Classical theories with entanglement

Abstract

We investigate operational probabilistic theories where the pure states of every system are the vertices of a simplex. A special case of such theories is that of classical theories, i.e. simplicial theories whose pure states are jointly perfectly discriminable. The usual Classical Theory satisfies also local discriminability. However, simplicial theories---including the classical ones---can violate local discriminability, thus admitting of entangled states. First, we prove sufficient conditions for the presence of entangled states in arbitrary probabilistic theories. Then, we prove that simplicial theories are necessarily causal, and this represents a no-go theorem for conceiving non-causal classical theories. We then provide necessary and sufficient conditions for simplicial theories to exhibit entanglement, and classify their system-composition rules. We conclude proving that, in simplicial theories, an operational formulation of the superposition principle cannot be satisfied, and that---under the hypothesis of $n$-local discriminability---no mixed state admits of a purification. Our results hold also in the general case where the sets of states fail to be convex.

The definitions of simplicial and classical theories have been relaxed, in order to include the non-convex cases; the results hold also in the non-convex cases. Improved and extended presentation. Added results on superposition and purification. Enhanced the 'Discussion and Conclusions' section. Minor corrections. Layout fixed and mention to Ref. [13] corrected

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• A no-go theorem on the nature of the gravitational field beyond quantum theory
• Entanglement and superposition are equivalent concepts in any physical theory
• Classicality without local discriminability: decoupling entanglement and complementarity
• Incompatibility of observables, channels and instruments in information theories
• Information and disturbance in operational probabilistic theories
• Measurement incompatibility is strictly stronger than disturbance
• Generalized dynamical theories in phase space and the hydrogen atom
• Shannon theory beyond quantum: information content of a source
• Bell Nonlocality in Classical Systems Coexisting with other System Types
• Causal influence in operational probabilistic theories
• A Post-Quantum Associative Memory
• Detecting genuine multipartite entanglement using moments of positive maps
• The signaling dimension of physical systems
• Optimal CHSH values for regular polygon theories in generalized probabilistic theories
• Minimal operational theories: classical theories with quantum features
• Decoupling local classicality from classical explainability: A noncontextual model for bilocal classical theory and a locally-classical but contextual theory
• Which entropy for general physical theories?