Entanglement dynamics of light's orbital angular momentum under a Lorentz boost

TL;DR

The paper tackles the relativistic dynamics of photonic OAM entanglement under Lorentz boosts between inertial observers. It develops a theoretical framework that maps azimuthal coordinates under length contraction, defines three motion models (Zero RM, Non-Zero RM1, Non-Zero RM2), and computes entanglement metrics such as entropy, purity, negativity, and effective dimensionality from joint amplitudes $A(k,m)$. The key finding is that OAM entanglement is observer-dependent: stationary observers see residual entanglement that diminishes toward a nonzero minimum as $\gamma(v)$ grows, while moving observers in the Non-Zero RM2 scenario experience complete entanglement degradation near the light cone. This challenges assumptions about invariance of OAM under spacetime transformations and suggests deep connections between relativity and quantum uncertainty, with experimentally testable proposals using SPDC and SLM-based mode distortion.

Abstract

In this study, we report on the evolution of photonic orbital angular momentum (OAM) entanglement in inertial reference frames under a Lorentz boost, covering the general cases of zero and non-zero relative motion between observers of the entangled state. We find that entanglement undergoes significant changes that are observer dependent, asymptotically approaching a minimum at very large velocities close to the light cone from the viewpoint of the stationary observers in the rest frame, and degrading completely from the viewpoint of the moving observer. Our results, as demonstrated through entanglement metrics such as entanglement entropy and purity, show that OAM and OAM entanglement are observer dependent, raising pertinent questions on the invariance of such entangled states to spacetime transformations.

Figures (5)

• Figure 1: (a) It shows Minkowski diagrams for different relativistic velocities corresponding to $\gamma(v)=1,5/4,5/3,5$. Photon pairs that are entangled in their OAM degree of freedom are generated in the stationary reference frame of $S(ct,x)$ sent to two independent detectors, Alice (A) and Bob (B) in the moving frame of $S'(ct',x')$. The relative velocity of these two frames is $v$, and the motion is only restricted to one dimension say $x$. In the rest frame, this is seen as a length contraction in the $x$-direction of the detectors, mapping the coordinates $(x, y) \rightarrow (x/ \gamma, y)$ and $\phi \rightarrow \phi' (\phi) = \arctan (\gamma \tan(\phi))$. The detectors project onto the OAM eigenstates $\ket{k}$ and $\ket{m}$, where Charlie (observer in $S$) observes their OAM joint detections as shown in the right column of (b) for Alice and Bob being at rest ($\gamma=1$, upper image) and moving at a relativistic speed ($\gamma(v)=5$, lower image). For $\gamma(v)=1$ the plot shows that the joint detection probability has OAM anti-correlations, i.e. non-zero detection probability for $k=-m$. However, as $\gamma(v)$ increases, the OAM dispersion appears and the width of the spectrum increases with the increment of the Lorentz factor ($\gamma(v)=5$). Right column in part (b) shows the joint detection probabilities for the Non-Zero RM1 (upper image) and Non-Zero RM2 (lower image) models and $\gamma(v)=5$.
• Figure 2: Joint detection probability ($P(k,m)$) is shown for $l_{max}=20$ and $\gamma(v)=1,20,100,10000$. Each row in subplots (a-c) shows one dynamical model. (a) relates to Zero RM that shows a drastic dispersion of OAM modes at high velocities. (b) displays Non-Zero RM1 in which the asymptotic behavior of entanglement is only due to three modes. (c) represents Non-Zero RM2 that includes only a single mode close to LC, hence, the state is separable and entanglement vanish entirely.
• Figure 3: Metrics for $l_{max}=20$ and $\gamma(v)=1-10000$ in $\text{log}_{10}$ basis. From the stationary observer's viewpoint the quantum state will remain entangled though reach a minimum degree asymptotically at the LC. However, from the viewpoint of the moving frame's observe, the entanglement will be vanished completely at the LC. Therefore, for this observer the state is separable.
• Figure 4: OAM marginals for $l_{max}=20$ and $\gamma(v)=1,20,100,10000$. (a) In the Zero RM, the information regarding the entanglement is scrambled in a way that no longer one can find the obvious track of its present. This is due to the extreme OAM dispersion. However, in the Non-Zero RMs, the entanglement behavior is clear. (b) displays the dynamics of entanglement in the Non-Zero RM1 that asymptotically approaches a minimum degree close to LC. Here, mainly five modes contribute to the joint detection. (c) illustrates the entanglement evolution in the Non-Zero RM2 that at LC will perfectly vanish.
• Figure 5: Schmidt decomposition coefficients related to the OAM entanglement evolution as a measure for the entanglement dynamics related to $\gamma(v)=1,20,100,10000$. Close to the LC, the Zero RM (a), and the Non-Zero RM1 (b) only have two effective modes. However, for the Non-Zero RM2 (c), the number of effective mode is only one asymptotically. This confirms the absence of entanglement in the case of Non-Zero RM2 model at LC.