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arXiv:2603.21347

Probabilistic theories stable under teleportation

Lionel J. Dmello, David Gross

quant-phSubmitted Mar 22, 2026• Announced Mar 24, 2026• 0 citations
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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.

Comments
18 pages, 7 figures

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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.
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.
A key goal in the foundations of quantum mechanics is to identify operational constraints characterizing physical theories. Bell inequalities do so for classical probability theory, while Tsirelson's bound provides a first step for quantum mechanics. Here, we address the dual question: Can one certify that all correlations predicted by quantum theory are actually realizable? In this work, we construct a finite number of two-body correlations such that the only probabilistic theory that (1) realizes them, and (2) does so in a way that is stable under iterated teleportation, is quantum theory. For nnn quantum systems of dimension ddd each, Condition (1) can be verified using poly⁡(d,n)\operatorname{poly}(d,n)poly(d,n) measurement settings. Condition (2) may be understood as a hierarchy of tests, one for each number NNN of teleportation steps. There is thus a sense in which finite-dimensional quantum theory can be self-tested. In particular, one can certify the existence of Bell inequality violations larger than any that have been directly observed.
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.
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.
We evaluate the maximal Clauser-Horne-Shimony-Holt (CHSH) violation for a generic (typically mixed) qubit-qudit state, obtaining easily computable expressions in arbitrary qudit dimension. This represents the optimal (2-2-2) Bell nonlocality for this kind of systems. The work generalizes the well-known Horodeckis result for a qubit-qubit setup. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues, namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. In addition, we show the impossibility of improving the amount of CHSH violation by embedding the qudit in a Hilbert space of larger dimension. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.
Abstract The Clauser–Horne–Shimony–Holt (CHSH) inequality test is widely used as a mean of invalidating the local deterministic theories. Most attempts to experimentally test nonlocality have presumed unphysical idealizations that do not hold in real experiments, namely, noiseless measurements. We demonstrate an experimental violation of the CHSH inequality that is free of idealization and rules out local models with high confidence. We show that the CHSH inequality can always be violated for any nonzero noise parameter of the measurement. Intriguingly, less entanglement exhibits more nonlocality in the CHSH test with noisy measurements. Furthermore, we theoretically propose and experimentally demonstrate how the CHSH test with noisy measurements can be used to detect weak entanglement on two-qubit states. Our results offer a deeper insight into the relation between entanglement and nonlocality.