  Given a tripartite quantum state on $A,B,C$ and the erasure channel on $C$,
the rotated Petz map is a recovery channel that acts on $B$ to recover the
erased quantum information. The infidelity of the best recovery is
upper-bounded by the conditional mutual information (CMI). In this work, we
study the infidelity of the rotated Petz map on several physically-relevant
long-range entangled quantum states. Specifically, we study three classes of
quantum phases: (i) steady states of measurement-induced phase transitions,
(ii) critical ground state under local measurements, and (iii) chiral states
under local measurements. We find that the average Petz map infidelity sharply
distinguishes the three classes: (i) and (ii) are distinguished by the scaling
of the infidelity with CMI and (iii) is characterized by an asymmetry of the
infidelity with the rotation parameter. We also study Petz map recovery for
topological order and find an operational interpretation of the topological
entanglement entropy. Our result indicates that the Petz map fidelity is a
useful diagnostic of quantum phases of matter.
