Quantum reservoir computing for photonic entanglement witnessing

TL;DR

This work introduces quantum reservoir computing via quantum extreme learning machines (QELMs) to witness entanglement from experimental data without detailed device modeling. By embedding polarization information into a large orbital angular momentum space through double quantum walks, the approach yields informationally complete single-setting measurements and a linear readout that learns to extract entanglement witnesses. The method demonstrates robust entanglement estimation, generalizes from separable training to entangled test states, and compares favorably with shadow tomography while remaining model-agnostic. Its platform-agnostic, self-calibrating nature and straightforward training make it a promising tool for scalable quantum-feature estimation across photonic and other quantum platforms.

Abstract

Accurately estimating properties of quantum states, such as entanglement, while essential for the development of quantum technologies, remains a challenging task. Standard approaches to property estimation rely on detailed modeling of the measurement apparatus and a priori assumptions on their working principles. Even small deviations can greatly affect reconstruction accuracy and prediction reliability. Here, we demonstrate that quantum reservoir computing embodies a powerful alternative for witnessing quantum entanglement and, more generally, estimating quantum features from experimental data. We leverage the orbital angular momentum of photon pairs as an ancillary degree of freedom to enable informationally complete single-setting measurements of their polarization. Our approach does not require fine-tuning or refined knowledge of the setup, at the same time outperforming conventional approaches. It automatically adapts to noise and imperfections while avoiding overfitting, ensuring robust reconstruction of entanglement witnesses and paving the way to the assessment of quantum features of experimental multiparty states.

Figures (12)

• Figure 1: Schematic overview of the QELM experiment.(a) The protocol describes the evolution of an input state through a fixed and unknown quantum channel called reservoir, which maps the state to a higher dimensional Hilbert space. Here, through projective measurements in a single set configuration, the output probability distribution $\{ p \}$ of the state over the basis $\boldsymbol{\mu} = \{\mu_b\}_{b=1}^{N_{\rm out}}$ is retrieved. These probabilities are used in the final training stage to reconstruct the expectation value of an observable on the state. (b) In the experimental realization, entangled and separable quantum states encoded in the polarization degree of freedom of photon pairs evolve through the reservoir dynamics implemented by a double quantum walk configuration. Through the evolution, the 4-dimensional input space is enlarged into the 25-dimensional space of the orbital angular momentum. By performing projective measurements on the computational basis of the latter, we obtain the output outcome probabilities on which the QELM model is trained to reconstruct a target entanglement witness.
• Figure 2: Experimental Setup.(a)Input state preparation --- Photon pairs are generated by spontaneous parametric down-conversion in a Type-II periodically poled potassium titanyl phosphate (PPKTP) crystal enclosed in a Sagnac interferometer. In the Sagnac interferometer, a dual-wavelength half-wave plate (DHWP) allows the nonlinear process to happen in each arm while compensating for the propagation delay acquired inside the crystal by the orthogonally polarized photons. The generated photons are separated by a dual-wavelength polarizing beam splitter (DPBS) while the pump laser is separated by a dichroic mirror (DM). After generation, each photon enters a layer consisting of a half-wave plate (HWP) and a quarter-wave plate (QWP) which encodes the input polarization state. (b)Reservoir evolution --- After the state preparation, each photon enters an independent discrete-time quantum walk (QW) consisting of a series of HWPs, QWPs, and q-plates (QPs), which transfer the polarization information into the OAM degree of freedom. This QW implements the reservoir dynamics needed by the QELM. The polarization of the photons exiting the QW is then projected with a HWP, a QWP, and a polarizing beam splitter (PBS). (c)Detection --- The final detection stage consists of a projective measurement in the OAM space, realized by a spatial light modulator (SLM) followed by coupling into single-mode fibers. This implements a projective measurement in the basis $\{\ket n:\, n=-2,\dots,2\}$. Finally, avalanche photodiode detectors (APDs) are used to collect the photons and detect the coincidence counts.
• Figure 3: Performance of entanglement witness estimation. Predicted versus true values of the witness $\mathcal{W}$ for the $\mathsf{E1}$ scenario under different training configurations. In panel (a), both training and test datasets contain separable and entangled states, while in panel (b), the training set includes only separable states and the test set only entangled ones. These results showcase how the model's linearity allows it to train on only states with $\expval{\mathcal{W}}>0$ and still accurately predict previously unseen states with $\expval{\mathcal{W}}<0$ (yellow area in the plot). The shaded grey region in each plot represents the estimation error, defined as the square root of the training MSE. Finally, panels (c) and (d) report the confusion matrices showing the accuracy in correctly identifying positive and negative values of $\expval\mathcal{W}$, corresponding to the data in (a) and (b), respectively.
• Figure 4: Dependence of estimation performances on noise and choice of reservoir. Panels (a) and (b) report the robustness of the witness estimate $\langle\mathcal{W}\rangle$ under noise, where the entangled reference state $\rho_{\rm ent}$ is replaced by statistical mixtures of the form $\rho_{noise} = (1 - p) \rho_{\mathrm{ent}} + p \rho_{\mathrm{sep}}$, where $p\in[0,1]$ is the noise parameter. Here, the studies are performed in the experimental scenario $\mathsf{E2}$. This scenario is not the optimal one but is obtained for a perturbed reservoir dynamic, swapping the QWP angles. In panel (a), the solid curve shows the average MSE on the test set as a function of $p$, and the shaded regions represent the standard deviations obtained by repeating the procedure with different random training instances. Each instance contains 150 training states and 150 test states. Panel (b) reports the corresponding accuracy in identifying negative values of the $\langle\mathcal{W}\rangle$, the dotted line indicates the $50\%$ accuracy value. Panel (c) compares experimental performances ($\mathsf{E1}$, $\mathsf{E2}$, $\mathsf{E3}$) with numerical simulations ($\mathsf S1$, $\mathsf S2$, $\mathsf S3$). Stars indicate the experimental MSEs, while solid lines represent the corresponding numerical results under the same conditions and color coding. In all cases, training is performed on separable states and testing on entangled states; the MSE is averaged over the possible target observables and plotted against the collected statistics (fixed as the same in both training and test). The shaded regions around each simulation curve indicate one standard deviation across sampling instances, while the boundaries of the gray shaded region mark the minimum and maximum averaged MSE obtained when training is performed with random reservoir configurations ($\mathsf S$-random). The inset details the behavior of the experimental results for $\mathsf{E2}$ and $\mathsf{E3}$, showing the consistency between numerical simulations and experimental data across varying statistics.
• Figure S1: Input states density matrix. Experimentally reconstructed real (left) and imaginary (right) parts of the density matrix associated with the states generated by the SPDC source: (a) entangled $\ket{\Psi_R^+}$, (b) partially entangled $\ket{\Psi_R^+}_{p_1}$ and (c) partially entangled $\ket{\Psi_R^+}_{p_2}$.