  We investigate how to coherently define entropy production for a process of
transient relaxation in the Quantum Brownian Motion model for harmonic
potential. We compare a form, called "Poised" (P), which after non-Markovian
transients corresponds to a definition of heat as the change in the system
Hamiltonian of mean force, with a recent proposal by Esposito (ELB) based on a
definition of heat as the energy change in the bath. Both expressions yield a
positive-defined entropy production and coincide for vanishing coupling
strength, but their difference is proved to be always positive (after
non-Markovian transients disappear) and to grow as the coupling strength
increases. In the classical over-damped limit the "Poised" entropy production
converges to the entropy production used in stochastic thermodynamics. We also
investigate the effects of the system size, and of the ensuing Poincar\'e
recurrences, and how the classical limit is approached. We close by discussing
the strong-coupling limit, in which the ideal canonical equilibrium of the bath
is violated.
