Classification of bipartite observables II: NIP observables and the Todorcevic trichotomy
Abstract
This is the second paper in a series that reads the dividing lines of Shelah's classification theory as properties of bipartite quantum observables. For an observable of two independently prepared systems, Alice's and Bob's, we study its idealized responses: the expectation values of Alice's limiting preparations, as functions of Bob's preparations, which serve as probes, with Bob's system idealized first. We show that the observable is NIP, that is, no sequence of Alice's preparations lets Bob's probes realize every finite pattern of high and low values at a fixed positive gap, exactly when all idealized responses are of the first Baire class, pointwise limits of sequences of continuous functions; they then form a separable Rosenthal compactum. Because Alice can mix preparations, this compactum is convex, and hereditary separability forces it to be metrizable, so that NIP observables occupy three of the four topological classes defined by metrizability and Todorčević's trichotomy. Comparisons of photon numbers, of positions and of events in spacetime realize these three classes, and trees of preparations realize the seven minimal families of the Argyros--Dodos--Kanellopoulos heptachotomy. For the ferromagnetic transverse-field Ising model at positive temperature, the finite-volume magnetizations with offset longitudinal fields form a stable family when the spontaneous magnetization vanishes, and a family whose closure is first countable but not hereditarily separable when it does not.
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BibTeX
@misc{duenez2026classification,
title = {{Classification of bipartite observables II: NIP observables and the Todorcevic trichotomy}},
author = {Eduardo Dueñez and José Iovino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12908},
eprint = {2610.12908},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12908},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of bipartite observables II: NIP observables and the Todorcevic trichotomy
AU - Eduardo Dueñez
AU - José Iovino
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.12908
UR - https://arxiv.org/abs/2610.12908
AB - This is the second paper in a series that reads the dividing lines of Shelah's classification theory as properties of bipartite quantum observables. For an observable of two independently prepared systems, Alice's and Bob's, we study its idealized responses: the expectation values of Alice's limiting preparations, as functions of Bob's preparations, which serve as probes, with Bob's system idealized first. We show that the observable is NIP, that is, no sequence of Alice's preparations lets Bob's probes realize every finite pattern of high and low values at a fixed positive gap, exactly when all idealized responses are of the first Baire class, pointwise limits of sequences of continuous functions; they then form a separable Rosenthal compactum. Because Alice can mix preparations, this compactum is convex, and hereditary separability forces it to be metrizable, so that NIP observables occupy three of the four topological classes defined by metrizability and Todorčević's trichotomy. Comparisons of photon numbers, of positions and of events in spacetime realize these three classes, and trees of preparations realize the seven minimal families of the Argyros--Dodos--Kanellopoulos heptachotomy. For the ferromagnetic transverse-field Ising model at positive temperature, the finite-volume magnetizations with offset longitudinal fields form a stable family when the spontaneous magnetization vanishes, and a family whose closure is first countable but not hereditarily separable when it does not.
ER -