Closed-form expressions for local extension maps

Determine whether closed-form expressions exist for local extension maps, analogous to the twirled Petz map for local recovery, that can be efficiently learned and applied in generic settings; if they exist, derive such expressions.

Background

The learning framework relies on two key structural properties: approximate Markovianity and local extendibility. For approximate Markovianity, local recovery maps can be constructed efficiently, and in many cases the twirled Petz map provides a closed-form near-optimal solution.

By contrast, for local extendibility (extending from a slightly larger region B' to BC while discarding redundant parts), no analogous closed-form map is known. Establishing such a formula would streamline learning and potentially reduce computational overhead.