Probabilistic theories stable under teleportation

TL;DR

This paper classifies all general probabilistic theories that maintain their Bell inequality violation levels under repeated entanglement swapping, revealing seven solutions and generalizing self-testing to GPTs.

Contribution

It provides a complete classification of GPTs with stable CHSH values under entanglement swapping, including a new representation-theoretic condition and generalization of self-testing.

Findings

Seven solutions to the classification problem.

The GPT from previous work has higher-dimensional local state spaces.

Generalized concept of self-testing for GPTs.

Abstract

A long-standing problem in the foundations of quantum mechanics is to identify a physical principle that explains why algebraically maximal violations of Bell inequalities can generally not be achieved in Nature. One recently proposed approach considers iterated Bell tests, where a Bell test is performed on a state that has undergone several rounds of entanglement swapping. Obtaining large violations in this scenario is more demanding, because it requires a theory to have both highly entangled states and highly entangled measurements. It has been conjectured that the maximal quantum mechanical Clauser-Horne-Shimony-Holt (CHSH)-value of $2\sqrt2$ might be optimal for any probabilistic theory which, like quantum mechanics, maintains its CHSH-value after an arbitrary number of rounds of entanglement swapping. However, in a previous paper, we have exhibited a first example of a probabilistic theory that can sustain a CHSH value of $4$ in this setting. In this work, further investigating this property, we give a classification of all general probabilistic theories (GPTs) whose CHSH value is stable in the above sense. The problem reduces to a representation-theoretic condition that allows for exactly seven solutions. The GPT from our previous work showed some counter-intuitive features, e.g. that the local state space had a higher dimension than seemed necessary to realize CHSH tests. The classification shows that this is necessarily so. Along the way, we generalize the concept of self-testing to GPTs.