Past observable dynamics of a continuously monitored qubit

Monitoring a quantum observable continuously in time produces a stochastic measurement record that noisily tracks the observable. For a classical process, such noise may be reduced to recover an average signal by minimizing the mean squared error between the noisy record and a smooth dynamical estimate. We show that for a monitored qubit, this usual procedure returns unusual results. While the record seems centered on the expectation value of the observable during causal generation, examining the collected past record reveals that it better approximates a moving-mean Gaussian stochastic process centered at a distinct (smoothed) observable estimate. We show that this shifted mean converges to the real part of a generalized weak value in the time-continuous limit without additional postselection. We verify that this smoothed estimate minimizes the mean squared error even for individual measurement realizations. We go on to show that if a second observable is weakly monitored concurrently, then that second record is consistent with the smoothed estimate of the second observable based solely on the information contained in the first observable record. Moreover, we show that such a smoothed estimate made from incomplete information can still outperform estimates made using full knowledge of the causal quantum state.

Figure 1

Partitioning of past collected readout. Given a temporal sequence of $N$ measurement results $\vec{r}=(r_1,...,r_N)$ collected at times $(t_1,...,t_N)$ separated by time steps $dt$, the best estimate of the monitored observable at an intermediate time $t_j$ depends on the distribution of the readout $r_j$ at that time, which depends on the future results ${\vec{r}}_f$ as well as the past history ${\vec{r}}_p$ because of the backaction of the measurement.

Figure 2

Evolution of observable estimates. Shown are single realizations of qubit Rabi oscillations in the $x\text{−}z$ plane with period $T_R$ while monitoring ${\sigma }_z$ with collapse time scale $\tau$ at periodic time steps $dt$. Compared are the expectation value $z$ (blue solid line), smoothed value $z_S$ (black dotted line), and weak value $z_w$ (red hashed line). The upper plot shows a weaker measurement regime with $\tau /T_R=2$, and with $dt/\tau =1/40$, so $z_S\approx z_w$. The lower plot shows a stronger measurement regime with $\tau /T_R=1/10$, and with $dt/\tau =1/2$, so $z_S$ deviates from sufficiently large values of $z_w$.

Figure 3

Estimate comparisons. Normalized histograms comparing the expectation value $z$ to the smoothed value $z_S$, for ${10}^5$ realizations monitoring ${\sigma }_z$ of a qubit oscillating with Rabi period $T_R$ for a duration $T$, with time steps $dt=T_R/100$. Short duration $T/T_R=5$ (red hashed line); long duration $T/T_R=50$ (blue circles). Top row: Relative mean squared error $Q(z_S,z)$; see Eq. (21). Bottom row: Hypothesis test logarithmic ratio $ln\left[R(z_S,z)\right]$ [see Eq. (22)] scaled by $2dt/T$ to show correspondence to $Q(z_S,z)$. Left column: Weaker measurement regime with $\tau /T_R=2$. Right column: Stronger measurement regime with $\tau /T_R=1/10$. For both figures of merit, positive values indicate that $z_S$ is a better estimate than $z$. The fraction of realizations for which $Q(z_S,z)>0$ (or $ln\left[R(z_S,z)\right]>0$) for $T/T_R=\{5,50\}$ is higher than $\{0.735,0.968\}$ for weaker measurements and higher than $\{0.976,0.999\}$ for stronger measurements, respectively.

Figure 4

Informationally incomplete estimate comparisons. Agents $\mathcal{Z}$ and $\mathcal{X}$ monitor ${\sigma }_z$ and ${\sigma }_x$, respectively, of a qubit oscillating with Rabi period $T_R$ for a duration $T/T_R=100$, with fixed time steps $dt/T_R=1/100$ and collapse time scale ${\tau }_z/T_R=1/10$. (a),(b) Normalized histograms with ${10}^4$ realizations, comparing the mean squared errors $Q$ relative to the reference expectation value $x$ that uses all information contained in the causal quantum state evolving from both measurement records $r_z$ and $r_x$. The ignorant expectation value $x^{\mathcal{Z}}$ (red solid line) and ignorant smoothed estimate $x_S^{\mathcal{Z}}$ (black circles) use incomplete information contained only in the record $r_z$, while the optimal smoothed estimate $x_S$ (blue starred line) also uses the information in $r_x$. (a) For weaker $\mathcal{X}$ monitoring, with ${\tau }_x/{\tau }_z=25$, the ignorant smoothed estimate $x_S^{\mathcal{Z}}$ still beats $x$ in more than $72\%$ of the realizations, while the best smoothed estimate $x_S$ beats $x$ in more than $98\%$ of the realizations and the ignorant expectation value $x^{\mathcal{Z}}$ beats $x$ in only $32\%$ of the realizations. (b) For stronger $\mathcal{X}$ monitoring, with ${\tau }_x/{\tau }_z=15$, the advantage of the ignorant smoothed estimate $x_S^{\mathcal{Z}}$ is reduced, with only $54\%$ of the realizations beating $x$. (c) Relative mean square error $Q$ as a function of the relative monitoring strength ${\tau }_x/{\tau }_z$, showing that for ${\tau }_x/{\tau }_z\gg 1$ the ignorant smoothed estimate $x_S^{\mathcal{Z}}$ reliably outperforms the expectation value $x$, despite having restricted information.