Shelah's dividing lines in quantum measurement: 32 variations in three parts
Preprints · José Iovino
A bipartite observable compares a preparation of one quantum system with a probe of another, and its expectation is a formula of real-valued logic. The paper is addressed to two communities, model theorists and operator theorists working in quantum information, and uses their common tool, the ultrapower, to translate between them. We show that three dividing lines of Shelah in model theory (stability, NIP and simplicity) classify bipartite observables, that each has a Fraïssé limit at its heart, and that each acquires a dictionary into physics. Stability is the independence of a measurement from the order of two idealized limits. NIP, the absence of the independence property, sorts physical orders, from photon numbers to the causal order of spacetime, into the classes of the trichotomy of Todorčević. Simplicity is met in quantum error correction, where forking separates harmless undetectable errors from logical ones. The three parts are organized around one theorem, the double limit criterion of Grothendieck, and each determines what mixing quantum states does: it realizes idealized preparations when the observable is stable, makes hereditary separability imply metrizability for NIP observables, and, for observables diagonal in a product basis, destroys simplicity in the presence of the independence property.
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