Classification of bipartite observables I: stable and uniformly stable observables

Abstract

When two quantum systems are prepared independently and each is driven to an idealized limit, the expectation of a bipartite observable may depend on which system is idealized first. We call an observable stable when it does not. We show that stability is equivalent to two operational properties. First, every idealized preparation of one system can be simulated to arbitrary accuracy, uniformly over all probes, by finite mixtures of actual preparations, provided that the probing system is idealized first. Second, any two idealized preparations have a unique, symmetric independent product. Under a uniform form of stability, we prove, the number of preparations that $n$ adaptive probes can tell apart at a fixed margin is bounded by a polynomial in $n$, and responses can be learned with a bounded number of mistakes; otherwise, at some margin, protocols with $n$ probes tell apart $2^n$ preparations. The swap test, Hong--Ou--Mandel coincidences and discrete equality tests turn out to be uniformly stable, with bounds independent of the dimension, while order comparisons of discrete spectra unbounded above are unstable. The proofs rest on the classical fact that a single double limit condition appears, under different names, in topology (Grothendieck), functional analysis (Krivine--Maurey), algebra (Arens), probability (Simons) and combinatorics (Pták), and in model theory as Shelah's stability, where the independent product corresponds to Harrington's lemma. Model theory thus serves as a Rosetta stone, pairing each physical question with its mathematical formulation.