Intermediate determinism in general probabilistic theories

Abstract

Quantum theory is indeterministic, but not completely so. When a system is in a pure state there are properties it possesses with certainty, known as actual properties. The actual properties of a quantum system (in a pure state) fully determine the probability of finding the system to have any other property. We call this feature intermediate determinism. In dimensions of at least three, the intermediate determinism of quantum theory is guaranteed by the structure of its lattice of properties. This observation follows from Gleason's theorem, which is why it fails to hold in dimension two. In this work we extend the idea of intermediate determinism from properties to measurements. Under this extension intermediate determinism follows from the structure of quantum effects for separable Hilbert spaces of any dimension, including dimension two. Then, we find necessary and sufficient conditions for a general probabilistic theory to obey intermediate determinism. We show that, although related, both the no-restriction hypothesis and a Gleason-type theorem are neither necessary nor sufficient for intermediate determinism.

References in corpus (24)

• In defense of the epistemic view of quantum states: a toy theory
• Informational derivation of Quantum Theory
• Quantum Theory From Five Reasonable Axioms
• Probabilistic theories with purification
• A derivation of quantum theory from physical requirements
• Quantum states and generalized observables: a simple proof of Gleason's theorem
• Generalized Probability Theories: What determines the structure of quantum theory?
• Limits on non-local correlations from the structure of the local state space
• General probabilistic theories: An introduction
• Generalized Probabilistic Theories Without the No-Restriction Hypothesis
• Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements
• Experimentally bounding deviations from quantum theory in the landscape of generalized probabilistic theories
• Entanglement is an inevitable feature of any non-classical theory
• Entanglement is necessary for emergent classicality in all physical theories
• If no information gain implies no disturbance, then any discrete physical theory is classical
• Simulability of observables in general probabilistic theories
• Contextuality of general probabilistic theories
• Self-testing of physical theories, or, is quantum theory optimal with respect to some information-processing task?
• A Gleason-type theorem for qubits based on mixtures of projective measurements
• An effect-theoretic reconstruction of quantum theory
• Sequential Product Spaces are Jordan Algebras
• Operational restrictions in general probabilistic theories
• General Probabilistic Theories with a Gleason-type Theorem
• Typical local measurements in generalised probabilistic theories: emergence of quantum bipartite correlations