Benchmarking Maximum-Likelihood State Estimation with an Entangled Two-Cavity State

The efficient quantum state reconstruction algorithm described by Six et al. [Phys. Rev. A 93, 012109 (2016)] is experimentally implemented on the nonlocal state of two microwave cavities entangled by a circular Rydberg atom. We use information provided by long sequences of measurements performed by resonant and dispersive probe atoms over timescales involving the system decoherence. Moreover, we benefit from the consolidation, in the same reconstruction, of different measurement protocols providing complementary information. Finally, we obtain realistic error bars for the matrix elements of the reconstructed density operator. These results demonstrate the pertinence and precision of the method, directly applicable to any complex quantum system.

Figure 1

Preparation and detection of a two-cavity entangled state. (a) Scheme of the experimental setup with two microwave cavities (${\mathtt{C}}_1$, ${\mathtt{C}}_2$) and circular Rydberg atoms; see the main text for details. (b) Quantum beat note between the two components of the nonlocal state $|\mathrm{\Psi }(t)⟩$ directly measured as $P_g(t)$. Error bars are statistical, and the solid line is a fit with $f(x)=y_0-C/2\cos (\delta x+\phi )$ ($0\le C\le 1$), leading to a $C_0=0.4$ contrast. (c) Time decay of $C$ revealing the decoherence of the two-cavity state. The experimental points are obtained by fitting the $P_g(t)$ oscillation signals around different $t$’s for two cavity temperatures: 0.8 (blue line) and 1.5 K (red line). The solid lines result from numerical predictions with $C_0$ as the only adjustable parameter. The shaded areas are numerical uncertainties resulting from the finite accuracy ($\pm 1\mathrm{ms}$) on the measured cavity lifetimes. The residual variance between the curves and the ensemble of points is compatible with the error bars, except for the longest time, $t=20\mathrm{ms}$. The measured frequency drift of the two-cavity setup of several tens of hertz per hour (approximate acquisition time for the measurement with $t=20\mathrm{ms}$) results in a phase diffusion, significantly reducing the oscillation contrast at this timescale.

Figure 2

Absolute values of density matrix elements in a $3\times 3$ two-cavity Hilbert space. Red guiding lines enclose areas with the same photon number in ${\mathtt{C}}_1$; $n_1$ and $n_2$ are photon numbers in ${\mathtt{C}}_1$ and ${\mathtt{C}}_2$, respectively. (a) Reconstruction with single resonant atom measurements. (b) Reconstruction with QND parity measurement. (c) Reconstruction with both resonant and dispersive data of (a) and (b). (d) Reconstruction with sequences of several resonant measurements. Error bars $\sigma \mathbf{(}\mathrm{abs}({\rho }_{\mathrm{ML}})\mathbf{)}$ in (a) are of the order of 0.5 and are not shown. Error bars smaller than ${10}^{-3}$ are not shown in plots (b)–(d). (e) Numerical prediction $\mathrm{abs}({\rho }_{\mathrm{TH}})$ of the prepared state. The exact values of all reconstructed density matrices in the full $5\times 5$ Hilbert space, together with their real and complex error bars, are given in the Supplemental Material [26].

Figure 3

Statistical properties of the phase uncertainty. Blue diamonds, dispersion ${\tilde{\sigma }}_{\varphi ,R}$ of the reconstructed $\varphi$ values vs the number of realizations $R$; solid line, fit with a statistical dispersion function; red circles, mean value of computed $\sigma (\varphi )$. Note that, having the limited set of all experimental realizations, we have fewer independent statistical sets of large size $R$. Therefore, there are fewer independent reconstructions, and hence the larger statistical variance of reconstruction results for large $R$. (Inset) Distribution of $\varphi$ from 800 reconstructions on samples with $R=2000$ realizations (histogram). Blue [red] line, Gaussian with width ${\tilde{\sigma }}_{\varphi ,R}$ [$⟨\sigma (\varphi ){⟩}_R$]; green band, independently measured confidence interval for ${\varphi }_a$.