Probabilistic theories with purification

TL;DR

This paper explores probabilistic theories with a purification principle, establishing a framework that reproduces key quantum features without relying on Hilbert space mathematics.

Contribution

It introduces a purification-based framework that generalizes quantum mechanics and derives fundamental quantum properties from this principle.

Findings

Reveals an isomorphism similar to Choi-Jamiolkowski in general probabilistic theories.

Derives key quantum phenomena such as no-cloning and teleportation from purification.

Shows that the purification principle implies a reversible realization of all physical processes.

Abstract

We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, namely that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamiolkowski isomorphism in quantum mechanics. Such an isomorphism allows one to prove most of the basic features of quantum mechanics, like e.g. existence of pure bipartite states giving perfect correlations in independent experiments, no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, no programming, no bit commitment, complementarity between correctable channels and deletion channels, characterization of entanglement-breaking channels as measure-and-prepare channels, and others, without resorting to the mathematical framework of Hilbert spaces.