Distillation by repeated measurements: Continuous spectrum case

Repeated measurements on one part of a bipartite system strongly affect the other part that is not measured, the dynamics of which is regulated by an effective contracted evolution operator. When the spectrum of this operator is discrete, the nonmeasured system is driven into a pure state, irrespective of the initial state, provided that the spectrum satisfies certain conditions. We show here that, even in the case of continuous spectrum, an effective distillation can occur under rather general conditions. We confirm it by applying our formalism to a simple model.

Figure 1

$P(N)$ (squares) vs $\Pi (N)$ (diamonds) as functions of the number of measurements $N$, when the coherent state $|\alpha \rangle$ of the cavity mode is repeatedly measured. The parameters are $\omega \tau =\pi /4$, $\sqrt{m/\hslash \omega }\hspace{0.5em}g=5$, $p_0/\sqrt{m\hslash \omega }=0.1$, $(\Delta p){}_0/\sqrt{m\hslash \omega }=0.2$, ${\Pi }_0=1/\sqrt{2}$, and $\left|\alpha \right|=1$, $\gamma =\pi$. This plot is independent of $x_0$ and $(\Delta x){}_0$. The final selected momentum is $p_{*}=2p_0$.

Figure 2

Exact (filled marks) and approximate (empty marks) behaviors of $P(N)$ (squares) vs $\Pi (N)$ (diamonds) as functions of the number of measurements $N$, when a number state $|1\rangle$ of the cavity mode is repeatedly measured. The parameters are the same as in Fig. 1, while $p_{*}=0$.