Minimal operational theories: classical theories with quantum features

TL;DR

This paper introduces minimal operational theories constrained by basic operations, demonstrating that classical theories can exhibit quantum-like features such as no-broadcasting and disturbance irreversibility, challenging traditional notions of non-classicality.

Contribution

It defines minimal strongly causal theories and constructs a classical toy model exhibiting quantum-like properties, highlighting that such features are not exclusive to quantum systems.

Findings

Minimal theories with certain constraints satisfy quantum no-go theorems.

Classical toy-theory exhibits quantum-like features without non-classicality.

Properties like irreversibility and no-broadcasting are not unique to quantum theory.

Abstract

We introduce a class of probabilistic theories, termed Minimal Strongly Causal Operational Probabilistic Theories, where system dynamics are constrained to the minimal set of operations consistent with the set of states and permitting conditional tests. Specifically, the allowed instruments are limited to those derived from compositions of preparations, measurements, swap transformations, and conditional operations. We demonstrate that minimal theories with conditioning and a spanning set of non-separable states satisfy two quantum no-go theorems: no-information without disturbance and no-broadcasting. As a key example, we construct Minimal Strongly Causal Bilocal Classical Theory, a classical toy-theory that lacks incompatible measurements, preparation uncertainty relations, and is noncontextual (both Kochen-Specker and generalised), yet exhibits irreversibility of measurement disturbance, no-information without disturbance, and no-broadcasting. Therefore, the latter three properties cannot be understood $\textit{per se}$ as signatures of non-classicality. We further explore distinctions between a theory and its minimal strongly causal counterpart, showing that while the minimal strongly causal version of quantum theory diverges from full quantum theory, the same does not hold for classical theory. Additionally, we establish the pairwise independence of the properties of simpliciality, strong causality, and local discriminability.