Topological robustness of orbital angular momentum entanglement in stochastic channels

Abstract

Orbital angular momentum (OAM) entanglement gives access to multiple qubit and high dimensional Hilbert spaces, but is unfortunately susceptible to disturbance, decaying in real-world noisy channels. Here, we show there is an underlying topology arising from OAM entanglement that is robust to such channels, which we demonstrate using atmospheric turbulence -- exemplary of stochastic or chaotic media. Using a quantum channel with various turbulence strengths, we find the OAM topological observable preserved even though the OAM itself is shown to be highly sensitive to the turbulence. We show this is true for mixed states too, with the OAM topology intact even as the purity of the state decreases due to decoherence. Our work offers a new perspective on OAM entanglement preservation, and may easily be extended to other spatial bases, degrees of freedom, as well as complex channels, whether static or dynamic.

Figures (11)

• Figure 1: (a) Reference characterisation of the undisturbed state: Normalised OAM spectrum showing power in individual OAM modes normalised to the total intensity. Density matrix $\rho_{AB}$ where the label $1$ represents state $|00\rangle$, $2 = |01\rangle$,$3 = |10\rangle$, and $4 = |11\rangle$. Coverage plot constructed from the Bloch vectors $b_{x}$, $b_{y}$ and $b_{z}$ and the calculated topological number. (b) Entangled photon pairs are generated with a defined topology; one photon is transmitted through a turbulent medium, inducing phase perturbations and entanglement decay. (c) Evolution of state properties under varying turbulence strengths. Panels show a representative subset of the ten turbulence screens used. The OAM spectra broaden, the density matrix degrades, local perturbations are seen as a rotation of the coverage sphere, and the topological invariant for each example is shown.
• Figure 2: (a) Average measured skyrmion number $N$ for three distinct topologies as a function of the distortion strength $\Omega$. Experimental data points represent the mean over 10 random turbulence realisations, with error bars indicating the standard deviation. Numerical simulations (dotted lines), averaged over 100 realisations, show excellent agreement with experiment; the shaded regions denote the standard deviation. (b) Reconstructed $b_x$ component of the Bloch field for the state $|\psi\rangle = \frac{1}{\sqrt{2}}(|0,0\rangle + |3,-3\rangle)$ across the investigated range of $\Omega$.
• Figure 3: (a) Schematic of the averaging process, where $n=10$ individual density matrices are summed to compute the ensemble-averaged matrix $\langle \rho \rangle$ for a single turbulence strength $\Omega$. (b) Workflow for the reconstruction of the spatially varying Bloch vector field $\mathbf{b}(\mathbf{r})$ and the skyrmion number $N$ from the mixed state $\langle \rho \rangle$. (c) Experimentally reconstructed density matrices and corresponding coverage plots for the state $|\psi\rangle = \frac{1}{\sqrt{2}}(|0,0\rangle + |1,-1\rangle)$ across the investigated range of turbulence strengths. (d) Normalised state purity $P = \text{Tr}(\langle \rho \rangle^2)$ as a function of turbulence strength, showing a decay toward the mixed-state limit. (e) Measured skyrmion number $N$ extracted from the ensemble-averaged states for various topologies. Markers represent experimental data using 10 realisations, with error bars representing one standard deviation. Dashed lines and shaded regions indicate numerical simulations using 100 realisations and their associated standard deviation.
• Figure 4: (a) Experimental setup for the generation and detection of OAM-OAM entangled states in turbulence. Abbreviations: mirror (M), non-linear crystal (NC), lens (L), band-pass filter (BPF), spatial light modulator (SLM), single-mode fibre (SMF), coincidence counter (CC). (b) Construction of phase screens displayed on SLM B, consisting of the spatial projective OAM masks, a grating phase screen and the turbulence phase screen.
• Figure 5: (a) QST (left), associated density matrix (right) and (b) bloch vector components (left) along with the full bloch vector field shown for the state $\ket{\Psi} = \frac{1}{\sqrt{2}}\left(\ket{0}_A \ket{0}_B + \ket{1}_A \ket{-1}_B\right)$.