Krein-de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We further develop this area by giving a description of canonical Hamiltonian systems whose spectral measures have logarithmic integral converging over the real line. This result can be viewed as a spectral version of the classical Szegô theorem in the theory of polynomials orthogonal on the unit circle. It extends the Krein-Wiener completeness theorem, a key fact in the prediction of stationary Gaussian processes.