⚛️ quantum physics

Probabilistic theories stable under teleportation

This paper classifies all general probabilistic theories whose CHSH values remain stable under arbitrary rounds of entanglement swapping, revealing exactly seven solutions that necessarily exhibit counter-intuitive features such as local state spaces with higher dimensions than strictly required for CHSH tests.

Original authors: Lionel J. Dmello, David Gross

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lionel J. Dmello, David Gross

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, complex video game. For decades, physicists have been trying to figure out the "source code" of this game. They know the rules of Quantum Mechanics (QM) work incredibly well, but they don't know why the universe chose these specific rules and not others.

One of the biggest mysteries is a rule about entanglement (spooky connections between particles). In the game, there's a limit to how strong these connections can be. Quantum mechanics hits a specific ceiling (called Tsirelson's bound, roughly 2.82). However, mathematically, you could imagine a universe where the connections are even stronger (up to a perfect 4). Why doesn't our universe go for the maximum power?

The "Teleportation" Test

To solve this, the authors of this paper proposed a new test. Imagine you have two players, Alice and Bob, who are far apart. They share a special "entangled" card. They play a game to test their connection strength.

Now, imagine a third party, Charlie, comes in. He takes Alice's card, performs a magic trick called entanglement swapping (essentially teleporting the connection), and passes a new card to a fourth player, Dave. Then Dave does it again, and again, and again.

The Question: If you keep teleporting this connection over and over, does the strength of the connection stay high, or does it fade away?
The Conjecture: Some scientists thought that only Quantum Mechanics could keep the connection strong forever. They believed that any "stronger" theory would eventually break down under this pressure.
The Twist: The authors previously found a weird, made-up theory (called "Oblate Stabilizer Theory") that could keep the connection strong forever, even stronger than Quantum Mechanics. This broke the idea that "teleportation stability" uniquely identifies our universe.

The Great Classification (The "Seven Families")

So, the authors asked: "Okay, if there are theories that can survive this teleportation marathon, what do they look like? Are there many, or just a few?"

They treated this like a detective story. They stripped away all the unnecessary details of these theories (the "irrelevant degrees of freedom") to see the core structure. They found that the math governing these theories is like a dance troupe.

The Dance Moves: Every time a teleportation happens, the state of the system changes. To keep the game going, the system must be able to "undo" these changes or rotate them back to the start. This means the system has a hidden symmetry group (a set of rules for how things can be shuffled).
The Seven Groups: The authors discovered that there are only seven specific ways (seven "families") to arrange these dance moves so that the connection never breaks.
- Think of it like a lock. There are only seven specific keys that can open the door to a theory that survives infinite teleportation.
- Most of these keys lead to theories that are very different from our universe.
- One of these keys leads to a theory that allows for the maximum possible connection strength (a score of 4).

The "Pancake" Problem

Here is the most surprising part. The authors found that for these theories to work, they cannot follow a rule called "Local Tomography."

Local Tomography (The "Lego" Rule): In our universe, if you want to know what a big, complex object is made of, you just look at its individual Lego bricks (local parts) and how they fit together. You don't need to look at the whole thing at once.
The "Pancake" Reality: The theories that survive the teleportation test are like pancakes. They are so flat and stretched out that you cannot understand the whole picture just by looking at the individual slices. You need to look at the whole pancake at once to understand it.
The Metaphor: Imagine trying to describe a 3D sculpture by only looking at its shadow on a wall. In our universe, the shadow tells you everything. In these "teleportation-stable" theories, the shadow is misleading; you need the full 3D object to make sense of it.

What Does This Mean for Us?

Quantum Mechanics is Special, but not Unique: The fact that Quantum Mechanics survives this teleportation test is cool, but it's not the only theory that can do it. There are six other "cousin" theories that can also do it.
Why We Might Not Be in a "Super-Strong" Universe: The theories that allow for super-strong connections (scoring 4) require these weird "pancake" structures that violate local tomography. Since our universe seems to follow the "Lego rule" (local tomography), we are likely stuck with the Quantum Mechanics version (scoring ~2.82) rather than the super-strong version.
A New Tool: The authors invented a new tool called "Self-Testing" for these theories. It's like a fingerprint scanner. If you see a specific pattern of results in an experiment, you can say, "Aha! This must be one of these seven specific types of universes."

Summary

The paper is a map of the "multiverse of possibilities." It says:

"If you want a universe where entangled particles stay connected forever, no matter how many times you teleport them, you are forced into one of seven specific mathematical shapes. Our universe is one of them (the Quantum one), but there are six others. Interestingly, the ones that are 'stronger' than ours require the universe to be 'flatter' and less local than we thought possible."

It doesn't prove Quantum Mechanics is the only answer, but it narrows the search down to a very small, very specific list of candidates.

1. Problem Statement

The foundational problem addressed is the identification of physical principles that explain why quantum mechanics (QM) achieves a specific maximal violation of Bell inequalities (the CHSH value of $2\sqrt{2}$ ) while strictly forbidding the algebraically maximal value of 4.

Context: Previous research (e.g., Weilenmann and Colbeck) proposed the "adaptive CHSH game," involving iterated entanglement swapping, as a potential discriminator. The conjecture was that only theories maintaining the CHSH value of $2\sqrt{2}$ after arbitrary rounds of teleportation would be physically viable, implying QM is optimal.
The Conflict: In a prior work, the authors constructed a General Probabilistic Theory (GPT) called Oblate Stabilizer Theory (OST) that maintains a CHSH value of 4 even after infinite rounds of entanglement swapping. This challenged the conjecture that stability under teleportation uniquely selects QM.
Goal: This paper aims to classify all GPTs that possess the property of maintaining a stable CHSH value under repeated entanglement swapping (teleportation) to determine the precise structural constraints this property imposes.

2. Methodology

The authors employ a rigorous framework combining General Probabilistic Theories (GPTs), representation theory, and semigroup analysis.

A. Framework and Definitions

GPT Formalism: States are linear functionals on effects within convex cones. The paper utilizes a "closure construction" to generate GPTs from specific sets of states and effects, explicitly breaking permutation symmetry between subsystems (Alice vs. Bob) to handle the directional nature of teleportation.
Teleportation Semigroup: The core mechanism is the "teleportation semigroup." When a bipartite state $\rho$ undergoes entanglement swapping via a measurement $\{\phi_k\}$ , it induces a map on the local state space. Iterating this process generates a semigroup of linear maps $\{R_k\}$ .
Self-Testing in GPTs: The authors generalize the concept of quantum self-testing. They define an Effective CHSH GPT by discarding all degrees of freedom unreachable from a specific CHSH instance. They prove that any instance achieving a CHSH value of 4 (or satisfying specific correlation conditions) leads to a unique effective GPT up to isomorphism.

B. The Iterated CHSH Game

The analysis focuses on a game where $N$ intermediate parties ("Bobs") perform entanglement swapping on a shared state before Alice and Charlie perform a final CHSH test.

Condition 1: The classification relies on instances where the CHSH value remains constant (equal to the initial value) and satisfies "self-testing" conditions (uniform marginals and specific correlator values) after any number of rounds $N$ .
Relabelling Group: The authors analyze the group of operations (relabeling settings and outcomes) required to restore the standard CHSH observable after teleportation. This group is identified as the Dihedral group $D_4$ (order 8).

C. Mathematical Reduction

Semigroup to Group: Using the theory of compact topological semigroups, the authors show that the closure of the teleportation semigroup contains a minimal ideal which forms a compact topological group ( $G_P$ ).
Representation Theory: The correction maps (relabellings) form a homomorphism from this compact group to $D_4$ . This induces a representation $\phi$ of a subgroup $H \subseteq D_4$ on the local state space.
Constraints: The physical requirement that the resulting maps correspond to valid states and effects imposes strict constraints on the character $\chi_\phi$ $χ_{ϕ}$ of this representation:
- The dimension must be $\ge 3$ (to violate CHSH).
- The character values must be non-negative (positivity of probabilities).
- The trivial representation must appear with multiplicity exactly 1 (related to the rank of the marginal state).

3. Key Contributions

Classification of Stable GPTs: The paper provides a complete classification of all GPTs that maintain a stable CHSH value under arbitrary entanglement swapping.
The "Seven-Fold Way": The authors prove that the representation-theoretic constraints reduce the solution space to exactly seven inequivalent families of GPTs. These correspond to specific representations of the groups $Z_4$ , $K_4$ (Klein four-group), and $D_4$ .
Generalized Self-Testing: They introduce a formal definition of self-testing for GPTs, establishing that high-correlation instances uniquely determine the effective theory up to isomorphism.
Operational Justification for Non-Locality: The work provides an operational argument for why Local Tomography (the assumption that global states are determined by local measurements) must be violated in theories supporting stable high CHSH values.

4. Key Results

A. The Seven Families

The valid effective GPTs fall into seven families defined by the character $\chi_\phi$ of the representation of the correction group $H$ :

Dimension 4:
- Regular representation of $Z_4$ ( $\chi^{(Z_4)}_{1234}$ ).
- Regular representation of $K_4$ ( $\chi^{(K_4)}_{1234}$ ).
- Three distinct representations of $D_4$ ( $\chi^{(D_4)}_{125}, \chi^{(D_4)}_{135}, \chi^{(D_4)}_{145}$ ).
Dimension 6:
- One representation of $D_4$ ( $\chi^{(D_4)}_{12345}$ ).
Dimension 8:
- Regular representation of $D_4$ ( $\chi^{(D_4)}_{123452}$ ).

B. Quantum Realizability

Quantum Mechanics (QM): QM corresponds to the family $\chi^{(K_4)}_{1234}$ . The authors show that the $Z_4$ family cannot be realized in QM because the required projective unitary representation would imply a trivial representation multiplicity of at least 2, violating the derived constraint (multiplicity 1).
Other Families: The paper constructs explicit GPTs for all seven families in the appendix. It leaves open whether the remaining families (specifically the $D_4$ variants) can be realized within QM, though it notes that some require POVMs rather than simple basis measurements.

C. Violation of Local Tomography

A critical finding is that none of the seven families are locally tomographic.

The "No-Pancake" Theorem: The authors prove a "GPT no-pancake theorem": In any locally tomographic GPT, the image of the local state space under the teleportation semigroup must have a directional vector space of dimension at least 3.
Implication: Since the effective GPTs for stable CHSH values have local effect spaces of dimension 3 but require higher-dimensional state spaces to support the necessary bipartite effects for swapping, they inherently violate local tomography. This justifies the existence of "redundant" degrees of freedom (as seen in OST) as a necessary feature for theories with stable high CHSH values.

5. Significance

Refining the Search for QM: The results demonstrate that "stability under teleportation" is a necessary but not sufficient condition to single out Quantum Mechanics. While it restricts the landscape to a finite set of seven families, it does not uniquely identify QM without further assumptions (like local tomography).
Understanding Non-Locality: The paper offers a profound insight into the relationship between entanglement swapping and local tomography. It suggests that the ability to perform robust entanglement swapping (teleportation) in a theory with strong non-local correlations necessitates the existence of non-local degrees of freedom that cannot be detected by local measurements alone.
Mathematical Rigor: By reducing a physical problem to a representation-theoretic classification, the authors provide a powerful tool for analyzing the structural limits of probabilistic theories beyond QM.

In summary, the paper establishes that while Quantum Mechanics is not the only theory capable of sustaining high CHSH values under teleportation, it is the only one among the seven mathematically possible families that satisfies the specific constraints of local tomography and standard quantum realizability (for the $K_4$ case). The other families represent "exotic" theories that require non-local state spaces to function.