Quantum Realizability of Probabilistic Theories Stable under Teleportation

TL;DR

This paper classifies which probabilistic theories stable under teleportation can be realized within quantum mechanics, identifying exactly two realizable families among seven.

Contribution

It completely resolves which GPT families stable under teleportation are compatible with standard quantum mechanics using elementary representation theory.

Findings

Exactly two families are quantum-realizable: ^{K_4}_{1234} and ^{D_4}_{125}.

Remaining five families admit no quantum realization.

Classification based on stability under teleportation and entanglement swapping.

Abstract

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 $K_4$, the cyclic group $\mathbb{Z}_4$, and the dihedral group $D_4$. 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 $\chi^{K_4}_{1234}$ and $\chi^{D_4}_{125}$, while the remaining five admit no quantum realization.