Spectral Localization Principle for Entanglement Harvesting

Abstract

We propose a unified physical principle for entanglement harvesting: the entanglement that two localized detectors can extract from a quantum field is determined solely by how localized the field's effective spectral density is. We demonstrate this in an analytically solvable model of two qubits coupled to a leaky single-mode cavity, which in turn couples to a continuous electromagnetic bath, and derive the maximum harvestable concurrence in closed form, $\mathcal{C}_{\max}(Q)=2e^{-π/(2Q)}(1+e^{-π/(2Q)})/(1+3e^{-π/Q})$, where $Q\equiv|Δ|/κ$ is the ratio of the qubit-cavity detuning $Δ$ to the cavity linewidth $κ$. In the high-$Q$ limit, $\mathcal{C}_{\max}\simeq1-π^{2}/(16Q^{2})$, so the entanglement is robust against cavity loss; in the low-$Q$ limit it decays exponentially to zero, consistent with the irreversible-reservoir character of a continuous field, where maximal entanglement is unattainable. Since $Q$ is proportional to the inverse participation ratio (IPR) of the effective spectral density, it is the single dimensionless parameter governing the crossover from deterministic gate-based entanglement ($Q\to\infty$) to vacuum harvesting ($Q\to0$). Our framework operationalizes the Reeh-Schlieder theorem by quantifying the fraction of vacuum correlations accessible to localized detectors. It also reveals a formal correspondence of the maximal concurrence with the IPR, analogous to the conductivity-participation-ratio relation in Anderson localization. The predicted $\mathcal{C}_{\max}(Q)$ curve is, in principle, directly observable in superconducting circuit QED experiments.