Vacuum Entanglement Harvesting

Entanglement harvesting is the transfer of pre-existing field correlations to initially uncorrelated local probes. The strongest version uses coupling supports that are entirely spacelike separated: microcausality then removes field-mediated signaling between the probes, while the vacuum two-point function can still correlate them. Whether those correlations overcome local excitation noise is a quantitative question, not an automatic property of the vacuum.

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 switching, smearing, and detector-domain conventions.

The chapter’s task map, protocol comparison, and failure controls give the shared causal context without repeating the overview figures here.

Local probes and the perturbative state

For two Unruh–DeWitt detectors $\nu\in\{A,B\}$, take the interaction-picture Hamiltonian

$$H_I(t)=\sum_{\nu=A,B}\lambda_\nu\chi_\nu(t)\mu_\nu(t) \int d^3\mathbf x\, F_\nu(\mathbf x-\mathbf x_\nu)\phi(t,\mathbf x),$$

with

$$\mu_\nu(t)=\sigma_\nu^+e^{i\Omega_\nu t} +\sigma_\nu^-e^{-i\Omega_\nu t}.$$

$\chi_\nu$ is the switching function, $F_\nu$ is the spatial profile, $\Omega_\nu\gt0$ is the detector gap, and $\lambda_\nu$ is a small dimensionless coupling in the benchmark below. Begin with $|g_Ag_B\rangle\langle g_Ag_B|\otimes\rho_\phi$, assume the field one-point function vanishes, expand the Dyson series, and trace out the field. In the ordered basis

$$\{|g_Ag_B\rangle,|e_Ag_B\rangle, |g_Ae_B\rangle,|e_Ae_B\rangle\},$$

the result through the displayed orders is

$$\rho_{AB}= \begin{pmatrix} 1-\mathcal L_{AA}-\mathcal L_{BB}&0&0&\mathcal M^*\\ 0&\mathcal L_{AA}&\mathcal L_{AB}&0\\ 0&\mathcal L_{BA}&\mathcal L_{BB}&0\\ \mathcal M&0&0&O(\lambda^4) \end{pmatrix} +O(\lambda^4).$$

Here $\mathcal L_{AA}$ and $\mathcal L_{BB}$ are local excitation probabilities of order $\lambda^2$. The cross term $\mathcal L_{AB}$ is a single-excitation coherence. The two-excitation coherence $\mathcal M$ comes from time-ordered pairs of local interactions. All are integrals of the Wightman function against the declared switching and smearing profiles; explicit general expressions appear in Pozas-Kerstjens and Martín-Martínez 2015, Eqs. (13)–(18), pp. 5–6.

For example, writing $dV=dtd^3\mathbf x$ and absorbing the detector trajectory into a spacetime profile $f_\nu(x)$,

$$\mathcal L_{\mu\nu} =\lambda_\mu\lambda_\nu \int dV\,dV'\, f_\mu(x)f_\nu(x') e^{-i\Omega_\mu t+i\Omega_\nu t'}W(x,x'),$$

where $W(x,x')=\operatorname{tr}[\rho_\phi\phi(x)\phi(x')]$. The time-ordering kernels in $\mathcal M$ are essential; simply replacing $W$ by an equal-time correlator generally gives the wrong state.

Negativity, concurrence, and perturbative order

This page uses the standard, undoubled negativity

$$\mathcal N(\rho)=\frac{\lVert\rho^{T_B}\rVert_1-1}{2},$$

so conventions in which “negativity” means $\lVert\rho^{T_B}\rVert_1-1$ are larger by a factor of two Vidal and Werner 2002, § II, pp. 2–3. The $2\times2$ partial-transpose block containing $\mathcal M$ has its smaller eigenvalue

$$E_-= \frac{\mathcal L_{AA}+\mathcal L_{BB} -\sqrt{(\mathcal L_{AA}-\mathcal L_{BB})^2+4|\mathcal M|^2}}{2}.$$

Consequently,

$$\mathcal N^{(2)}= \max\!\left\{0, \frac{\sqrt{(\mathcal L_{AA}-\mathcal L_{BB})^2+4|\mathcal M|^2} -(\mathcal L_{AA}+\mathcal L_{BB})}{2} \right\}.$$

It is positive exactly when $|\mathcal M|\gt\sqrt{\mathcal L_{AA}\mathcal L_{BB}}$. For identical detectors, $\mathcal L_{AA}=\mathcal L_{BB}=\mathcal L$, this reduces to

$$\mathcal N^{(2)}=\max\{0,|\mathcal M|-\mathcal L\}.$$

These are the perturbative eigenvalue and negativity formulas of Pozas-Kerstjens and Martín-Martínez 2015, Eqs. (66)–(68), pp. 18–19. If the difference $|\mathcal M|-\sqrt{\mathcal L_{AA}\mathcal L_{BB}}$ is only of order $\lambda^4$, the omitted terms can reverse its sign; second order then does not settle whether the exact state is entangled.

Concurrence requires another convention check. For an exact two-qubit $X$ state,

$$C=2\max\!\left\{ 0, |\rho_{14}|-\sqrt{\rho_{22}\rho_{33}}, |\rho_{23}|-\sqrt{\rho_{11}\rho_{44}} \right\}.$$

The displayed second-order detector matrix determines the first candidate,

$$C_{\mathcal M}^{(2)} =2\max\{0,|\mathcal M|-\sqrt{\mathcal L_{AA}\mathcal L_{BB}}\}.$$

For identical detectors when this branch wins, $C_{\mathcal M}^{(2)}=2\mathcal N^{(2)}$. It does not follow that one may set $\rho_{44}=0$ in the other branch: $\rho_{44}=O(\lambda^4)$, but $\sqrt{\rho_{11}\rho_{44}}=O(\lambda^2)$. A full claim about that concurrence branch needs the fourth-order population. This is a common perturbative-order trap in an otherwise correct $X$-state calculation; the exact concurrence definition is due to Wootters 1998, Eqs. (9)–(10), pp. 2246–2247.

The threshold has a simple physical interpretation. $|\mathcal M|$ is the nonlocal two-excitation amplitude that can make $|gg\rangle$ coherent with $|ee\rangle$, whereas $\sqrt{\mathcal L_{AA}\mathcal L_{BB}}$ is the geometric mean of the two local noise scales. A very quiet detector cannot compensate indefinitely for a noisy partner: the asymmetric formula retains both local probabilities separately. Nonzero $\mathcal L_{AB}$ or mutual information may still show that the probes became correlated when the negativity is zero, but those observations do not establish entanglement. The finite two-probe output is valuable precisely because its partial transpose is an ordinary matrix even when factorization of continuum field degrees of freedom into spatial subsystems is subtle.

The initial-state assumption matters as much as the witness. Starting from correlated detectors, or from detectors sharing a quantum ancilla, can produce a positive final negativity without transferring any field entanglement. A harvesting benchmark therefore records the initial product state and repeats the calculation with the detector–field coupling removed. Likewise, local filtering after the interaction may reveal hidden detector entanglement, but a postselected filter must be reported with its success probability and cannot be folded into the unconditional $\rho_{AB}$ above.

A strictly spacelike vacuum benchmark

Work in $3+1$ dimensional Minkowski spacetime with a massless scalar vacuum, natural units, and identical pointlike inertial detectors at rest. Pointlike here means a spatial delta profile with a compact worldline segment; narrow smooth compact smearings provide a regulated spatial version. Set the switching time $T=1$, gap $\Omega T=1$, coupling $\lambda=0.1$, and separation $R=3T$. Use the peak-one smooth compact switching

$$\chi(t)= \begin{cases} \exp\!\left(1-\dfrac1{1-t^2}\right),&|t|\lt1,\\[4pt] 0,&|t|\geq1. \end{cases}$$

Every pair of points in the two coupling supports obeys $|t-t'|\leq2T\lt R$, so the supports are strictly spacelike. The Pauli–Jordan commutator therefore vanishes between them; any nonzero second-order cross term comes from the vacuum’s state-dependent correlations, not a causal signal.

With the Fourier convention $\widetilde\chi(\omega)=\int dt\,\chi(t)e^{i\omega t}$ and $\operatorname{sinc}z=\sin z/z$, even simultaneous switchings give

$$\mathcal L= \frac{\lambda^2}{4\pi^2} \int_0^\infty dk\,k |\widetilde\chi(\Omega+k)|^2,$$ $$\mathcal M= -\frac{\lambda^2}{4\pi^2} \int_0^\infty dk\,k\, \operatorname{sinc}(kR) \widetilde\chi(\Omega-k)\widetilde\chi(\Omega+k).$$

Numerically Fourier-transforming the stated bump and applying Gauss–Legendre quadrature on $0\leq k\leq120/T$ gives

$$\frac{\mathcal L}{\lambda^2}=0.04515365, \qquad \frac{|\mathcal M|}{\lambda^2}=0.003639458.$$

Doubling the nodes and extending the cutoff to $160/T$ leaves the quoted digits stable. Thus

$$\mathcal L=4.515365\times10^{-4}, \qquad |\mathcal M|=3.639458\times10^{-5},$$

and

$$\mathcal N^{(2)} =\max\{0,-4.151419\times10^{-4}\}=0.$$

This is a useful null benchmark: the vacuum has nonzero spacelike correlations, yet these parameters do not harvest entanglement because local noise is larger. Positive regions for other gaps, durations, separations, and smearings are mapped in Pozas-Kerstjens and Martín-Martínez 2015, §§ III–IV, pp. 7–17; Gaussian examples there must not be relabeled as exact compact-support separations.

As a correlation-removal control, use a regulated two-region field model with the same local two-point blocks but zero $A$–$B$ cross block. Then $\mathcal L$ is unchanged while $\mathcal M=\mathcal L_{AB}=0$, and the probes remain separable at this order. Such a covariance replacement is a diagnostic comparison, not a claim that an arbitrary surgically modified covariance is a continuum vacuum state.

Controls that can change the conclusion

Support tails. Gaussian switching and smearing never have exact compact support. A center-to-center spacelike separation does not remove causal overlap. Either use compact profiles or bound the contribution of the tails to every density-matrix element entering the witness.

Ultraviolet dependence. Repeat the calculation while varying smearing width and switching smoothness. If the positive region vanishes as the regulator is refined, the justified conclusion is about that detector model, not an intrinsic scale-free vacuum resource.

Causal exchange. When the supports are causally connected, the field commutator contributes and probe entanglement may be created by communication. Tjoa and Martín-Martínez 2021, Eqs. (22)–(26) and §§ III–IV, pp. 5–12 separate commutator and anticommutator contributions and show why “field-mediated entanglement” is broader than genuine harvesting.

Omitted orders. Demand a leading positive margin much larger than an estimate of $O(\lambda^4)$ corrections, or compute the next order. A curve obtained by setting the second-order expression exactly to zero is not a sharp physical phase boundary without that check.

Exercises

1. An asymmetric positive example. Suppose a controlled calculation gives $\mathcal L_{AA}=3.0\times10^{-4}$, $\mathcal L_{BB}=7.0\times10^{-4}$, and $|\mathcal M|=6.0\times10^{-4}$. Compute the second-order negativity and the $\mathcal M$-branch concurrence.

Solution

Substitution gives

$$\mathcal N^{(2)}= \frac{\sqrt{(-4.0\times10^{-4})^2 +4(6.0\times10^{-4})^2}-10^{-3}}{2} =1.3246\times10^{-4}.$$

Since $\sqrt{\mathcal L_{AA}\mathcal L_{BB}}=4.5826\times10^{-4}$,

$$C_{\mathcal M}^{(2)} =2(6.0-4.5826)\times10^{-4} =2.8348\times10^{-4}.$$

The two values are not related by a factor of two because the detectors are asymmetric.

2. Certifying strict spacelike separation. Two pointlike detectors switch on compact intervals of half-width $T$ centered at the same coordinate time and are separated by $R$. Find a simple condition ensuring every pair of coupling events is spacelike.

Solution

The largest possible time separation is $2T$. With the detectors at fixed spatial separation, every event pair is spacelike if

$$R\gt2T.$$

$R=2T$ permits null-related endpoints, and $R\lt2T$ permits causal pairs. For extended spatial smearings, replace $R$ by the minimum distance between the two spatial supports.

3. Perturbative scaling. If $|\mathcal M|-\mathcal L=2.0\times10^{-3}\lambda^2$ for identical detectors and the unknown fourth-order correction is bounded in magnitude by $0.2\lambda^4$, for which $\lambda$ does the leading positive margin exceed the bound?

Solution

Require

$$2.0\times10^{-3}\lambda^2\gt0.2\lambda^4.$$

For nonzero $\lambda$, this is $\lambda^2\lt10^{-2}$, or $|\lambda|\lt0.1$. Equality does not provide a strict robustness margin.

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.
• Vidal, G., and Werner, R. F. (2002). “Computable Measure of Entanglement.” Physical Review A 65, 032314. DOI. Open PDF.
• Wootters, W. K. (1998). “Entanglement of Formation of an Arbitrary State of Two Qubits.” Physical Review Letters 80, 2245–2248. DOI. Open PDF.