Does entanglement harvesting require cooling the environment to sufficiently low temperatures? I stumbled upon this webpage, which describe a process they call "entanglement harvesting". I was wondering whether to observe this effect it is necessary to cool the environment to sufficiently low temperatures so that the electromagnetic field really is in the vacuum state. Or is the vacuum entangled even for example a "thermal" state.
 A: This paper (https://arxiv.org/abs/1508.01209) talks about photon mediated coupling between two identical detectors - Alice and Bob. The detectors start off in the ground state $\left | g \right >$ as opposed to the excited state $\left | e \right >$ while the EM field starts off in the vacuum $\left | 0 \right >$. The punch line seems to be that, after a long time, you can trace out the EM field to find that Alice and Bob are entangled.
The most important part is being able to treat $\{ \left | g \right >, \left | e \right > \}$ as a quantum system rather than one whose wavefunction has "collapsed". For this, the environment will almost certainly need to be cooled to prevent decoherence - the coupling of Alice and Bob to the EM field should dominate over their coupling to everything else. An approximate two-level system like this could be an atom in a magnetic trap. And these typically require low temperatures.
But I don't think any additional cooling would be needed to make the field well behaved. Atoms interacting with "the vacuum" is usually a very safe assumption and it has been used to model other things which have been seen experimentally like the Lamb shift. But to answer your question, a thermal electromagnetic field would still be entangled because it would have a density matrix like
\begin{equation}
\rho = \frac{1}{Z} \sum_n e^{-\beta E_n} \left | n \right >.
\end{equation}
Here $\left | n \right >$ is a basis of states the field near the detector can be in. This means that if you also included states like $\left | f \right >$ having to do with the field "far away", you would see a total wavefunction $\psi$ where the $n$ and $f$ parts are entangled. That's the only way a mixed state like $\rho$ can show up after restricting to just the near part.
As an aside, this overall entangled wavefunction is hard to pin down. There are many possible ways to take a given $\rho$ for a subsystem and "lift" it to some $\psi$ for the whole system. One possible choice which is often useful is called the thermofield double.
A: Here's a small supplement to Connor Behan's more thorough answer: In relativistic quantum field theory, any state of finite energy is entangled with respect to location.$^\dagger$ The vacuum state is just one example. This is part of the Reeh-Schlieder theorem, which is stated and proven in section 1.3.1 of ref 1. For a more extended review, see ref 2.
$^\dagger$  More precisely, the theorem holds any state that is analytic in the energy, using the language of ref 1. 

*

*Horuzhy (1990, english edition), Introduction to Algebraic Quantum Field Theory, Kluwer Academic


*Witten, Notes on Some Entanglement Properties of Quantum Field Theory (https://arxiv.org/abs/1803.04993)
