A machine learning approach to tomographic pattern generation and classification of quantum states of light

Abstract

Optical tomograms can be envisaged as patterns. The Wasserstein generative adversarial network (WGAN) algorithm provides a platform to train the machine to compare patterns corresponding to input and generated tomograms. Using a deep-learning framework with two convolutional neural networks and WGAN, we have trained the machine to generate tomograms of Fock states, coherent states (CS) and the single photon added CS ($1$-PACS). The training process was continued until the Wasserstein distance between the input and output tomographic patterns levelled off at a low value. The mean photon number, variances and higher moments were extracted directly from the generated tomograms, to distinguish between different Fock states and also between the CS and the $1$-PACS, without using an additional classifier neural network. The robustness of our results has been verified using two error models and also with different colormaps that define the tomographic patterns. We have examined if the training program successfully reflected some of the findings in a recent experiment in which state reconstruction was carried out to establish that the fidelities between an amplified CS, an optimal CS and a $1$-PACS were close to unity, over a range of parameter values. By training the machine to reproduce tomograms corresponding to these specific states, and comparing the mean photon numbers of these states obtained directly from the tomograms, we have established that the variations in these observables reflect the experimental trends. State reconstruction from tomograms could be challenging, in general, since the Hilbert space associated with quantized light is large. The tomographic approach provides a viable alternative to detailed state reconstruction. Our work demonstrates the use of machine learning to generate optical tomograms from which the states can be directly characterized.

Figures (32)

• Figure 1: Schematic of the procedure used for machine learning. In the flowchart above, (a) the tomogram corresponding to a quantum state of light can be represented as a pattern. Here a collection of optical tomograms of each of three states, namely, the photon number states, coherent states, and $1$-photon added coherent states are separately used for pattern training. (b) The generator and the discriminator (also called a critic) are trained simultaneously with one of these three collections of tomograms. Training is through several convolutional layers. Wasserstein distance quantifies the extent of divergence between input and generated patterns. Feedback loop between discriminator and generator till discriminator acknowledges that generator has been reasonably trained. (c) After reasonably successful training, output tomographic patterns are available. Random samples from a large number of output tomograms used for step (d). (d) Directly compute with ease the mean photon number $\langle \hat{n} \rangle$ and quadrature variance $\Delta {\hat{X}}_{\theta}^{2}$, corresponding to quadrature angle $\theta$, from each output tomogram. Compare with corresponding quantities computed from input patterns to estimate extent of errors. (e) Quantum state tomography (QST) to obtain the state $\hat{\rho}$ and the Wigner function $W(x, p)$ from output tomograms, (for characterizing the state) is circumvented here. QST could pose challenges for continuous variable systems. Further, from the quantities $\langle \hat{n} \rangle$, $\Delta {\hat{X}}_{\theta}^{2}$ and higher-order moments computed readily from output tomographic patterns, classification is straightforward without an additional neural network. Our approach is contained in the blue dashed box, and provides a viable alternative to the QST procedure.
• Figure 2: Optical tomograms $w(X_{\theta}, \theta)$ corresponding to six single-mode states of the radiation field. Here, (a), (b) and (c) are the tomograms for the $0$, $1$ and $2$-photon states respectively, and (d), (e) and (f) correspond to a CS $|\alpha\rangle$, the $1$-photon added CS $|\alpha,1\rangle$ and the $2$-photon added CS $|\alpha,2\rangle$, with $\alpha=0.5$. For this value of $\alpha$ and the range of values selected for the machine learning procedure, the probability distribution functions and hence the tomograms, are predominantly in the range $X_{\theta} \in [-5, 5]$. We note that the single vertical cut in (b) and (e) corresponds to one photon, and two vertical cuts in (c) and (f) to two photons. The number of zeroes of the PDFs in the different quadratures and their location determine the number and location of the cuts.
• Figure 3: Training with the Wasserstein generative adversarial network (WGAN) algorithm: Schematic of the WGAN framework. The WGAN algorithm is composed of two deep neural networks, a generator $G$ and a discriminator $D$. ($G$ and $D$ are two separate convolutional neural networks (CNNs) in this work.) $G$ is initialized with a random input $z$ which is sampled from a uniform/ Gaussian distribution. $G$ learns the 'real' (input training) tomograms corresponding to (a) Fock states, (b) CS and (c) $1$-PACS, through feedback from $D$. $D$ on the other hand, compares the output from $G$ (which are the 'fake' tomograms) and the real tomograms, and gives feedback to $G$. (The feedbacks are backpropagations for adjusting weights.) The discriminator's (generator's) objective is to maximize (minimize) the Wasserstein distance (Eq. (\ref{['eq:Kantorovich_Rubinstein_duality']})). The learning process is complete when Eq. (\ref{['eq:Kantorovich_Rubinstein_duality']}) has saturated (equivalently when $G$ generates samples mimicking the real input tomograms). Other architectural details of the CNNs and the values of the hyperparameters used are given in Tables \ref{['tab:generator_params']} and \ref{['tab:discriminator_params']}, and Sec. \ref{['sec:wgan_network_arch_dataset_prep']}, respectively.
• Figure 4: Representative generated samples for the Fock states $|n\rangle$ obtained after training with the WGAN algorithm for $25000$ epochs. Here $n=0,1,2,3,4$ and $5$ in (a), (b), (c), (d), (e) and (f), respectively. It is evident that the intensity distribution does not change with $\theta$ in these tomograms. This feature is discussed further in Sec. \ref{['sec:Fock']}.
• Figure 5: The mean photon number $\langle \hat{n} \rangle$ (black circles) for the Fock states $|n\rangle~(n = 0, 1, \dots, 5)$ computed from $500$ randomly chosen generated tomograms. $\langle \hat{n} \rangle$ has been calculated from the reconstructed PDFs along $\theta=0, \pi/3$ and $2\pi/3$, by setting $m=n=1$ in Eq. (\ref{['eq:mean_ph_num_Wunsche']}). The red dashed lines are the theoretical values of $\langle \hat{n} \rangle$ corresponding to the Fock states considered. The computed value of $\langle \hat{n} \rangle$ is within $4\%$ error tolerance of the theoretical value for each generated state whose tomogram can be identified to clearly correspond to a particular photon number. Spurious samples (typically black circles midway between two consecutive horizontal red lines) do not fall under this category. These arise due to inherent issues in the generation process.