Quantum state tomography with noninstantaneous measurements, imperfections, and decoherence

Tomography of a quantum state is usually based on a positive-operator-valued measure (POVM) and on their experimental statistics. Among the available reconstructions, the maximum-likelihood (MaxLike) technique is an efficient one. We propose an extension of this technique when the measurement process cannot be simply described by an instantaneous POVM. Instead, the tomography relies on a set of quantum trajectories and their measurement records. This model includes the fact that, in practice, each measurement could be corrupted by imperfections and decoherence, and could also be associated with the record of continuous-time signals over a finite amount of time. The goal is then to retrieve the quantum state that was present at the start of this measurement process. The proposed extension relies on an explicit expression of the likelihood function via the effective matrices appearing in quantum smoothing and solutions of the adjoint quantum filter. It allows us to retrieve the initial quantum state as in standard MaxLike tomography, but where the traditional POVM operators are replaced by more general ones that depend on the measurement record of each trajectory. It also provides, aside from the MaxLike estimate of the quantum state, confidence intervals for any observable. Such confidence intervals are derived, as the MaxLike estimate, from an asymptotic expansion of multidimensional Laplace integrals appearing in Bayesian mean estimation. A validation is performed on two sets of experimental data: photon(s) trapped in a microwave cavity subject to quantum nondemolition measurements relying on Rydberg atoms, and heterodyne fluorescence measurements of a superconducting qubit.

Figure 1

MaxLike quantum state tomography based on the QND photon measurements reported in Fig. 4 of [15] for a photon creation quantum jump induced at $t=0$. The latter is not taken into account in the measurement model. Top: solid red line corresponds to the MaxLike mean photon number $\text{Tr}\left[{\rho }_{\text{ML}}\left(t\right){\mathit{a}}^{†}\mathit{a}\right]$ of the tomography target $\overline{\rho }\left(t\right)$, and based on measurements between $t$ and $+172$ ms. The two dashed blue lines correspond to $\text{Tr}\left({\rho }_{\text{ML}}{\mathit{a}}^{†}\mathit{a}\right)\pm 2{\sigma }_{\text{ML}}\left({\mathit{a}}^{†}\mathit{a}\right)$, a $95\%$ confidence interval. The black line corresponds to the mean photon number ${\langle {\mathit{a}}^{†}\mathit{a}\rangle }_t=\text{Tr}\left[\overline{\rho }\left(t\right){\mathit{a}}^{†}\mathit{a}\right]$ given by (12). Bottom: $2{\sigma }_{\text{ML}}\left({\mathit{a}}^{†}\mathit{a}\right)$ based on (10). See the text for more explanations.

Figure 2

Comparison of MaxLike reconstruction for $x$ (left plot: solid and dashed curves) and $y$ (right plot: solid and dashed curves) exploiting the measurement data set associated with $N=4\times {10}^4$ experimental trajectories, with an ensemble average of the normalized fluorescence signals (black circle) exploiting a much larger data set associated with $N^{*}=3\times {10}^6$ experimental trajectories. The red solid lines correspond to $x_{\text{ML}}$ and $y_{\text{ML}}$, while the blue dashed lines correspond to the confidence intervals $x_{\text{ML}}\pm 2{\sigma }_{\text{ML}}\left({\sigma }_x\right)$ and $y_{\text{ML}}\pm 2{\sigma }_{\text{ML}}\left({\sigma }_y\right)$ obtained from (10). The black circles are generally admitted to represent reliable estimates of $\text{Tr}\left({\overline{\rho }}_t{\sigma }_x\right)$ and $\text{Tr}\left({\overline{\rho }}_t{\sigma }_y\right)$; see the text for more explanations.

Figure 3

MaxLike reconstruction for $z$ exploiting a measurement data set with $N=4\times {10}^4$ experimental trajectories; the red solid line corresponds to $z_{\text{ML}}$ and the blue dashed line to the confidence interval $z_{\text{ML}}\pm 2{\sigma }_{\text{ML}}\left({\sigma }_z\right)$ obtained from (10); see the text for more explanations.

Figure 4

Similar to Fig. 2, but the ensemble average of the normalized fluorescence signals (black circles) uses the same dataset as the MaxLike reconstruction. The MaxLike reconstruction appears to be significantly less noisy, illustrating the efficiency of our method; see the text for more explanations.