Feedback policies for measurement-based quantum state manipulation

In this paper, we propose feedback designs for manipulating a quantum state to a target state by performing sequential measurements. In light of Belavkin's quantum feedback control theory, for a given set of (projective or nonprojective) measurements and a given time horizon, we show that finding the measurement selection policy that maximizes the probability of successful state manipulation is an optimal control problem for a controlled Markovian process. The optimal policy is Markovian and can be solved by dynamical programming. Numerical examples indicate that making use of feedback information significantly improves the success probability compared to classical scheme without taking feedback. We also consider other objective functionals including maximizing the expected fidelity with the target state as well as minimizing the expected arrival time. The connections and differences among these objectives are also discussed.

Figure 1

The probabilities of successfully reaching $|1\rangle$ from the initial state $|0\rangle$ using optimal feedback-free policy ${\pi }^{\mathrm{n}}$ and optimal feedback policy ${\pi }^{☆}$, respectively.

Figure 2

The probabilities of successfully reaching $|1\rangle$ from the initial state $|0\rangle$ using different sizes of measurement set by feedback strategy.

Figure 3

The minimized average number of steps it takes to arrive at the target state $|1\rangle \langle 1|$ from the initial state $|0\rangle \langle 0|$ employing control set $\mathcal{E}$ of size $T$.