Markov Constraints Enhance Identifiability in Quantum Shadow Inversion

Abstract

We study quantum shadow inversion under Markovian locality constraints for four-partite systems arranged along the chain $A$--$B$--$C$--$D$. The goal is to reproduce the expectation value of a fixed endpoint observable $O_{AD}$ after an unknown global unitary, without requiring full unitary inversion. We formulate the task using Markov-admissible supermaps and introduce the Markov-implementable centralizer to describe the remaining endpoint gauge freedom. We show that unrestricted endpoint post-processing is too broad, and impose an endpoint-local refinement. Under this condition, every implementable endpoint unitary must factorize across $A|D$, so the Markov constraint strictly reduces the centralizer-induced shadow ambiguity whenever the full centralizer contains non-product unitaries. This provides a structural mechanism by which Markov locality enhances identifiability in quantum shadow inversion.