Experimental certification of high-dimensional entanglement with randomized measurements

Abstract

High-dimensional entangled states offer higher information capacity and stronger resilience to noise compared with two-dimensional systems. However, the large number of modes and sensitivity to random rotations complicate experimental entanglement certification. Here, we experimentally certify three-dimensional entanglement in a five-dimensional two-photon state using 800 Haar-random measurements implemented via a 10-plane programmable light converter. We further demonstrate the robustness of this approach against random rotations, certifying high-dimensional entanglement despite arbitrary phase randomization of the optical modes. This method, which requires no common reference frame between parties, opens the door for high-dimensional entanglement distribution through long-range random links.

Figures (6)

• Figure 1: Scheme and experimental setup. (a) Photon pairs are generated via spontaneous parametric down-conversion (SPDC) by pumping a nonlinear crystal using a classical pump beam. We set the phase-matching conditions of the SPDC process to generate spatially entangled photons over five pairs of spots defined by ten apertures placed at the far field of the SPDC crystal. An arbitrary quantum link can potentially manifest as unknown rotations that act on the five-dimensional space of each photon. A 10-plane light converter consisting of a phase-only spatial light modulator (SLM) and mirror (gray) is programmed to apply different Haar-random unitary transformations $(U_a^{(i)} \otimes U^{(i)}_b)$ on the modes of each photon. Coincidence counting $C^{(i)}_{m,n}$ between all modes $m,n$ at the output of the light converter, measured for 800 transformations ($i= 1,\dots,800$), provides access to the distribution of randomized correlations in the state. Entanglement is then certified using the second- and fourth-order moments of this distribution (see text). (b) A simulation showing the total intensity of all modes at each plane of the light converter illustrates the five modes encoding the state of each photon and their random mixing in the light converter, for one of the transformations $(U_a^{(i)} \otimes U^{(i)}_b)$ used. The maximal intensity at each plane is normalized to one.
• Figure 2: Entanglement certification with randomized measurements. The solid lines in different colors represent lower bounds corresponding to different Schmidt numbers $r$. A point below the $r$-th curve certifies $(r+1)$-dimensional entanglement. The black point represents the ideal maximally entangled state on the $(\bar{\mathcal{S}}_{\varrho}^{(2)} , \bar{\mathcal{S}}_{\varrho}^{(4)})$ plane. The red point, with error bars representing 2 standard deviations, shows experimental data from randomized measurements. The blue point represents a simulation using the reconstructed density matrix $\varrho_{\text{tomo}}$ from tomography and the same $800$ chosen unitaries. Inset: Zoom around the data point.
• Figure 3: Robustness to arbitrary state rotations. Experimental data points with (green) and without (red) random phases. Error bars represent 2 standard deviations. With the added random phases, it seems as if even the Schmidt number $r=4$ can be certified. The finite number of sampled unitaries explains the difference between the two results (green and red). The histogram of $10^6$ samples represents a simulation of results from random phases. The densities are normalized to the highest value (yellow = 1). White represents zero density.
• Figure S1: The tomography result of $\varrho_{\text{tomo}}$. The axis ticks represent the order of the state basis $\ket{ij}$ and $\bra{kl}$. The left panel shows the real part, while the right panel shows the imaginary part.
• Figure S2: The solid piecewise lines are the boundaries for all Schmidt numbers. The red point represents the reconstructed state with simulated measurements. The error bars are the $2\sigma$ uncertainty region. The green point corresponds to the result of the randomized measurement.