On state vs. channel quantum extension problems: exact results for UxUxU symmetry

Understand

We develop a framework which unifies seemingly different extension (or "joinability") problems for bipartite quantum states and channels.

• This includes well known extension problems such as optimal quantum cloning and quantum marginal problems as special instances.
• Central to our generalization is a variant of the Choi-Jamiolkowski isomorphism between bipartite states and dynamical maps which we term the "homocorrelation map": while the former emphasizes the preservation of the positivity constraint, the latter is designed to preserve statistical correlations, allowing direct contact with entanglement.
• In particular, we define and analyze state-joining, channel-joining, and local-positive joining problems in three-party settings exhibiting collective UxUxU symmetry, obtaining exact analytical characterizations in low dimension.