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arXiv:2405.13819 · Physical Review A

Entanglement swapping in generalized probabilistic theories and iterated Clauser-Horne-Shimony-Holt games

Lionel J. Dmello, Laurens T. Ligthart, David Gross

Published Aug 28, 2024• 2 citations
arXivHTMLPDF

Abstract

While there exist theories that have states more strongly entangled than quantum theory, in the sense that they show Clauser-Horne-Shimony-Holt (CHSH) values above Tsirelson's bound, all known examples of such theories have a strictly smaller set of measurements. Therefore, in tasks that require both bipartite states and measurements, they do not perform better than quantum mechanics. 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 generalized probabilistic theories (GPTs). In particular, we introduce the iterated CHSH game, which measures the power of a GPT to preserve nonclassical 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.

DOI
10.1103/physreva.110.022225

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While there exist theories that have states more strongly entangled than quantum theory, in the sense that they show Clauser-Horne-Shimony-Holt (CHSH) values above Tsirelson's bound, all known examples of such theories have a strictly smaller set of measurements. Therefore, in tasks that require both bipartite states and measurements, they do not perform better than quantum mechanics. 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 generalized probabilistic theories (GPTs). In particular, we introduce the iterated CHSH game, which measures the power of a GPT to preserve nonclassical correlations, in terms of the largest CHSH value obtainable after nnn 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.
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 222\sqrt222​ 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 444 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.
Correlation self-testing of a theory addresses the question of whether we can identify the set of correlations realizable in a theory from its performance in a particular information processing task. Applied to quantum theory it aims to identify an information processing task whose optimal performance is achieved only by theories realizing the same correlations as quantum theory in any causal structure. In M. Weilenmann and R. Colbeck [Phys. Rev. Lett. 125, 060406 (2020)], we introduce a candidate task for this---the adaptive Clauser-Horne-Shimony-Holt (CHSH) game. Here, we analyze the maximum probability of winning this game in different generalized probabilistic theories. We show that theories with a joint state space given by the minimal or the maximal tensor product are inferior to quantum theory, before considering other tensor products in theories whose elementary systems have various two-dimensional state spaces. For these, we find no theories that outperform quantum theory in the adaptive CHSH game and prove that it is impossible to recover the quantum performance in various cases. This is the first step towards a general solution that, if successful, will have wide-ranging consequences, in particular, enabling an experiment that could rule out all theories in which the set of realizable correlations does not coincide with the quantum set.
Generalized probabilistic theories (GPTs) provide a framework in which a range of possible theories can be examined, including classical theory, quantum theory, and those beyond. In general, enlarging the state space of a GPT leads to fewer possible measurements because the additional states give stronger constraints on the set of effects, the constituents of measurements. This can have implications for information processing. In box world, for example, a GPT in which any no-signaling distribution can be realized, there is no analog of a measurement in the Bell basis and hence the analog of entanglement swapping is impossible. A comprehensive study of measurements on multiple systems in box world has been lacking. Here we consider such measurements in detail, distinguishing those that can be performed by interacting with individual systems sequentially (termed wirings), and the more interesting set of those that cannot. We compute all the possible box-world effects for cases with small numbers of inputs, outputs, and parties, identifying those that are wirings. The large state space of box world leads to a small effect space and hence the effects of box world are widely applicable in GPTs. We also show some possible uses of nonwirings for information processing by studying state discrimination, nonlocality distillation, and the box-world analog of nonlocality without entanglement. Finally, we connect our results to the study of logically consistent classical processes and to the composition of contextuality scenarios. By enhancing understanding of measurements in box world, our results could be useful in studies of possible underlying principles on which quantum theory can be based.
The classification of general probabilistic theories (GPTs) whose CHSH value is stable under arbitrary rounds of teleportation and entanglement swapping was obtained in Dmello and Gross work and consists of seven families, indexed by characters of the Klein four-group K4K_4K4​, the cyclic group Z4\mathbb{Z}_4Z4​, and the dihedral group D4D_4D4​. The question of which of these families admits a realization within standard quantum mechanics was left open. In this work we resolve this question completely. Using elementary representation theory, we prove that exactly two families are quantum-realizable, namely χ1234K4χ^{K_4}_{1234}χ1234K4​​ and χ125D4χ^{D_4}_{125}χ125D4​​, while the remaining five admit no quantum realization.
Correlation self-testing of quantum theory involves identifying a task or set of tasks whose optimal performance can be achieved only by theories that can realise the same set of correlations as quantum theory in every causal structure. Following this approach, previous work has ruled out various classes of generalised probabilistic theories whose joint state spaces have a certain regularity in the sense of a (discrete) rotation symmetry of the bipartite state spaces. Here we consider theories whose bipartite state spaces lack this regularity. We form them by taking the convex hull of all the local states and a finite number of non-local states. We show that a criterion of compositional consistency is needed in such theories: for a measurement effect to be valid, there must exist at least one measurement that it is part of. This goes beyond previous consistency criteria and corresponds to a strengthening of the no-restriction hypothesis. We show that quantum theory outperforms these theories in a task called the adaptive CHSH game, which shows that they can be ruled out experimentally. We further show a connection between compositional consistency and Tsirelson's bound.
Abstract In this study, we consider generalized probabilistic theories (GPTs) and focus on a class of theories called regular polygon theories, which can be regarded as natural generalizations of a two-level quantum system (a qubit system). In the usual CHSH setting for quantum theory, the CHSH value is known to be optimized by maximally entangled states. This research will reveal that the same observations are obtained also in regular polygon theories. Our result gives a physical meaning to the concept of ‘maximal entanglement’ in regular polygon theories.