Non-locality in theories without the no-restriction hypothesis

TL;DR

This paper explores the effects of relaxing the no-restriction hypothesis in generalized probabilistic theories, showing that it does not necessarily lead to increased non-locality, and introduces a generalized maximal tensor product.

Contribution

It introduces the generalized maximal tensor product, extending the GPT framework without the no-restriction hypothesis, and analyzes its implications for non-locality.

Findings

Relaxing the no-restriction hypothesis does not necessarily increase non-locality.

The generalized maximal tensor product recovers the standard one when at least one system obeys the no-restriction hypothesis.

Under certain conditions, relaxing the hypothesis does not enable stronger non-local correlations.

Abstract

The framework of generalized probabilistic theories (GPT) is a widely-used approach for studying the physical foundations of quantum theory. The standard GPT 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. In Janotta and Lal [Phys. Rev. A 87, 052131 (2013)], it was shown how this assumption can be relaxed, and how such an approach can be used to describe new classes of probabilistic theories. This involves introducing a new, more general, definition of maximal joint state spaces, which we call the generalised maximal tensor product. Here we show that the generalised maximal tensor product recovers the standard maximal tensor product when at least one of the systems in a bipartite scenario obeys the no-restriction hypothesis. We also show that, under certain conditions, relaxing the no-restriction hypothesis for a given state space does not allow for stronger non-locality, although the generalized maximal tensor product may allow new joint states.