  In classical thermodynamics energy always flows from the hotter system to the
colder one. However, if these systems are initially correlated, the energy flow
can reverse, making the cold system colder and the hot system hotter. This
intriguing phenomenon is called ``anomalous energy flow'' and shows the
importance of initial correlations in determining physical properties of
thermodynamic systems. Here we investigate the fundamental limits of this
effect. Specifically, we find the optimal amount of energy that can be
transferred between quantum systems under closed and reversible dynamics, which
then allows us to characterize the anomalous energy flow. We then explore a
more general scenario where the energy flow is mediated by an ancillary quantum
system that acts as a catalyst. We show that this approach allows for
exploiting previously inaccessible types of correlations, ultimately resulting
in an energy transfer that surpasses our fundamental bound. To demonstrate
these findings, we use a well-studied quantum optics setup involving two atoms
coupled to an optical cavity.
