Spectral measures and sharpness of quantum speed limits

Abstract

Quantum speed limits bound the time required for a given state to evolve into a given subspace of the Hilbert space under specific energy constraints. We prove that for several standard orthogonalization time bounds, the unique spectral measure associated with the initial vector that saturates the bound is a sum of two Dirac measures. Furthermore, we constrain the families of spectral measures that come arbitrarily close to these bounds and construct examples with measures which are purely absolutely continuous and purely singular continuous with respect to the Lebesgue measure. As a complementary perspective, we discuss the geometric intuition behind quantum speed limits provided by the numerical range of the evolution operator.