  We perform a study on quantum entropy production, different kinds of
correlations, and their interplay in the driven Caldeira-Leggett model of
quantum Brownian motion. The model, taken with a large but finite number of
bath modes, is exactly solvable, and the assumption of a Gaussian initial state
leads to an efficient numerical simulation of all desired observables in a wide
range of model parameters. Our study is composed of three main parts. We first
compare two popular definitions of entropy production, namely the standard
weak-coupling formulation originally proposed by Spohn and later on extended to
the driven case by Deffner and Lutz, and the always-positive expression
introduced by Esposito, Lindenberg and van den Broeck, which relies on the
knowledge of the evolution of the bath. As a second study, we explore the
decomposition of the Esposito et al. entropy production into system-environment
and intra-environment correlations for different ranges of couplings and
temperatures. Lastly, we examine the evolution of quantum correlations between
the system and the environment, measuring entanglement through logarithmic
negativity.
