Entanglement-swapping in generalised probabilistic theories, and iterated CHSH games

TL;DR

This paper explores entanglement swapping within generalized probabilistic theories (GPTs), introducing an iterated CHSH game to measure non-classical correlations, and constructs a GPT capable of surpassing quantum limits after multiple rounds.

Contribution

It introduces an algorithmic method to construct multipartite GPTs supporting entanglement swapping and demonstrates a GPT achieving maximal CHSH value after multiple rounds.

Findings

Constructed a GPT with CHSH value of 4 after any number of rounds

Developed an algorithm to extend bipartite GPTs to multipartite GPTs supporting entanglement swapping

Addressed the optimality question of quantum theory in iterated CHSH games

Abstract

While there exist theories that have states "more strongly entangled" than quantum theory, in the sense that they show CHSH values above Tsirelson's bound, all known examples of such theories have a strictly smaller set of measurements. Therefore, in tasks which require both bipartite states and measurements, they do not perform better than QM. One of the simplest information processing tasks involving both bipartite states and measurements is that of entanglement swapping. In this paper, we study entanglement swapping in generalised probabilistic theories (GPTs). In particular, we introduce the iterated CHSH game, which measures the power of a GPT to preserve non-classical correlations, in terms of the largest CHSH value obtainable after $n$ rounds of entanglement swapping. Our main result is the construction of a GPT that achieves a CHSH value of $4$ after an arbitrary number of rounds. This addresses a question about the optimality of quantum theory for such games recently raised by Weilenmann and Colbeck. One challenge faced when treating this problem is that there seems to be no general framework for constructing GPTs in which entanglement swapping is a well-defined operation. Therefore, we introduce an algorithmic construction that turns a bipartite GPT into a multipartite GPT that supports entanglement swapping, if consistently possible.