Figure 1

Schematic diagram showing the transformation of the quantum state of a qubit as a result of a nonprojective measurement. The double arrow represents the measurement axis. The tail of the single-headed arrow represents the premeasurement state, and the head represents the postmeasurement state. The quantum state “moves” in the plane defined by the premeasurement state and the measurement axis. The distance that the quantum state moves depends on the different parameters in the problem, including in particular the measurement strength $\epsilon$.Reuse & Permissions

Figure 2

The probability of obtaining the target state $|{\psi }_T\rangle =\cos \frac{\pi }{8}{e}^{-i\pi /8}|\uparrow \rangle +\sin \frac{\pi }{8}{e}^{i\pi /8}|\downarrow \rangle$ in $N$ measurement steps starting from the initial state $|{\psi }_I\rangle =(e^{-i\pi /4}|\uparrow \rangle +e^{i\pi /4}|\downarrow \rangle )/\sqrt{2}$ and following the procedure explained in Sec. 3a. For this particular choice of states, the average number of measurement steps is 10.5, and the probability of successful state preparation in under 20 steps is 0.86.Reuse & Permissions

Figure 3

The distribution of states obtained from repeated measurement using a SIC-POVM. The different panels correspond to different measurement strengths: $\epsilon =0.1$ (a), 0.5 (b), 0.9 (c) and 0.99 (d). In each panel 400 points are plotted. Each point represents the values of the spherical coordinates $\theta$ and $\varphi$ of the quantum state $|\psi \rangle$, plotted using a maplike scheme designed not to distort the areas of different regions on the surface of the Bloch sphere. To obtain a distribution that is representative of the long-time steady state, each simulation involves a sequence of ${10}^5$ measurement steps. The 400 points in each panel represent the quantum state at equally spaced measurement steps; that is, we plot the coordinates of the state after measurement step number 250, 500, 750, etc. The results shown in this figure are independent of the initial state used in the calculations.Reuse & Permissions

Figure 4

Schematic diagram showing how a given initial state can be transformed into a given target state after two applications of the SIC-POVM with appropriately chosen measurement strengths. The first measurement brings the quantum state into the plane defined by the target state and one of the SIC-POVM states $|{\psi }_i\rangle$, and the second measurement results in the target state. It should be emphasized that the success of each one of these two steps is probabilistic.Reuse & Permissions