Identifying nonclassicality from experimental data using artificial neural networks

The fast and accessible verification of nonclassical resources is an indispensable step toward a broad utilization of continuous-variable quantum technologies. Here, we use machine learning methods for the identification of nonclassicality of quantum states of light by processing experimental data obtained via homodyne detection. For this purpose, we train an artificial neural network to classify classical and nonclassical states from their quadrature-measurement distributions. We demonstrate that the network is able to correctly identify classical and nonclassical features from real experimental quadrature data for different states of light. Furthermore, we show that nonclassicality of some states that were not used in the training phase is also recognized. Circumventing the requirement of the large sample sizes needed to perform homodyne tomography, our approach presents a promising alternative for the identification of nonclassicality for small sample sizes, indicating applicability for fast sorting or direct monitoring of experimental data.

Figure 1

Homodyne detection scheme. The signal field $\widehat{\rho }$ and the reference coherent beam (LO) are mixed using a 50:50 beam splitter (BS) before measuring light intensity.

Figure 2

Nonclassicality prediction of the neural network (NN) on the training states [coherent, thermal, and mixed coherent states as classical ones; Fock, squeezed-coherent, and single-photon-added coherent states (SPACS) as nonclassical ones], each in its corresponding state-parameter domain. $\alpha$ is the coherent amplitude, $n$ is the number of photons, and $\overline{n}$ is the mean number of photons. The gray horizontal line corresponds to the nonclassicality threshold $t=0.9$. Note that, for the squeezed-coherent states, the squeezing parameter $\xi$ is chosen randomly in $\xi \in [0.5,1]$ and is not shown in this plot. For each Fock state, the NN prediction is tested for four different simulations of the quadrature measurements. For details on the state parameters, see Appendix pp2.

Figure 3

Bottom: Nonclassicality prediction of the neural network (NN; teal) and quadrature variance of the corresponding distribution (yellow) for experimental data from squeezed states, dependent on the homodyne phase setting. Shaded regions are indicating nonclassicality for the corresponding quantity. Top: For $\varphi =0$ and $\pi /2$, the exact quadrature distribution $p\left(x\right)$ is shown (solid) in comparison with the quadrature distribution of the vacuum state (dashed).

Figure 4

(a) (bottom) Binary neural network (NN) prediction for experimentally generated single-photon-added coherent states (SPACS; ${\widehat{a}}^{†}|\alpha \rangle$ with 14 different values of $\alpha$ measured along 11 quadrature angles $\varphi$). Yellow (teal) points indicate that the prediction $r$ of the NN is above (below) the threshold $t=0.9$. (top) Additionally, we exemplarily display the quadrature distribution $p\left(x\right)$ for some representative $\alpha$ and $\varphi$. (b) Cut of the network prediction for SPACS along $sin\varphi =0$. (c) Prediction of the NN for experimental data from coherent states for the same parameters as in (b).

Figure 5

Nonclassicality prediction of a network trained without single-photon-added coherent states (SPACS; $r_{\mathrm{noSPACS}}$, teal) for simulated SPACS as a function of $\alpha$ ($\varphi =0$) together with the quadrature variance of the corresponding distributions (yellow). Shaded regions are indicating nonclassicality seen by the network (teal) and by the variance criterion (yellow).

Figure 6

Prediction $r$ of the neural network (NN) for simulated quadrature measurements of the cat state $|{\alpha }_{-}\rangle$ as a function of $\alpha$, for (a) $\varphi =\pi /2$ and (b) $\varphi =\pi /4$. Above the subplots, we show the quadrature distribution $p\left(x\right)$ for different $\alpha$ (solid) in comparison with the quadrature distribution using the same parameters but a quantum efficiency $\eta =1$ (dashed).

Figure 7

Prediction $r$ of the neural network (NN) for experimental data from single-photon-added coherent states (SPACS; yellow) and coherent states (teal) for $\alpha =0.32$ and $\varphi =0$ as a function of the measurement sample size. The dashed vertical line indicates the sample size $16000$ that was used in the training phase.