Entropy increase in $K$-step Markovian and consistent dynamics of closed quantum systems

We consider sequences of measurements implemented by positive operator valued measures (POVMs). Starting from the assumption that these sequences may be described as consistent and Markovian, even and especially for closed quantum systems, we identify properties of the equilibrium state that coincide with the properties of typical pure quantum states. We define a physical entropy that converges against the standard entropies in the approach to equilibrium. Furthermore, strict limits to its possible decrease are derived on the basis of Renyi entropies. It is demonstrated that Landauer's principle follows directly from these limits. Since the above assumptions are rather strong, we exemplify the fact that they may nevertheless apply by checking them numerically for some transition paths in a concrete model.

Figure 1

Comparison of the dynamics of the occupation probabilities $p_1\left(t\right)$ and $p_2\left(t\right)$ of the first and second band, respectively, as resulting from the Schrödinger equation with the same dynamics as resulting from the rate equation (39). Obviously there is a good agreement. For specific model parameters, see text.

Figure 2

Numerical results on the measure of nonconsistency in the “two-macrostate” model (37). Central symbols indicate means, vertical bars spreadings for different random realizations of the model. Obviously, nonconsistency decreases with increasing the density of states. For specific model parameters, see text.

Figure 3

Numerical results on the measure of non-Markovianity in the “two-macrostate” model (37). Also non-Markovianity decreases with increasing the density of states. For specific model parameters, see text.