Phase-space contraction and quantum operations

We give a criterion to differentiate between dissipative and diffusive quantum operations. It is based on the classical idea that dissipative processes contract volumes in phase space. We define a quantity that can be regarded as “quantum phase space contraction rate” and which is related to a fundamental property of quantum channels: nonunitality. We relate it to other properties of the channel and also show a simple example of dissipative noise composed with a chaotic map. The emergence of attractor-like structures is displayed.

Figure 1

Sechematic representation of the Husimi function $\Gamma (z,z^{*})$ for the noise used in Ref. 16. Dissipation takes place where $\Gamma (z,z^{*})\ne 0$.

Figure 2

Graphical representation of the structure of the matrix $\left[p_{ij}\right]$ with $N=32$ and $\alpha =0.5$. The color scheme is as follows: white means null value and a black square equals one element of value equal to 1.

Figure 3

(Color online) The contracting parameter $\eta$ as a function of $\alpha$ for the dissipation model of Eq. (34) for two different values of $ϵ$: (∗) $ϵ=0.5$, (▴) $ϵ=0.75$.

Figure 4

Plot of the Husimi function $\Gamma (z,z^{*})$ for $N=64$ (in gray scale, black is maximum and white minimum) and different values of $\alpha$. Lighter regions indicate local contractive regions.

Figure 5

(Color online) Left: histogram representation of the attractor for the classical standard map [Eq. (42) with $k=0.065$, $\delta =0.6$] with dissipation. Right: Husimi representation for invariant state for the quantum standard map with the dissipative noise described in Sec. 5 ($\alpha \sim 0$, $ϵ=0.4$).

Figure 6

(Color online) Husimi representation (same color scheme as Fig. 4) of the invariant state for the standard map with dissipation for different values of $\alpha$ and $ϵ$ and $N=64$. The difference between $\alpha \approx 0$ (dissipation) and $\alpha \approx 1$ (diffusion) can be observed. (Axes and units are the same as in Figs. 4, 5)