Correlation Extraction versus Causal Exchange

Two initially uncorrelated probes can end correlated for three physically different reasons: they sampled correlations already present in the field state, one probe influenced the other through the field, or their apparatus shared noise or preparation data. The final density matrix does not remember which explanation is correct. Support geometry, the field’s symmetric and antisymmetric kernels, and deliberate interventions supply the missing causal information.

Required background. Field communication defines the induced channel, and vacuum entanglement harvesting defines the strictly spacelike extraction claim.

Helpful background. Signaling and causal composition supplies the receiver-side intervention test.

The chapter’s task map, protocol comparison, and failure controls show where these tests enter the full protocol without repeating the shared figures.

Two kernels answer different questions

For a Hermitian scalar field in a state $\rho_\phi$, define

$$W(x,x')=\operatorname{tr}[\rho_\phi\phi(x)\phi(x')],$$

The discussion below assumes a vanishing one-point function, $\langle\phi(x)\rangle=0$. For a coherent or otherwise displaced state, replace $W$ and $H$ by their connected kernels after subtracting $\langle\phi(x)\rangle\langle\phi(x')\rangle$; the subtracted mean is a first-order local drive and should not be counted as an extracted cross correlation.

$$H(x,x')=\langle\{\phi(x),\phi(x')\}\rangle, \qquad [\phi(x),\phi(x')]=i\Delta(x,x').$$

Then

$$W(x,x')=\frac12H(x,x')+\frac{i}{2}\Delta(x,x').$$

The Hadamard kernel $H$ is symmetric and state-dependent. It describes fluctuations and statistical correlations sampled by the probes. The Pauli–Jordan kernel $\Delta$ is antisymmetric and, for a free field, state-independent. Microcausality gives $\Delta(x,x')=0$ at spacelike separation. With the convention above, the retarded fundamental solution is

$$G_{\mathrm{ret}}(x,x')=-\theta(t-t')\Delta(x,x').$$

Thus $H\neq0$ does not by itself permit signaling, while a nonzero retarded smearing identifies a possible causal response.

The distinction appears directly in the two-detector perturbation series. For ground-state probes, the time-ordered coherence can be written schematically but exactly at second order as

$$\begin{aligned} \mathcal M={}&-\lambda_A\lambda_B \int dV\,dV'\,f_A(x)f_B(x')e^{i(\Omega_A t+\Omega_B t')}\\ &\times\left[ \theta(t-t')W(x,x')+\theta(t'-t)W(x',x) \right]. \end{aligned}$$

Substitution of the kernel decomposition gives

$$\mathcal M=\mathcal M_H+\mathcal M_\Delta,$$ $$\mathcal M_H=-\frac{\lambda_A\lambda_B}{2} \int dV\,dV'\,f_Af_Be^{i(\Omega_A t+\Omega_B t')}H(x,x'),$$ $$\mathcal M_\Delta=-\frac{i\lambda_A\lambda_B}{2} \int dV\,dV'\,f_Af_Be^{i(\Omega_A t+\Omega_B t')} \operatorname{sgn}(t-t')\Delta(x,x').$$

For completely spacelike supports, $\mathcal M_\Delta=0$ exactly, whereas $\mathcal M_H$ can remain. In a causally connected arrangement, both terms can contribute to final probe entanglement. This split, including its time-ordering dependence, is developed for detector harvesting in Tjoa and Martín-Martínez 2021, Eqs. (22)–(26) and Appendix A, pp. 5–7, 16–19.

A signaling test is more direct than inspecting $\mathcal M$. Deliberately vary detector $A$’s preparation or coupling while leaving detector $B$’s local procedure fixed. At leading cross order, the resulting change in a $B$ observable contains a retarded smearing of $\Delta$. If every point in $A$’s support is spacelike to every point in $B$’s support, that contrast vanishes even when $H$ and the final mutual information are nonzero. Relativistic detector channels make this intervention criterion explicit Cliche and Kempf 2010, §§ III–IV, pp. 4–8.

Matched compact-support benchmark

The following calculation separates the two kernels without changing the field, detector shape, or spatial separation. Work in $3+1$ dimensional Minkowski spacetime with a massless scalar vacuum and natural units. Place pointlike stationary probes a distance $R=3T$ apart and set $T=1$. Use the peak-one compact switching

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

At spatial separation $R$, the vacuum Wightman distribution is

$$W_0(\tau,R)= \frac{1}{4\pi^2\left[R^2-(\tau-i0)^2\right]}.$$

The corresponding kernels are

$$H_0(\tau,R)=\frac1{2\pi^2} \operatorname{PV}\frac1{R^2-\tau^2},$$ $$\Delta_0(\tau,R)= -\frac{\delta(\tau-R)-\delta(\tau+R)}{4\pi R}.$$

First switch both probes on over $[-T,T]$. Since $|t_B-t_A|\leq2T\lt R$, all support pairs are spacelike. Define the dimensionless smeared kernels

$$H_{AB}=\int dt_A\,dt_B\, \chi_A(t_A)\chi_B(t_B)H_0(t_B-t_A,R),$$ $$\Delta_{AB}=\int dt_A\,dt_B\, \chi_A(t_A)\chi_B(t_B)\Delta_0(t_B-t_A,R).$$

Direct quadrature gives

$$H_{AB}^{\mathrm{space}}=0.0085171771, \qquad \Delta_{AB}^{\mathrm{space}}=0.$$

This is the extraction side of the comparison: there is a nonzero vacuum correlation kernel and an exact commutator control.

Next keep probe $A$ centered at $t=0$ and center probe $B$ at $t=R+\delta$ with $\delta=0.5T$. The support regions are now causally connected, although not every event pair is timelike. In $3+1$ dimensional massless theory, Huygens propagation makes the retarded contribution come from null-related event pairs within those supports. Principal-value quadrature for $H$ and the light-cone delta function for $\Delta$ give

$$H_{AB}^{\mathrm{causal}}=-0.0117285322,$$ $$|\Delta_{AB}^{\mathrm{causal}}| =\frac1{4\pi R}\int dt\,\chi(t)\chi(t-\delta) =0.0185692533,$$

where

$$\int dt\,\chi(t)\chi(t-0.5)=0.7000443570.$$

The signs of response terms also contain the detector coupling convention, so the robust causal diagnostic is the nonzero retarded magnitude and intervention contrast. The signed $H$ value, by contrast, is the Hadamard distribution evaluated on the stated test functions; it is not a probability.

For reproducibility, reduce the double integrals to the switching autocorrelation

$$C(s)=\int dt\,\chi(t)\chi(t+s),$$

integrate $C(s)/(R^2-s^2)$ over $-2\leq s\leq2$ for the spacelike case, and take the principal value of $C(s)/[R^2-(R+\delta+s)^2]$ for the delayed case. Increasing the working precision and quadrature subdivisions leaves the displayed digits stable.

Interventions distinguish the mechanisms

The matched benchmark supports three independent controls.

Sender intervention. Toggle $\lambda_A$, or change $A$’s initial state, without changing $B$’s procedure. In the simultaneous spacelike arrangement, microcausality forces the receiver’s intervention contrast to zero. In the delayed arrangement, the nonzero $\Delta_{AB}$ permits a contrast. Correlation in the final joint state is therefore not the same observable as influence on $B$.

Correlation removal. In a regulated two-region Gaussian model, replace the initial covariance by a positive comparison covariance with the same local $A$ and $B$ blocks and a zero cross block. This removes the corresponding $H_{AB}$ contribution while retaining the field algebra and hence the same $\Delta_{AB}$. Spacelike extraction should disappear; delayed causal exchange need not. The construction must be checked for positivity and should not be advertised as an arbitrary continuum-state surgery.

Apparatus control. Turn off both probe–field couplings but retain the preparation, clocks, readout electronics, and analysis pipeline. Correlation surviving this control is not evidence for either vacuum extraction or field exchange. A shared random calibration variable can correlate two readouts while leaving both $H_{AB}$ and $\Delta_{AB}$ irrelevant.

These controls also clarify terminology. “Field-mediated probe entanglement” states only that the field interaction appears in the protocol. “Harvested entanglement” additionally requires evidence that pre-existing field correlations, rather than causal communication, supplied the relevant nonlocal resource. Causally connected detector entanglement can be communication-generated in important massless flat-spacetime regimes Tjoa and Martín-Martínez 2021, §§ III–IV, pp. 8–13; this conclusion should retain the models and regimes stated there.

A practical causal decision procedure

Begin with geometry, before calculating a detector witness. List the complete spacetime supports of both interactions, including switching and spatial smearing. If their causal hulls overlap, the protocol is not a strict spacelike extraction test even if the detector centers are spacelike. Compute or bound the smeared commutator on those supports. An exact microcausal zero is stronger than a small number obtained after a regulator or numerical cancellation.

Next compute the state-dependent kernel with the same profiles. A nonzero $H_{AB}$ says that these particular probes can sample a cross correlation; it does not say that the final detector state is entangled. Detector phases, local noise, and time ordering still enter the reduced state. Conversely, a zero detector witness can result from destructive interference even when both kernels are nonzero. Kernel values diagnose available mechanisms, whereas the reduced state diagnoses the protocol’s output.

Then define an intervention as a pair of complete experimental choices at $A$, such as “coupling off” and “coupling on with fixed input,” while holding the local channel and measurement at $B$ fixed. Compare the unconditional probability distribution at $B$, not a distribution conditioned on a record that is unavailable there. A nonzero contrast supports causal influence from the changed operation. A zero contrast in a spacelike arrangement is required by microcausality; a zero contrast in a causal arrangement can instead mean that the chosen detector observable or switching profile is insensitive to the available retarded response.

Finally cross the intervention result with the correlation-removal and apparatus controls. If the joint witness disappears only when the initial cross block is removed and the receiver contrast is zero, extraction is supported. If it survives correlation removal but tracks the sender intervention, causal exchange is supported. If it remains with both field couplings off, common apparatus structure is sufficient. Mixed outcomes are not failures of the method: they show that more than one mechanism contributes, so the result should be reported as a decomposition or bounded mixture rather than forced into a single label.

Higher orders and imperfect localization

A leading commutator null is decisive only when the complete supports are spacelike. Gaussian tails may contain null- or timelike-related pairs even when the switching centers are spacelike. Bound the tail-smeared retarded kernel or replace the profiles by compact ones. At higher perturbative orders, list every path connecting the probes, including common ancillas and repeated field interactions. If the bound on an unresolved causal or apparatus contribution is comparable to the witnessed correlation, the justified description is simply “probe correlation under the stated model.”

The split $W=H/2+i\Delta/2$ is invariant, but informal labels such as “vacuum fluctuations” and “radiation reaction” can depend on operator ordering. The spacelike commutator null and the receiver intervention are preferable because they have direct operational consequences.

Exercises

1. Derive the kernel split. Starting from the time-ordered bracket in $\mathcal M$, show why the commutator appears with $\operatorname{sgn}(t-t')$.

Solution

Use

$$W(x,x')=\frac12(H+i\Delta), \qquad W(x',x)=\frac12(H-i\Delta).$$

Then

$$\begin{aligned} &\theta(t-t')W(x,x')+\theta(t'-t)W(x',x)\\ &\quad=\frac12H +\frac{i}{2}[\theta(t-t')-\theta(t'-t)]\Delta\\ &\quad=\frac12H+\frac{i}{2}\operatorname{sgn}(t-t')\Delta, \end{aligned}$$

apart from the measure-zero convention at $t=t'$. Multiplication by the overall Dyson-series factor yields $\mathcal M_H$ and $\mathcal M_\Delta$ above.

2. Evaluate the light-cone smearing. For the delayed benchmark, use the future light-cone term in $\Delta_0$ to reduce the double integral to one dimension.

Solution

Write $\chi_A(t_A)=\chi(t_A)$ and $\chi_B(t_B)=\chi(t_B-R-\delta)$. The future delta function imposes $t_B-t_A=R$, so

$$\begin{aligned} |\Delta_{AB}| &=\frac1{4\pi R}\int dt_A\, \chi(t_A)\chi(t_A-\delta). \end{aligned}$$

For $R=3$, $\delta=0.5$, and the stated bump, the overlap is $0.7000443570$, giving $|\Delta_{AB}|=0.0185692533$.

3. Classify a control result. A joint probe witness is nonzero. It survives removal of initial field cross correlations, disappears when $A$’s coupling is turned off, and changes when $A$’s input state changes. Which mechanism is supported?

Solution

The witness does not require the initial cross correlations, but it does require an active sender and responds to a sender intervention. Those observations support causal exchange. They do not prove that the Hadamard kernel makes no contribution in the original state; they show that pre-existing cross correlation is not necessary for the observed effect under this control.

References

• Cliche, M., and Kempf, A. (2010). “The Relativistic Quantum Channel of Communication through Field Quanta.” Physical Review A 81, 012330. DOI. Open PDF.
• 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.