Figure 1

(a) Location of the Zeno-like fixed points in the three-dimensional parameter space. Plotted are the neighborhoods in which the eigenvalue ${\lambda }_2^{\prime }$ differs from 1 by at most ${10}^{-2}$. The thickness of these neighborhoods illustrates the level of fine-tuning required to realize Zeno-like fixed points with this accuracy. (b) Top view of the Bloch sphere. Locations of Zeno-like fixed-point states (where ${\lambda }_1^{\prime }=1$ or ${\lambda }_2^{\prime }=1)$ are in blue. They are highly concentrated on the $xy$ plane. States preserved by the conventional Zeno effect are shown as a horizontal bold red line. (c) Bloch sphere locations of target states preserved by Brouwer fixed points $({\lambda }_0=1)$ when the initial state of the detectors is arbitrarily varied, while being held fixed for each set of repeated measurements. We see that by varying the choice for the initial state of the detectors, the fixed points cover the full sphere isotropically about the $z$ axis. For comparison, the bold red line on the $x$ axis shows the states that can be preserved by the conventional QZE.