Classical theories with entanglement

Marco Erba (Universita di Pavia), Giacomo Mauro D’Ariano (Universita di Pavia), Paolo Perinotti (Universita di Pavia)

We present the framework of Operational Probabilistic Theories (OPTs), where the probabilistic structure hinges upon braided strict monoidal categories. We then focus on classical theories, namely theories where the classes of states are simplexes. We show that, in general, a classical theory may admit entangled states, i.e. states which cannot be prepared with local operations and classical communication. This feature is found to be equivalent to the failure of local discriminability or, equivalently in the classical case, of atomicity of state-composition. We then construct an explicit example—complete with the set of operations—of such a theory. The OPT is causal and allows for both reversible permutability and perfect discriminability of pure states. Interestingly, this theory provides a “counterexample” to the notion of purity defined in the context of process theories (via the so-called ‘leaks’), meaning that the definition fails to apply to the present case. Entanglement, on the other hand, is found to be a ubiquitous feature for any simplectic theory which is not standard classical theory. Finally, the OPT constructed is well-suited for cryptographic and computational protocols.