Double Dilation $\neq$ Double Mixing

TL;DR

This paper demonstrates that the iterative processes of mixing and dilation for density operators, though similar in construction, produce fundamentally different mathematical objects, with implications for quantum and process theories.

Contribution

It shows that double dilation and double mixing are not equivalent, clarifying their distinct mathematical and physical properties through diagrammatic proofs.

Findings

Dilation yields all symmetric bipartite states after iteration.

Mixing only produces disentangled states after iteration.

Results apply broadly beyond quantum theory to 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.