Vacuum Entanglement Harvesting

Entanglement harvesting is the transfer of field correlations to initially uncorrelated localized probes. For the strongest spacelike claim, the complete coupling supports must be spacelike, local excitation noise must be separated from the nonlocal coherence, and causal-exchange terms must be absent within controlled tail and perturbative errors.

Required background. Field communication supplies the localized two-probe channel.

Helpful background. Mutual information and correlations distinguishes total correlation from entanglement. Localized detector models supplies the switching and smearing data.

Two-probe reduced state

Take identical two-level detectors initially in $|g_Ag_B\rangle$ and couple them weakly to a scalar field. To leading nontrivial order, the reduced state has the schematic $X$ form

$$\rho_{AB}= \begin{pmatrix} 1-P_A-P_B&0&0&X^*\\ 0&P_B&L^*&0\\ 0&L&P_A&0\\ X&0&0&O(\lambda^4) \end{pmatrix} +O(\lambda^4).$$

$P_A,P_B$ are local excitation noise, while $X$ and $L$ contain nonlocal two-point-function integrals with the two spacetime profiles. At this order, the potentially negative partial-transpose eigenvalue gives

$$\mathcal N^{(2)}= \max\!\left\{0, \frac{\sqrt{(P_A-P_B)^2+4|X|^2}-(P_A+P_B)}{2} \right\},$$

so harvesting requires $|X|>\sqrt{P_AP_B}$. For the symmetric case $P_A=P_B=P$, this reduces to $\max(0,|X|-P)$. Which coherence enters depends on initial states and coupling choices, so the matrix definition must accompany the formula.

Harvesting is a branch in which two initially uncorrelated probes couple locally and are compared later. It is distinct from deliberately encoding a message at one probe. The diagram is schematic.

Strictly spacelike supports

Let the spacetime smearing functions $f_A,f_B$ have compact supports $K_A,K_B$ with $K_A\subset K_B^\perp$. The field commutator vanishes between the supports, excluding causal exchange through the field at the coupling order. Vacuum anticommutator correlations can still contribute to $X$ and $L$.

Compare with a controlled separable field state or a model in which initial cross correlations are removed while local spectra are matched. If probe entanglement disappears in the spacelike vacuum case under that control, the result supports extraction of pre-existing field correlations. Final probe entanglement alone is insufficient when supports overlap or higher-order exchange paths are unresolved.

Pozas-Kerstjens and Martín-Martínez 2015, §§ II–IV analyze the dependence on dimension, switching, size, gap, and perturbative order. Tjoa and Martín-Martínez 2021, §§ II–IV show why field-mediated detector entanglement need not always warrant the label “harvesting,” sharpening the need for causal and correlation controls.

Tails and ultraviolet stability

Gaussian profiles have tails, so nominally separated detector centers do not establish exact spacelike support. Truncate to compact profiles or bound the tail contribution to every matrix element used in $\mathcal N$. Vary spatial smearing and switching smoothness; a harvesting region that disappears under ultraviolet refinement supports only a regulator-specific claim.

The perturbative negativity must also dominate omitted orders. Near the threshold $|X|=\sqrt{P_AP_B}$, an $O(\lambda^4)$ remainder can change the sign. Quote an uncertainty interval or compute the next order rather than presenting a sharp phase boundary.

Support tails move a nominal harvesting protocol toward causal exchange; ultraviolet sensitivity can instead invalidate the detector model. Both controls are needed before final probe entanglement is attributed to pre-existing spacelike field correlations. The map is schematic.

Evidence boundary

As assessed through 2026-08-10, the cited literature provides rigorous model definitions and substantial analytic and numerical evidence for harvesting in regulated detector models. It does not license a generic statement that every weak local detector pair harvests entanglement, nor does it by itself establish an experimental platform realization.

References

• Pozas-Kerstjens, A., and Martín-Martínez, E. (2015). “Harvesting Correlations from the Quantum Vacuum.” Physical Review D 92, 064042. DOI. Open PDF.
• Tjoa, E., and Martín-Martínez, E. (2021). “When Entanglement Harvesting Is Not Really Harvesting.” Physical Review D 104, 125005. DOI. Open PDF.