Effective method to estimate multidimensional Gaussian states

A simple and efficient method for characterization of multidimensional Gaussian states is suggested and experimentally demonstrated. Our scheme shows analogies with tomography of finite-dimensional quantum states, with the covariance matrix playing the role of the density matrix and homodyne detection providing Stern-Gerlach-like projections. The major difference stems from a different character of relevant noises: while the statistics of Stern-Gerlach-like measurements is governed by binomial statistics, the detection of quadrature variances corresponds to ${\chi }^2$ statistics. For Gaussian and near Gaussian states the suggested method provides, compared to standard tomography techniques, more stable and reliable reconstructions. In addition, by putting together reconstruction methods for Gaussian and arbitrary states, we obtain a tool to detect the non-Gaussian character of optical signals.

Figure 1

Experimental setup.

Figure 2

Raw homodyne data for the vacuum state (top) and for the OPO output close to threshold (bottom).

Figure 3

Hilbert-Schmidt distances of state reconstructions done in $N$-dimensional Fock spaces with respect to the reference state reconstructed in dimension $N=30$.

Figure 4

Wigner functions (upper row) and cuts along the $y$ axis (bottom row) of the OPO state reconstructed in Fock spaces of dimensions (a) $N=8$, (b) $N=15$, and (c) $N=25$, respectively.

Figure 5

(Color online) Variance (circles), $W_{\text{JB}}$ (triangles), and the $p$ value of $W_{\text{SW}}$ [$p\left(W_{\text{SW}}\right)$, squares] associated with the homodyne data of Figs. 2. Top: vacuum data. Bottom: OPO close to threshold data [here $p\left(W_{\text{SW}}\right)$ has been magnified by a factor $\times 1.5$]. The dashed line is the threshold value 0.05 for $p\left(W_{\text{SW}}\right)$ ($\times 1.5$ in the bottom plot); the dot-dashed line is the threshold value 5.99 for $W_{\text{JB}}$. If $W_{\text{JB}}\ge 5.99$ and/or $p\left(W_{\text{SW}}\right)\le 0.05$, then the null hypothesis (Gaussian states) is rejected. See the text for details. Results indicate that vacuum data (top) are consistent with the Gaussian hypothesis whereas for data coming from the OPO close to threshold the Gaussian hypothesis should be rejected.