Single-shot parameter estimation via continuous quantum measurement

We present filtering equations for single-shot parameter estimation using continuous quantum measurement. By embedding parameter estimation in the standard quantum filtering formalism, we derive the optimal Bayesian filter for cases when the parameter takes on a finite range of values. Leveraging recent convergence results [R. van Handel, e-print arXiv:0709.2216], we give a condition which determines the asymptotic convergence of the estimator. For cases when the parameter is continuous valued, we develop quantum particle filters as a practical computational method for quantum parameter estimation.

Figure 1

(Color online) Continuous measurement of single qubit precessing in an external magnetic field.

Figure 2

(Color online) (Bottom) Simulated typical measurement trajectory for continuous $Z$ measurement, $\kappa =1$, $B=0$. (Top) Filtered values of ${\pi }_t\left[{\sigma }_x\right]$ and ${\pi }_t\left[{\sigma }_z\right]$ for simulated trajectory.

Figure 3

(Color online) (a) Filtered $p_t^{\left(i\right)}$ for $B∊\left\{2\kappa ,5\kappa ,8\kappa ,12\kappa \right\}$. The filter tracks to the true underlying value of $B=2\kappa$. (b) Filtered $p_t^{\left(i\right)}$ for $B∊\left\{-\kappa ,+\kappa \right\}$. The filter does not track to $B=+\kappa$ with probability one, though it is the most probable parameter value.

Figure 4

(Color online) Rate of convergence $\left(I_{0.95}\right)$, averaged over 1000 trajectories. The filters are for cases when possible $B$ values are either all larger or all smaller than the measurement strength $\kappa$.

Figure 5

(Color online) Kernel density reconstruction of $p_t\left(B\right)dB=P\left(B\mid M_{\left[0,t\right]}\right)dB$ for $N=1000$ particle filter set with $dB=10\kappa /150$, $a=0.98$, $h={10}^{-3}$, and resampling threshold of 2/3. The true magnetic field was $B=5\kappa$.