  We study the role of coherence in closed and open quantum batteries. We
obtain upper bounds to the work performed or energy exchanged by both closed
and open quantum batteries in terms of coherence. Specifically, we show that
the energy storage can be bounded by the Hilbert-Schmidt coherence of the
density matrix in the spectral basis of the unitary operator that encodes the
evolution of the battery. We also show that an analogous bound can be obtained
in terms of the battery's Hamiltonian coherence in the basis of the unitary
operator by evaluating their commutator. We apply these bounds to a 4-state
quantum system and the anisotropic XY Ising model in the closed system case,
and the Spin-Boson model in the open case.
