Quantum discord in predictive processes

We generalize a recent result for classical systems [Phys. Rev. Lett. 109, 120604 (2012)] to the quantum regime: A system which responds to a changing environment can be thought of as performing an implicit computation of the environment's variables. We argue that the change in the entropy of the system, conditioned on the relevant part of its surroundings, is a meaningful measure of the implicit computation's thermodynamic inefficiency. By considering a simple model of a system that can make predictions regarding a changing external signal, we further show that this can be thought of as a measure of ineffective use of information, in complete analogy with the classical case. We use quantum discord to quantify any quantum advantage or disadvantage in the efficiency, and show that quantum correlations can give an improved thermodynamic operation quantified by a negative "quantum part" of the dissipated work.