Simultaneous continuous measurement of noncommuting observables: Quantum state correlations

We consider the temporal correlations of the quantum state of a qubit subject to simultaneous continuous measurement of two noncommuting qubit observables. Such qubit state correlators are defined for an ensemble of qubit trajectories, which has the same fixed initial state and can also be optionally constrained by a fixed final state. We develop a stochastic path integral description for the continuous quantum measurement and use it to calculate the considered correlators. Exact analytic results are possible in the case of ideal measurements of equal strength and are also shown to agree with solutions obtained using the Fokker-Planck equation. For a more general case with decoherence effects and inefficiency, we use a diagrammatic approach to find the correlators perturbatively in the quantum efficiency. We also calculate the state correlators for the quantum trajectories which are extracted from readout signals measured in a transmon qubit experiment, by means of the quantum Bayesian state update. We find an excellent agreement between the correlators based on the experimental data and those obtained from our analytical and numerical results.

Figure 1

Conditional averages for ideal $XZ$ measurement. Panel (a) shows the qubit state averaged over a subensemble of quantum trajectories, which begin at the initial pure state, ${\mathit{q}}_{\mathrm{in}}=\{cos(\pi /4),0,sin(\pi /4)\}$ at time $t=0$ and end at the final pure state ${\mathit{q}}_f=\{sin(7\pi /8),0,cos(7\pi /8)\}$ at time $t=T$. We show the subensemble average trajectories for three different periods $T={\tau }_m,3.5{\tau }_m$, and $10{\tau }_m$. Solid lines depict our analytical results, Eq. (33), and the crosses depict Monte Carlo simulation results. The dashed line, connecting the initial state and the fully mixed state, shows the conventional exponential decay of the (total) ensemble average state due to $XZ$ measurement. In panel (b), solid lines show our analytical results for the quantum state correlators, Eqs. (22, 23) and (29). The boundary conditions on the quantum trajectories are the same as those in panel (a). The crosses represent Monte Carlo simulation results.

Figure 2

Covariance and variance functions of the Bloch coordinates for nonideal $XZ$ measurements. The analytical results obtained from the diagrammatic perturbative expansion are presented with solid curves and the Monte Carlo simulation results are presented with markers. The results from the diagrammatic expansion agree with the numerical data much better for the case of small measurement quantum efficiency $\eta =0.05$, as shown in panel (b), than for the case with $\eta =0.5$, shown in panel (a). The insets show the results for the variance functions. We assume both measurement channels have the same quantum efficiency $\left({\eta }_x={\eta }_z=\eta \right)$.

Figure 3

Two-sided probability distribution of the polar coordinate $\theta$ at time $t$. Such distribution takes into account only quantum trajectories with fixed initial and final states, parametrized by the angles ${\theta }_{\mathrm{in}}=\pi /4$ and ${\theta }_f=7\pi /8$ at times $t=0$ and $T=10{\tau }_m$, respectively. The distribution is shown for various times: $t=0.5{\tau }_m,2.5{\tau }_m,5{\tau }_m,7.5{\tau }_m$, and $9.5{\tau }_m$. The circles represent Monte Carlo simulation results.

Figure 4

Comparison of theory and experiment for quantum state correlators: (a) covariances and (b) variances, for the qubit under the simultaneous measurement of ${\sigma }_x$ and ${\sigma }_z$. The correlators derived from experimental qubit trajectories are shown by solid curves, whereas the Monte Carlo simulation results are shown by markers, showing excellent agreement with the experimental data. As expected from Fig. 2, the first-order perturbative solutions, shown by dashed gray lines, while giving the correct qualitative behavior, deviate from the experimental results in the moderately high efficiency limit $({\eta }_z=0.54$ and ${\eta }_x=0.41)$.