Success probability of positive quantum spectral filtering ($P=\Vert A|b⟩{\Vert }^2$)

For $0\le A<I$ and normalized $|b⟩={\sum }_j{\beta }_j|u_j⟩$, filtering by $h(\lambda )=\lambda$ succeeds withThe inequality follows by replacing each nonnegative ${\lambda }_j^2$ by its minimum and using ${\sum }_j|{\beta }_j{|}^2=1$. Equality holds exactly when all occupied eigenvectors belong to the minimum-eigenvalue eigenspace. If ${\lambda }_{\min }=0$, success is positive exactly for inputs outside the kernel of $A$.