Classical predictability and coarse-grained evolution of the quantum baker’s map

We investigate how classical predictability of the coarse-grained evolution of the quantum baker’s map depends on the character of the coarse-graining. Our analysis extends earlier work by Brun and Hartle [Phys. Rev. D 60, 123503 (1999)] to the case of a chaotic map. To quantify predictability, we compare the rate of entropy increase for a family of coarse-grainings in the decoherent histories formalism. We find that the rate of entropy increase is dominated by the number of scales characterizing the coarse-graining.

Figure 1

A schematic illustration of the projectors $P_{\mathit{y}}^{(l,r)}$ and $P_{{\mathit{y}}^1,{\mathit{y}}^2}^{(l,m_l,m_r,r)}$. (a) Let $n-l=2$, $N-l-r=4$, and $\mathit{y}=y_1\dots y_4=1110$. The projector $P_{\mathit{y}=1110}^{(l,r)}$ is then in approximate correspondence with the phase-space region shaded in gray. (b) Let $n-l=4$, $m_l=2$, $m_r=2$, $\mathit{y}=101$, and ${\mathit{y}}^2=01$. The projector $P_{{\mathit{y}}^1=10,{\mathit{y}}^2=01}^{(l,2,2,r)}$ is then in approximate correspondence with the disconnected phase-space region given by the 16 black cells.