Sparsity-Driven Entanglement Detection in High-Dimensional Quantum States

Abstract

The characterization of high-dimensional quantum entanglement is crucial for advanced quantum computing and quantum information algorithms. Traditional methods require extensive data acquisition and suffer from limited visibility due to experimental noise. Here, we introduce a sparsity-driven framework to enhance the detection and certification of high-dimensional entanglement in spatially entangled photon pairs. By applying $\ell_1$-regularized reconstruction to sample covariance matrices obtained from measurements on photons produced via spontaneous parametric down-conversion (SPDC) measurements, we enhance the visibility of the correlation signal while suppressing noise. We demonstrate, using a position-momentum Einstein-Podolsky-Rosen (EPR) entanglement criterion, that this approach enables certification of an entanglement dimensionality that cannot be achieved without regularization. Our method is scalable, simple to use and compatible with existing quantum-optics platforms, thus paves the way for efficient, real-time analysis of high-dimensional quantum states.

Figures (10)

• Figure 1: Schematic of the experimental setup. Spatially entangled photon pairs are generated via SPDC and imaged in both imaging, $L_2$, and Fourier, $L_1$, configurations using an EMCCD camera. Here, $f_1 = 100[mm]$ and $f_2 = 50[mm]$ are the focal length of $L_1$ and $L_2$ respectively.
• Figure 2: EMCCD SNR as a function of the number of frames acquired for unoptimized and optimized sample covariance matrices, (a) Sum-coordinate projection of the covariance matrix (b) Minus-coordinate projection of the covariance matrix. (c-d) are the Sum-coordinate projection of the covariance matrix using $\sim10^5$ frames: c unoptimize data and d optimize data.
• Figure 3: SPAD camera results in Fourier basis after $10^7$ frames in a logarithmic scale. (a,c) Unoptimized full covariance matrix and conditional image (correlation of a single exemplary pixel to the whole frame), both reshaped into an image. (b,d) Same as (a,c) but optimized using Eq. \ref{['eq:objective']}.
• Figure 4: Entanglement dimension witness calculation as a function of the number of images.
• Figure 5: The dominant eigenvectors from spectral analysis of the sample covariance matrices reshaped into image form. (a) & (b) first and second eigenvectors of the raw data multiply by their compatible eigenvalues in absolute value and reshaped into image form. (c) & (d) are the same for the optimized matrix.