Failure mechanism of iterative unextended Petz tomography

Determine why iterating the unextended Petz recovery map for quantum measurement tomography can approach rank-deficient states without reaching the maximum-likelihood estimator.

Background

The paper contrasts classical Jeffrey updating, which converges to the maximum-likelihood estimator under suitable nonsingularity and support conditions, with the unextended quantum Petz iteration. For quantum measurement tomography, the unextended Petz update is monotonic in likelihood but may converge to a rank-deficient state rather than the maximum-likelihood estimator, particularly when the target state is near a rank-deficient subspace or has high purity. The authors provide a heuristic explanation in Appendix B ("Iterative Petz vs Iterative Mirrored Petz"), but explicitly leave the underlying reason for this behavior unresolved.