Minimal operational theories: classical theories with quantum features

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 $\text{per se}$ 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.

30+11 pages; no figures, lots of diagrams

References in corpus (33)

• Quantum cryptography: Public key distribution and coin tossing
• Measurement-based quantum computation with cluster states
• Hidden Variables and the Two Theorems of John Bell
• In defense of the epistemic view of quantum states: a toy theory
• Contextuality for preparations, transformations, and unsharp measurements
• Informational derivation of Quantum Theory
• Probabilistic theories with purification
• No-local-broadcasting theorem for quantum correlations
• A derivation of quantum theory from physical requirements
• A generalized no-broadcasting theorem
• An Invitation to Quantum Incompatibility
• Universal quantum computation using only projective measurement, quantum memory, and preparation of the 0 state
• Limited Holism and Real-Vector-Space Quantum Theory
• General probabilistic theories: An introduction
• Generalized Probabilistic Theories Without the No-Restriction Hypothesis
• The Characterization of Noncontextuality in the Framework of Generalized Probabilistic Theories
• The Feynman problem and Fermionic entanglement: Fermionic theory versus qubit theory
• Fermionic computation is non-local tomographic and violates monogamy of entanglement
• The computational landscape of general physical theories
• Contextuality without incompatibility
• Classical theories with entanglement
• A structure theorem for generalized-noncontextual ontological models
• Testing axioms for Quantum Mechanics on Probabilistic toy-theories
• Classicality without local discriminability: decoupling entanglement and complementarity
• No-free-information principle in general probabilistic theories
• Incompatibility of observables, channels and instruments in information theories
• Post-processing of quantum instruments
• Fermionic state discrimination by local operations and classical communication
• Cellular automata in operational probabilistic theories
• Information and disturbance in operational probabilistic theories
• Measurement incompatibility is strictly stronger than disturbance
• No-broadcasting characterizes operational contextuality
• Causal influence in operational probabilistic theories

Cited by in corpus (1)

• Decoupling local classicality from classical explainability: A noncontextual model for bilocal classical theory and a locally-classical but contextual theory