Learning to detect optical nonclassicality

Abstract

Nonclassicality, defined in the quantum optical sense, serves as a resource for photon-based quantum technologies. Therefore, certifying the nonclassicality of a quantum state is crucial for gauging its potential for quantum advantage. However, traditional nonclassicality witnesses that assume perfect knowledge of the witness observables often fail in realistic scenarios with limited statistics and finite-resolution photon detectors. Furthermore, these witnesses do not exploit the fact that certain states are unlikely to be observed in a given experiment. Here, we train a variational model to distinguish classical from nonclassical states using finitely many measurement samples of multimode quantum states that are probed with different photon-number-resolving detection schemes. The learned decision rule is then an indicator of nonclassicality, tailored to a given set of physically relevant states. Our approach is both data-driven and interpretable in the sense that the learned analytical decision rule can be extracted. Training the model on experimental data measured with (i) a superconducting nanowire single-photon detector and (ii) a time-bin multiplexing detection scheme demonstrates the versatility of the approach, paving the way for efficient nonclassicality detection.

Figures (14)

• Figure 1: Conceptualization of the classification task. A multimode quantum state is measured with a specific detection scheme (here, the inset depicts a finite PNR detector and a time-bin multiplexing scheme) and stored with the correct binary label: classical or nonclassical. The set of $M$ measurements is input to the algebraic classifier that learns to assign the correct label to the data. First, the encoder (violet boxes at the top) learns to extract relevant moments from the data, then the decoder constructs a polynomial decision rule from these moments. After training, the model is capable of predicting whether a state is nonclassical. Additionally, extracting the learned decision rule allows one to judge and interpret its reliability.
• Figure 2: Structure of the algebraic classifier. The set of sampled photon numbers/clicks $\left \lbrace x_{\nu}^{\alpha}\right\rbrace$ is input to the encoder that learns to extract relevant moments up to order $L$. These weighted sums of moments are stored in $\lbrace \boldsymbol{x}^{(i)}\rbrace_{i=1,...,L}$ and input to the decoder. The decoder constructs a polynomial of order $L$ and optimizes its coefficients.
• Figure 3: Visualization of the dataset simulated with a perfect PNR detector and $10^3$ samples per state: Orange circles (purple stars) represent classical (nonclassical) states. The shaded blue area visualizes the set of states that exhibit sub-Poissonian statistics. (a) To account for statistical errors of the moments, a bias can be added to the witness. Geometrically, this corresponds to a vertical shift of the decision boundary (blue arrow). The magenta line represents the learned decision rule of the model that is trained with a single encoding layer, a constant learning rate and no regularization $\lambda=0$. The hatched area visualizes the set of states that is identified as nonclassical by the model. (b) Zoom of the regime with small $\langle \hat{n}\rangle$ and $\langle\hat{n}^2\rangle$ marked as gray box in (a). Errorbars represent the standard error.
• Figure 4: Performance of model on the dataset simulated with a perfect PNR detector and $10^3$ measurements per state. The dashed-dotted lines represent traditional witnesses: The (moment-based) Mandel $Q$ parameter, its third-order generalization and Klyshko's witness (probability-based). The solid lines show the performance of the model trained with one and two encoding layers, corresponding to second- and third-order witnesses: Pluses represent the model being trained with a constant learning rate of $10^{-2}$; the best epoch was chosen according to the best accuracy during training. Crosses represent the model trained with a scheduler that halved the learning rate once a plateau was reached. To visualize the increase of the regularization strength $\lambda$, that implicitly parametrizes the curve, the markers are connected with solid lines. The red circle show the point of $\lambda=0.8$ where the decision rule \ref{['eq: learned decision rule']} is extracted.
• Figure 5: Performance of model on the dataset simulated with a finite resolution photon-number detector and $10^3$ samples per state. The dashed-dotted lines represent traditional nonclassicality witnesses: the Mandel $Q$ parameter, its third-order generalization (both moment-based), Klyshko's witness and the generalized Klyskho witness (both probability-based). The solid red lines correspond to the model being trained with one and two encoding layers, corresponding to a second- and third-order criterion respectively.