Classification of bipartite observables III: simplicity, mixtures and quantum error correction
Abstract
This is the third paper in a series that reads the dividing lines of Shelah's classification theory as properties of bipartite quantum observables. Here we study simplicity through two countable structures with simple unstable theories: the Rado graph, and the symplectic space of labels of Pauli strings of finite support, in which addition records the products of the strings and the symplectic form records their commutation. In the second structure, forking characterizes the logical errors of stabilizer codes: an error undetectable by the checks of a code is logical exactly when its type over the stabilizer labels and a complete set of logical-operator labels forks over the stabilizer labels, and the independence theorem becomes a rule for gluing syndromes. A two-valued relation between countable sets of labels defines a diagonal bipartite observable, whose real-valued theory describes Alice's probability mixtures of labels and their expected responses at Bob's probes. Mixing changes the classification: if the relation has the independence property, this theory has the tree property of the second kind. For the observable of the symplectic form, actual mixtures of checks witness this property for the response itself; for the observable of the Rado graph, every array that witnesses it for a quantifier-free formula in Bob's probe contains idealized preparations, which lie outside the original structure.
Related
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- Classification of bipartite observables II: NIP observables and the Todorcevic trichotomy
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BibTeX
@misc{duenez2026classification,
title = {{Classification of bipartite observables III: simplicity, mixtures and quantum error correction}},
author = {Eduardo Dueñez and José Iovino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22205},
eprint = {2610.22205},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22205},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of bipartite observables III: simplicity, mixtures and quantum error correction
AU - Eduardo Dueñez
AU - José Iovino
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.22205
UR - https://arxiv.org/abs/2610.22205
AB - This is the third paper in a series that reads the dividing lines of Shelah's classification theory as properties of bipartite quantum observables. Here we study simplicity through two countable structures with simple unstable theories: the Rado graph, and the symplectic space of labels of Pauli strings of finite support, in which addition records the products of the strings and the symplectic form records their commutation. In the second structure, forking characterizes the logical errors of stabilizer codes: an error undetectable by the checks of a code is logical exactly when its type over the stabilizer labels and a complete set of logical-operator labels forks over the stabilizer labels, and the independence theorem becomes a rule for gluing syndromes. A two-valued relation between countable sets of labels defines a diagonal bipartite observable, whose real-valued theory describes Alice's probability mixtures of labels and their expected responses at Bob's probes. Mixing changes the classification: if the relation has the independence property, this theory has the tree property of the second kind. For the observable of the symplectic form, actual mixtures of checks witness this property for the response itself; for the observable of the Rado graph, every array that witnesses it for a quantifier-free formula in Bob's probe contains idealized preparations, which lie outside the original structure.
ER -