Krein–de Branges spectral theory provides a correspondence between canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We revisit this area by giving a description of canonical Hamiltonian systems whose spectral measures have logarithmic integral converging over the real line. Our result can be viewed as a spectral version of the classical Szegő theorem in the theory of polynomials orthogonal on the unit circle. It extends Krein–Wiener completeness theorem, a key fact in the prediction of stationary Gaussian processes.