  Clausius inequality has deep implications for reversibility and the arrow of
time. Quantum theory is able to extend this result for closed systems by
inspecting the trajectory of the density matrix on its manifold. Here we show
that this approach can provide an upper and lower bound to the irreversible
entropy production for open quantum systems as well. These provide insights on
the thermodynamics of the information erasure. Limits of the applicability of
our bounds are discussed, and demonstrated in a quantum photonic simulator.
