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.

References

Unlike local recovery maps, it is still unknown whether local extension maps have closed-form expressions like twirled Petz maps that can be learned efficiently in generic cases.

— Learning and Generating Mixed States Prepared by Shallow Channel Circuits  (2604.01197 - Hu et al., 1 Apr 2026) in Section 2.4 (Local extendibility)

Regarding the crossed product $#1{A}\rtimes_{\sigma}{$ associated with a type $\text{III}{1}$ von~Neumann algebra and a faithful state $(#1{A},\omega)$, if $\mathcal{E}:#1{A}\rtimes{\sigma}{\to#1{A}$ denotes the restriction of the map on $#1{A}\overline{\otimes}\mathcal{B}(L2({))\to#1{A}$ uniquely determined by sending $a\otimes T$ to $\xi(T) a$ for some fixed faithful state $\xi$ on $\mathcal{B}(L2({))$, is there a more explicit description of the Petz recovery map of $\mathcal{E}$ in this case?

— Bayesian inference and retrodiction for faithful states on von Neumann algebras  (2608.20001 - Karmakar et al., 20 Aug 2026) in Item (4), Section 9, “Summary and discussion” (page number unavailable)