arXiv subjects

José Iovino

Publications and source records attributed to José Iovino.

8 recordsLinked to original sources

arXiv2026

Classification of bipartite observables I: stable and uniformly stable observables

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.

math.FA↗

arXiv2026

Classification of bipartite observables III: simplicity, mixtures and quantum error correction

The paradigm of a simple unstable theory in the sense of Shelah is the theory of the random graph, the Fraïssé limit of the finite graphs. Quantum error correction has a paradigm of its own, and we show that it is the counterpart of the random graph: the Pauli group on countably many qubits, taken modulo phases and with commutation as its relation, is the Fraïssé limit of the finite alternating spaces over the two-element field, and its form computes the syndromes of stabilizer codes. Seen through this correspondence, simplicity theory turns out to be a theory of error correction. Clifford encoding circuits implement the homogeneity of the Pauli group, forking recognizes the logical errors, and the independence theorem of Kim and Pillay is the gluing of syndromes. The two paradigms look alike as first-order structures, and they part company only when observables enter. Each defines a bipartite observable, and we prove that, once preparations are mixed, both acquire the tree property of the second kind in real-valued logic. They reach it by different roads, however, and we show that the probes decide which. The probes of the Pauli observable are the Pauli channels; for them the indicator of each syndrome sector is the difference of the responses of two uniform mixtures, and actual mixtures already witness the tree. The probes of the Rado observable see only marginals; conditions given by actual mixtures remain compatible on disjoint blocks, and the tree must be sought among idealized preparations.

math.LO↗

arXiv2026

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

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.

quant-ph↗

arXiv2026

Complexity of deep computations via topology of function spaces

We use topological methods to study complexity of deep computations and limit computations. We use topology of function spaces, specifically, the classification of Rosenthal compacta, to identify new complexity classes. We use the language of model theory, specifically, the concept of \emph{independence} from Shelah's classification theory, to translate between topology and computation. We use the theory of Rosenthal compacta to characterize approximablility of deep computations, both deterministically and probabilistically.

math.LO↗

arXiv2014

Model Theory and the mean ergodic theorem for abelian unitary actions

We explore connections between von Neumann's mean ergodic theorem and concepts of model theory. As an application we present a proof Wiener's generalization of von Neumann's result in which the group acting on the Hilbert space $\mathcal{H}$ is any abelian group of unitary transformations of $\mathcal{H}$.

math.LO↗

arXiv2013

Ultraproducts and metastability

Given a convergence theorem in analysis, under very general conditions a model-theoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory.

math.FA↗

arXiv2012

Omitting uncountable types, and the strength of $[0,1]$-valued logics

We study $[0,1]$-valued logics that are closed under the Łukasiewicz-Pavelka connectives; our primary examples are the the continuous logic framework of Ben Yaacov and Usvyatsov \cite{Ben-Yaacov-Usvyatsov:2010} and the Łukasziewicz-Pavelka logic itself. The main result of the paper is a characterization of these logics in terms of a model-theoretic property, namely, an extension of the omitting types theorem to uncountable languages.

math.LO↗

arXiv2008

Model theoretic forcing in analysis

We present a framework for model theoretic forcing in a non-first-order context, and present some applications of this framework to Banach space theory.

math.LO↗