Double Dilation $\neq$ Double Mixing (extended abstract)

TL;DR

This paper demonstrates that iterated density operator constructions via mixing and dilation are fundamentally different, with dilation producing entangled states and mixing only producing disentangled states, challenging their assumed equivalence.

Contribution

It reveals that double dilation and double mixing are not equivalent processes, providing a diagrammatic proof that dilation can generate entangled states while mixing cannot.

Findings

Dilation yields arbitrary symmetric bipartite states.

Mixing only produces disentangled states.

Results hold in general process theories.

Abstract

Density operators are one of the key ingredients of quantum theory. They can be constructed in two ways: via a convex sum of 'doubled kets' (i.e. mixing), and by tracing out part of a 'doubled' two-system ket (i.e. dilation). Both constructions can be iterated, yielding new mathematical species that have already found applications outside physics. However, as we show in this paper, the iterated constructions no longer yield the same mathematical species. Hence, the constructions 'mixing' and 'dilation' themselves are by no means equivalent. Concretely, when applying the Choi-Jamiolkowski isomorphism to the second iteration, dilation produces arbitrary symmetric bipartite states, while mixing only yields the disentangled ones. All results are proven using diagrams, and hence they hold not only for quantum theory, but also for a much more general class of process theories.