainty in GPTs.

Uncertainty relations in GPTs mirror those in quantum theory.

The results suggest a universal link beyond quantum mechanics.

Abstract

In quantum theory, it is known for a pair of noncommutative observables that there is no state on which they take simultaneously definite values, and that there is no joint measurement of them. They are called preparation uncertainty and measurement uncertainty respectively, and research has unveiled that they are not independent from but related with each other in a quantitative way. This study aims to reveal whether similar relations to quantum ones hold also in generalized probabilistic theories (GPTs). In particular, a certain class of GPTs is considered which can be characterized by transitivity and self-duality and regarded as extensions of quantum theory. It is proved that there are close connections expressed quantitatively between two types of uncertainty on a pair observables also in those theories: if preparation uncertainty exists, then measurement uncertainty also exists, and they are described by similar inequalities. Our results manifest that their correspondences are not specific to quantum theory but more universal ones.