Quantum measurement of a mesoscopic spin ensemble

We describe a method for precise estimation of the polarization of a mesoscopic spin ensemble by using its coupling to a single two-level system. Our approach requires a minimal number of measurements on the two-level system for a given measurement precision. We consider the application of this method to the case of nuclear-spin ensemble defined by a single electron-charged quantum dot: we show that decreasing the electron spin dephasing due to nuclei and increasing the fidelity of nuclear-spin-based quantum memory could be within the reach of present day experiments.

Figure 1

(Color online) Illustration of the first two steps of the measurement procedure. The original distribution $P\left(A_z\right)$, with rms width ${\Delta }_0$ is shown at the top. After the first measurement, with result $r_M=0,1$, the conditional distribution (middle plots) reflects the knowledge of the least significant bit. The next measurement result $r_{M-1}$ further reduces the distribution (bottom plots).

Figure 2

(Color online) Estimation in the presence of preparation and measurement error $p$: the logarithm of the improvement $\Delta A_{z,\mathrm{est}}∕{\Delta }_0$ is plotted versus the number $M$ of binary digits measured. The solid lines represent different error rates ($p=0$: black circles; $p={10}^{-4}$ blue crosses; $p={10}^{-2}$: red triangles; and $p={10}^{-1}$: green diamonds). The broken red curves (triangles) demonstrate the benefit of simple error correction (for $p={10}^{-2}$): strategy I (3 repetitions per digit: use the majority result, either for all digits or only for leading $M∕2$ digits); (dash-dotted, dotted—almost undistinguishable); strategy II (increase number of repetitions for more significant digits to 7 repetitions for leading $M∕8$ digits, 5 for next $M∕8$, and 3 for next $M∕4$); dashed. We see that the latter provides a better accuracy than the uncorrected $p={10}^{-4}$ curve.