Classification of bipartite observables II: NIP observables and the Todorcevic trichotomy

Abstract

A bipartite observable is stable when its idealized responses are continuous. We show that it is NIP exactly when they are of the first Baire class. They then form a Rosenthal compactum, and the dividing lines of Shelah in model theory, together with the classification of Rosenthal compacta of Todorčević, acquire a physical meaning. For observables, mixtures of states make every hereditarily separable compactum of idealized responses metrizable, and the Todorčević trichotomy becomes a classification into three classes. Each class contains a physical order: the comparison of photon numbers, the comparison of positions, and the causal order of spacetime, whose idealized responses are not first countable because a light cone can shrink to a point; the comparison of two phases on a circle joins it there. Each of the seven minimal families of the heptachotomy, which Argyros, Dodos and Kanellopoulos extracted from the methods of Todorčević, is realized by a tree of preparations of photons, of a particle in a box, or of a spin chain read by a probe; the spin chain also shows that NIP, unlike its uniform version, is not symmetric between the two parties. Finally, for the Ising model in a transverse field we show that, in the ordered phase, the responses of the two pure phases form split pairs whose midpoints, the responses of the symmetric state, form an uncountable discrete set; the proof uses a Lee--Yang theorem for this model.