  We consider quantum batteries given by composite interacting quantum systems
in terms of the thermodynamic work cost of local random unitary processes. We
characterize quantum correlations by monitoring the average energy change and
its fluctuations in the high-dimensional bipartite systems. We derive a
hierarchy of bounds on high-dimensional entanglement (the so-called Schmidt
number) from the work fluctuations and thereby show that larger work
fluctuations can verify the presence of stronger entanglement in the system.
Finally, we develop two-point measurement protocols with noisy detectors that
can estimate work fluctuations, showing that the dimensionality of entanglement
can be probed in this manner.
