In this paper, we deal with a special class of passive systems, which possess the characteristic property of having no finite spectral zeros. We call these systems strongly passive. It is well known that, for these systems, storage functions, i.e. solutions to the linear matrix inequality (LMI) arising from the Kalman–Yakubovich–Popov (KYP) lemma, cannot be obtained by the conventional approach of algebraic Riccati equations (AREs) and Hamiltonian matrices. In this paper, we first show that a strongly passive system always admits a unique storage function. We then provide a closed-form expression for this unique storage function. Using the closed-form formula of the unique storage function we characterise the ‘lossless’ trajectories of strongly passive systems and show that such systems admit impulsive lossless trajectories on the half-line; we call them fast lossless trajectories. This adds to the existing notion that such systems do not admit any ‘slow’ lossless trajectories.