Benign Projective Landscapes for Measured Quantum Divergences
Abstract
We study nonconvex optimization of measured quantum f-divergences over rank-one projective measurements. For smoothly operator-Fenchel liftable generators and faithful states, every projective local maximum and every second-order stationary point is globally optimal over all POVMs. The criterion is blockwise: a critical PVM is optimal exactly when the compressed states are proportional on each equal-score block; otherwise an explicit two-vector rotation has positive ascent curvature. Operator-convex generators admit a positive atomic curvature resolution, and the quadratic χ2 case yields two-sided residual bounds. For binary accessible information, this framework proves the known adaptive-capacity equality and shows that every nonidentical qubit ensemble has exactly two stationary projective measurements, proving conjectures of Keil and Thai--Dall'Arno. A rare-prior limit connects weighted Jensen--Shannon information to relative entropy and yields a finite counterexample to the proposed equivalence between observational-entropy and all-ensemble mutual-information orders. The landscape theorem also covers measured Rényi divergences of finite positive order and measured relative entropy.
Related
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BibTeX
@misc{salazar2026benign,
title = {{Benign Projective Landscapes for Measured Quantum Divergences}},
author = {Domingos S. P. Salazar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27767},
eprint = {2609.27767},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27767},
}
RIS
TY - UNPB
TI - Benign Projective Landscapes for Measured Quantum Divergences
AU - Domingos S. P. Salazar
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27767
UR - https://arxiv.org/abs/2609.27767
AB - We study nonconvex optimization of measured quantum f-divergences over rank-one projective measurements. For smoothly operator-Fenchel liftable generators and faithful states, every projective local maximum and every second-order stationary point is globally optimal over all POVMs. The criterion is blockwise: a critical PVM is optimal exactly when the compressed states are proportional on each equal-score block; otherwise an explicit two-vector rotation has positive ascent curvature. Operator-convex generators admit a positive atomic curvature resolution, and the quadratic χ^2 case yields two-sided residual bounds. For binary accessible information, this framework proves the known adaptive-capacity equality and shows that every nonidentical qubit ensemble has exactly two stationary projective measurements, proving conjectures of Keil and Thai--Dall'Arno. A rare-prior limit connects weighted Jensen--Shannon information to relative entropy and yields a finite counterexample to the proposed equivalence between observational-entropy and all-ensemble mutual-information orders. The landscape theorem also covers measured Rényi divergences of finite positive order and measured relative entropy.
ER -