Can Randomly Structured Metasurfaces Be Used for Quantum Tomography of High-Dimensional Spatial Qudits?

TL;DR

This work investigates whether randomly structured metasurfaces can support high-dimensional quantum state tomography of spatial qudits. By framing tomography as an inverse problem with an instrument matrix and analyzing the conditioning via the matrix condition number, the authors show that large-scale, random freeform metasurfaces can yield well-conditioned measurements when detector redundancy is sufficient and the metasurface parameters are carefully chosen. Demonstrations include high-dimensional single-photon (10D) and two-photon entangled (49D) reconstructions with realistic SNRs, using Hermite-Gaussian basis and MLE reconstruction to ensure physical density matrices. The study highlights practical pathways for miniaturized, robust quantum-measurement platforms and provides design guidelines for metasurface thickness, average meta-atom size, and detector counts to enable scalable quantum tomography.

Abstract

Reconstructing the density matrix of the quantum state of photons through a tomographically complete set of measurements, known as quantum state tomography, is an essential task in nearly all applications of quantum science and technology, from quantum sensing to quantum communications. Recent advances in optical metasurfaces enable the design of ultrathin nanostructured optical elements performing such state tomography tasks, promising greater simplicity, miniaturization, and scalability. However, reported metasurfaces on this goal were limited to a small Hilbert dimension, e.g., polarization qubits or spatial qudits with only a few states. When scaling up to higher-dimensional qudit tomography problems, especially those involving spatial qudits, a natural question arises: whether a metasurface with randomized nanostructures is sufficient to perform such qudit tomography, achieving optimal conditions. In this work, we attempt to answer this question through a set of numerical experiments with random metasurfaces, utilizing large-scale simulations of over 16,000 distinct metasurfaces each exceeding 200 wavelengths in size. We show that with sufficient redundancy in the number of detectors, random metasurfaces perform reasonably well in quantum photonic spatial qudit tomography encoded in Hermite-Gaussian states for up to approximately 10 states. Furthermore, we discuss additional considerations for optimizing metasurfaces in multiphoton cases. Our work opens a pathway toward computationally efficient, miniaturized, and error-tolerant quantum measurement platforms.

Figures (8)

• Figure 1: Schematic of the spatial qudit tomography of photons with metasurfaces. Single- or multi-photon spatial state encoded under a finite $D$-dimensional Hermite-Gaussian (HG) basis is refracted by a metasurface onto a single-photon detector array, which measures average photon counts $\bm{\Gamma}^{(1)}$ of each pixel for a single-photon state reconstruction or an $N$-fold correlation $\bm{\Gamma}^{(N)}$ among pixels for an $N$-photon state. The reconstruction algorithm recovers the density matrix of the input knowing the transformation of the device.
• Figure 2: Numerical experiment setup. (a) The single-photon simulation setup of a metasurface. The single photon input state, $\bm{\rho}^{(1)}$, is scattered by the metasurface and propagated to the far-field detector pixels, resulting the expected photon counts of each pixel. This relation between the flattened input density matrix, $\bm{\rho}^{(1)}$, and the expected average photon count vector, $\bm{\Gamma}^{(1)}$, is linearly related by the instrument matrix, $\mathbf{M}^{(1)}$. (b) Parametrization of a freeform metasurface structure. A 1-dimensional freeform metasurface can be parametrized by the structure thickness, $h$, and an array, $\mathbf{d}$, representing the widths of the ridges and gaps.
• Figure 3: Effects on condition number, $\kappa$ from (a) different orders of HG states of reconstruction, or equivalently, Hilbert dimension, $D$, and (b) different number of measurement pixels, $L$, and (c) structure randomness, $\sigma_\mathrm{param}$. Here, no minimum feature size constraint is imposed, and the thickness is fixed at $800\ \mathrm{nm}$. In (a), all 987 detector pixels are used; the theoretical condition number of $\kappa(\mathbf{M}_\mathrm{SIC}^{(1)})=\sqrt{D+1}$ is plotted as a red line for comparison. In (b), the average meta-atom size is set to 222 $\mathrm{n m}$; ill-posed points ($L<D^2$) are omitted. In (c), all 987 detector pixels are used with Hilbert dimension $D=10$. The condition number reduces monotonically with fewer HG states and more measurements, and it decreases with structure randomness in general.
• Figure 4: Plots of condition number, $\kappa$, and average transmission, $T_{\mathrm{avg}}$, with respect to metasurface thicknesses, $h$. The Hilbert dimension is set to 10 and all 987 detector pixels. The left plots (a) are without minimum feature size constraint $d_\text{min}=0$, and the right plots (b) impose a minimum feature size constraint of $d_\text{min}=100\ \mathrm{n m}$.
• Figure 5: Reconstruction of an example 10-HG-order input state using a selected random metasurface $h=200\ \mathrm{nm}$, $\mu=667\ \mathrm{nm}$, and $d_\text{min}=100 \ \mathrm{nm}$, whose structure is shown in (a). The ground truth density matrix of the input is shown in (b). With SNRs of 12.58 and 0.50, the reconstructed density matrices are reconstructed to have fidelities of 0.995 as in (c) and 0.821 as in (d) respectively.