Sharp Entropy Cost of Coherent Leakage Across a Block Decomposition

Abstract

Let $\mathcal{H}=P\mathcal{H}\oplus Q\mathcal{H}$ and let $\Pi\rho=P\rho P\oplus Q\rho Q$ be the two-block pinching of a density matrix. Write $A=P\rho P$, $B=P\rho Q$, and $C=Q\rho Q$. We study the block-coherence entropy $D(\rho\|\Pi\rho)$, the relative-entropy cost of removing cross-block coherence. First, we prove a midpoint Bogoliubov--Kubo--Mori coercivity estimate, giving the operator-kernel lower bound $D(\rho\|\Pi\rho)\ge \operatorname{Tr}[B^*\Omega^{-1}_{A,C}(B)]$ for strictly positive diagonal blocks. The proof uses the block-sign involution $U=P-Q$ to obtain a sign-symmetric midpoint lower bound for the BKM Hessian. Second, we determine the exact minimum in the associated constrained entropy-minimization problem: at fixed protected-sector floor, leakage mass, and quadratic coherence, the minimum can be achieved by a single active two-level block, independently of the ambient dimension. Third, under a uniform floor $A\ge a_0P$ and $0<\varepsilon_Q\le a_0/2$, we obtain the logarithmic boundary law $D(\rho\|\Pi\rho)\ge \|B\|_F^2\log(a_0/\varepsilon_Q)$, showing that the lower-bound coefficient multiplying the quadratic coherence diverges logarithmically as leakage mass tends to zero. As a finite-dimensional application, the estimates give an entropy certificate for coherent leakage from a protected subspace: $P\rho Q$ measures protected--leakage coherence, and the constrained entropy minimum is attained by a state supported on a single effective two-level transition. The BKM estimate also yields a dissipation lower bound along an idealized uniform block-dephasing orbit. Finally, the midpoint mechanism extends to finite-dimensional trace-preserving $\mathbb{Z}_2$-graded automorphisms and bounded finite-trace semifinite corners.