Classicality without local discriminability: decoupling entanglement and complementarity

TL;DR

This paper constructs a classical probabilistic theory where all pure states are entangled, demonstrating entanglement can exist without local discriminability, and explores its implications for quantum information phenomena.

Contribution

It introduces a novel operational probabilistic theory with entanglement in classical systems, decoupling entanglement from local discriminability, and develops tools for analyzing such theories.

Findings

Entanglement exists without local discriminability.

The theory exhibits phenomena like cloning, entanglement swapping, and dense coding.

It violates several quantum information principles, such as local discriminability and purification.

Abstract

An operational probabilistic theory where all systems are classical, and all pure states of composite systems are entangled, is constructed. The theory is endowed with a rule for composing an arbitrary number of systems, and with a nontrivial set of transformations. Hence, we demonstrate that the presence of entanglement is independent of the existence of incompatible measurements. We then study a variety of phenomena occurring in the theory -- some of them contradicting both Classical and Quantum Theories -- including: cloning, entanglement swapping, dense coding, additivity of classical capacities, non-monogamous entanglement, hypersignaling. We also prove the existence, in the theory, of a universal processor. The theory is causal and satisfies the no-restriction hypothesis. At the same time, it violates a number of information-theoretic principles enjoyed by Quantum Theory, most notably: local discriminability, purity of parallel composition of states, and purification. Moreover, we introduce an exhaustive procedure to construct generic operational probabilistic theories, and a sufficient set of conditions to verify their consistency. In addition, we prove a characterisation theorem for the parallel composition rules of arbitrary theories, and specialise it to the case of bilocal-tomographic theories. We conclude pointing out some open problems. In particular, on the basis of the fact that every separable state of the theory is a statistical mixture of entangled states, we formulate a no-go conjecture for the existence of a local-realistic ontological model.