Entropy production in the quantum walk

We explore the notion of generated entropy in open quantum systems. We focus on the study of the discrete-time quantum walk on the line, from the entropy production perspective. We argue that the evolution of the coin can be modeled as an open two-level system that exchanges energy with the lattice at some effective temperature that depends on the initial state. The entropy balance shows that there is a positive-entropy production during the evolution, in accordance with the second law of thermodynamics.

Figure 1

Time evolution of the Bloch vector associated with the coin state. The initial states considered are Gaussian-distributed walkers with width $\sigma =10$ and initial chirality $|+\rangle$ (green small circle) and $\frac{1}{\sqrt{2}}\left(\right|+\rangle +|-\rangle )$ (red large circle), respectively. The Bloch vectors are represented by the black dots and converge to the asymptotic state (the center of each circle). The same asymptotic state is reached for all the initial coin states in each circumference.

Figure 2

Dimensionless thermodynamic function for (a) von Neumann entropy, (b) entropy change due to heat transfer, and (c) generated entropy, corresponding to an initially Gaussian-distributed walker, centered at the origin, with $\sigma =30$ (black thick line), 20 (green thin line), 10 (blue dot-dashed line), and 5 (red dotted line). The initial chirality is $|+\rangle$ in all cases.

Figure 3

Similar to Fig. 2 for a walker that starts in a uniform superposition of 101 position states centered at the origin. The initial chirality state is $|+\rangle$. The dimensionless thermodynamic function is shown for (a) von Neumann entropy, (b) entropy change due to heat transfer, and (c) generated entropy.