Midpoint BKM Estimates and Boundary Coherence

TL;DR

This paper derives a noncommutative lower bound for quantum relative entropy between a density matrix and its block-diagonal part using midpoint estimates of the BKM metric, revealing spectral and coherence properties.

Contribution

It introduces a novel midpoint estimate for the BKM Hessian that captures joint spectral and coherence information in quantum states.

Findings

Established a lower bound involving the BKM kernel for quantum relative entropy.

Derived explicit logarithmic bounds under spectral gap conditions.

Connected the BKM metric to the Hessian of quantum relative entropy.

Abstract

We study lower bounds for the quantum relative entropy between a density matrix and its block-diagonal part. For a block matrix with diagonal blocks A,C>0 and off-diagonal coherence block B, we prove a lower bound expressed through the associated Bogoliubov--Kubo--Mori (BKM) kernel. The proof uses a midpoint estimate for the BKM Hessian along the affine interpolation between the matrix and its block-diagonal projection. The resulting estimate is genuinely noncommutative and retains information about the joint spectral structure of the diagonal blocks and the coherence term. As a consequence, under a spectral gap condition on A relative to C, we obtain an explicit logarithmic lower bound proportional to the squared Frobenius norm of the coherence block. The appearance of the BKM metric is natural in this setting because it coincides with the Hessian of quantum relative entropy.