Bridge the Markov property of mixed states and nondestructive ground-state tomography

Establish a principled connection between the local Markov property of quantum Gibbs (mixed) states—specifically, the existence of quasi-local recovery maps under local noise—and nondestructive, catalytic tomography of gapped ground states, determining whether and how the former implies the latter for measuring local observables efficiently and quasi-locally.

Background

The paper contrasts its catalytic tomography protocol with the local Markov property of thermal (Gibbs) states, where local noise can be recovered by quasi-local dynamics. While those recovery maps enjoy quasi-locality, their runtime depends on local mixing times and does not directly yield tomography of observables. The authors note that such recovery in mixed states—classical or quantum—does not automatically imply a nondestructive measurement capability for ground states.

This open problem asks for a formal bridge between these two notions: whether the structural Markov properties of mixed states can be leveraged to obtain nondestructive, quasi-local tomography procedures for gapped ground states, potentially unifying recovery-based dynamics and catalytic measurement frameworks.