Closed-form expressions for smeared bi-distributions of a massless scalar field: non-perturbative and asymptotic results in relativistic quantum information

TL;DR

This work derives closed-form expressions for smeared two-point functions of a massless scalar field in Minkowski spacetime using Gaussian spacetime test functions, enabling analytic evaluation of local operations by Unruh-DeWitt detectors. It provides complete, non-perturbative treatments for gapless detectors via a Magnus-expansion approach and yields explicit, closed-form formulas for entanglement harvesting between Gaussian-smeared detectors, including asymptotic scaling laws at large separations. The results reveal how local noise and nonlocal field correlations compete to enable or suppress entanglement, and they recover known literature results as special cases while offering new analytic tools for relativistic quantum information. Overall, the paper delivers practical, analytic expressions for smeared field bi-distributions and their use in probing quantum fields and harvesting entanglement, with broad implications for local operations in quantum field theory and relativistic quantum information processing.

Abstract

Using spacetime Gaussian test functions, we find closed-form expressions for the smeared Wightman function, Feynman propagator, retarded and advanced Green's functions, causal propagator and symmetric propagator of a massless scalar field in the vacuum of Minkowski spacetime. We apply our results to localized quantum systems which interact with a quantum field in Gaussian spacetime regions and study different relativistic quantum information protocols. In the protocol of entanglement harvesting, we find a closed-form expression for the entanglement that can be acquired by probes which interact in Gaussian spacetime regions and obtain asymptotic results for the protocol. We also revisit the case of two gapless detectors and show that the detectors can become entangled if there is two-way signalling between their interaction regions, providing closed-form expressions for the detectors' final state.

Figures (4)

• Figure 1: The negativity and signalling contribution for two detectors interacting with a massless scalar field in Gaussian spacetime regions separated by $|\bm L| = 5 T$, with detector sizes $\sigma = 0.01T$.
• Figure 2: The solid lines correspond to the eigenvalues of the partial transpose of the detectors final state $\hat{\rho}_\textsc{ab}^{t_\textsc{b}}$, when the detectors start both in their ground state, as a function of the interaction time $T$, scaled by the detectors' separation $|\bm L|$. The dashed lines are the eigenvalues of $\hat{\rho}_\textsc{ab}^{t_\textsc{b}}$ if it were to evolve only according to the unitary $\hat{U}_\textsc{c} = e^{- \frac{\mathrm{i}}{2}\hat{\mu}_\textsc{a} \hat{\mu}_\textsc{b} \Delta_\textsc{ab}}$. We picked $\sigma = 0.05|\bm L|$ for these plots.
• Figure 3: The squared Hilbert-Schmidt norm of the difference between the density operators $\hat{\rho}_\textsc{ab}$ and $\hat{\rho}_\textsc{c}$ for $\sigma = 0.05 |\bm L|$ considering detectors that start in their ground state. The dashed lines correspond to their asymptotic limit as $T\to\infty$.
• Figure 4: The limit of the asymptotic behaviour of the squared Hilbert-Schmidt norm of the operator $\rho_{\textsc{ab}} - \hat{\rho}_\textsc{c}$. This limit only depends on $\lambda$.