Experimental Quantum State Tomography of Multimode Gaussian States

Abstract

Multimode Gaussian states are a versatile resource for quantum information technologies and have been realized across a wide range of physical platforms. Recent progress in the large-scale generation of such states provides a key ingredient for scalable quantum technologies. Despite the importance of accurately characterizing these states, conventional tomography methods are often impractical because they require large sample sizes and can yield unphysical states. Here we present a reliable and efficient tomography method for multimode Gaussian states based on maximum-likelihood estimation. By directly operating on covariance matrices, the method avoids the exponential overhead associated with density-matrix reconstruction. We consider two commonly used detection schemes--single and joint homodyne detection--and systematically analyze the reconstruction performance. Our method outperforms conventional approaches by ensuring physical covariance matrices and achieving better agreement with the true states. To demonstrate the experimental applicability of the method, we experimentally generate various multipartite entangled states--six-mode graph states with different connectivity, a six-mode GHZ state, and a fully connected ten-mode graph state--and reconstruct their covariance matrices. Using the reconstructed covariance matrices, we quantify fidelities, detect entanglement, and reveal the multimode structure of squeezing and noise. Our technique offers a practical diagnostic tool for developing scalable quantum technologies.

Figures (8)

• Figure 1: Measurement schemes for characterizing multimode Gaussian states. (a) Single homodyne detection. The quadrature of a selected mode (either an individual mode or a superposition of modes) is measured using a single homodyne detector after a mode mixer. (b) Joint homodyne detection. The quadratures of $M$ modes are measured simultaneously using multiple homodyne detectors, each with an adjustable phase $\theta_m$.
• Figure 2: Illustration of the maximum likelihood estimation method for reconstructing a covariance matrix. A set of quadrature outcomes $\boldsymbol{\chi}$ is obtained using either single homodyne detection or joint homodyne detection (see Fig. \ref{['fig:homodyne_detection']}). The covariance matrix is parametrized as $\boldsymbol{V}(\boldsymbol{\kappa}^{(t)}, \boldsymbol{T}^{(t)})$ to express a general multimode Gaussian state while satisfying the physical condition. The likelihood function is computed in each step $t$ based on the parameters $\boldsymbol{\kappa}^{(t)},\boldsymbol{T}^{(t)}$ and the quadrature outcomes $\boldsymbol{\chi}$. The parameters are updated to maximize the likelihood function. After all iterations, the final parameters $\boldsymbol{\kappa}^{(t_\mathrm{max})}$ and $\boldsymbol{T}^{(t_\mathrm{max})}$ are used to reconstruct the covariance matrix $\boldsymbol{V}(\boldsymbol{\kappa}^{(t_\mathrm{max})}, \boldsymbol{T}^{(t_\mathrm{max})})$.
• Figure 3: Minimum symplectic eigenvalues ($\lambda_{\min}$) of reconstructed covariance matrices obtained with (a) single homodyne detection and (b) joint homodyne detection. A two-mode cluster state with 6-dB pure squeezing and 30% optical loss is considered. Histograms of $\lambda_{\min}$ are obtained from 100 Monte Carlo simulations, each generating a total of $N_t=10,000$ measurement outcomes. The same simulation data are used for the direct method (blue bars) and the MLE method (purple bars). Gray regions indicate unphysical reconstructions ($\lambda_{\min}<1$), and red lines indicate the true value of $\lambda_{\min}$.
• Figure 4: Performance of covariance-matrix reconstruction methods. (a-c) single and (d-f) joint homodyne detection. Open and filled markers denote the direct and MLE reconstruction methods, respectively, and blue circles and green squares represent the fidelity (with respect to the true state) and the fraction of failures ($\lambda_{\min}<1$), respectively. The MLE method does not produce any failure cases. (a,d) Reconstruction of a two-mode cluster state as a function of the total number of outcomes $N_t$. (b,e) Reconstruction of linear cluster states as the mode number $M$ increases. The number of repeated measurements $N_r$ is fixed at 5,000, and the total number of outcomes is $N_t = N_r M(2M+1)$ for single homodyne detection and $N_t = N_r M (\lfloor\log_2 M \rfloor +3)$ for joint homodyne detection. (c,f) Reconstruction of six-mode entangled states with various connectivities. The x-axis labels indicate different correlation types: red graphs for graph states and the blue graph for a GHZ state. The number of repeated measurements is $N_r=$5,000. In all simulations, the quantum states are constructed from 6-dB pure squeezing and 30% optical loss (Appendix A). Error bars and failure fractions are obtained from repeated simulations: 50 runs for (a,c,d,f) and 25 runs for (b,e).
• Figure 5: Entanglement detection based on covariance-matrix reconstruction. The covariance matrices of the GHZ state from Fig. \ref{['fig:performance']}(c,f) are used for (a) single and (b) joint homodyne detection, where the unphysical covariance matrices are excluded in the calculation. The minimum eigenvalue under partial transposition ($\lambda_{\textrm{PT}}$) is plotted for the bipartition represented by the pink and green shaded areas in each graph diagram. Open and filled circles correspond to the results obtained with the direct and MLE methods, respectively. Black dashed lines indicate the true values.