Generalized Probabilistic Theories Without the No-Restriction Hypothesis

TL;DR

This paper extends the framework of generalized probabilistic theories by removing the no-restriction hypothesis, enabling the description of new theories including those with intrinsic noise and maximal nonlocal correlations.

Contribution

It introduces a generalized framework for GPTs without the no-restriction hypothesis, including a self-dualization procedure and a new tensor product for joint states.

Findings

Describes new classes of probabilistic theories with intrinsic noise.

Characterizes the convex closure of the Spekkens toy theory.

Shows the unrestricted Spekkens toy theory as 'boxworld' with maximal nonlocality.

Abstract

The framework of generalized probabilistic theories (GPTs) is a popular approach for studying the physical foundations of quantum theory. The standard framework assumes the no-restriction hypothesis, in which the state space of a physical theory determines the set of measurements. However, this assumption is not physically motivated. We generalize the framework to account for systems that do not obey the no-restriction hypothesis. We then show how our framework can be used to describe new classes of probabilistic theories, for example those which include intrinsic noise. Relaxing the restriction hypothesis also allows us to introduce a 'self-dualization' procedure, which yields a new class of theories that share many features of quantum theory, such as obeying Tsirelson's bound for the maximally entangled state. We then characterize joint states, generalizing the maximal tensor product. We show how this new tensor product can be used to describe the convex closure of the Spekkens toy theory, and in doing so we obtain an analysis of why it is local in terms of the geometry of its state space. We show that the unrestricted version of the Spekkens toy theory is the theory known as 'boxworld' that allows maximal nonlocal correlations.