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The ground state of a relativistic QFT has nonzero correlation between field operators at spatially separated points. A way to interpret this is through entanglement between different spacetime regions.

In Quantum Mechanics entanglement is between for example, two qubits which is the mathematical representation of for example, entanglement between spins of two electrons.

What is the physical interpretation of entanglement in the vacuum of relativistic QFT? I understand that the entanglement is between regions of spacetime but what are the states of the two regions which are entangled?

Update: I would like to make this questions a bit more precise using Rindler decomposition of Minkowski space. The Minkowski space can be decomposed into left and right rindler wedges. The vacuum state wavefunction can be written as (the derivation can be seen in for example Daniel Harlow's lecture notes, section 3.3) $$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_i e^{-\pi\omega_i}|i^{*}\rangle_L|i\rangle_R$$

where $|i^{*}\rangle_L$ and $|i\rangle_R$ are $\mathrm{eigenstates^{*}}$ of the restriction of Lorentz boost operator to the right wedge ($K_r$). In this the entanglement between two pieces of the Minkowski space is evident. But, is there an operational interpretation for this entanglement? What would be a measurement procedure which would destroy this entanglement?

*Actually, $|i^{*}\rangle_L$ is not an eigenstate of $K_r$. $|i^{*}\rangle_L=\Phi |i\rangle _L$ where $|i\rangle _L$ is an eignestate of $K_r$ and $\Phi$ is a antiunitary operator that exists in all quantum field theories and is usually called CPT. (details can be found in the reference given above)

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The vacuum of a quantum field theory (QFT) is the ground state of the QFT. So the "entanglement in the vacuum" means the entanglement in the ground state of the QFT. There is a famous saying: vacuum is not empty, it is filled with fluctuation quantum fields. The ground state of a QFT is a quantum mechanical superposition of a lot of different field configurations throughout the space. So it is possible to talk about how the quantum field fluctuation at one place is correlated/entangled with the fluctuation at another place. From the modern point of view, matters are made of quantum information. The QFT is an effective description of the collective motion of the underlying qubits, and the vacuum state is the ground sate of these qubits, so it makes perfect sense to talk about quantum entanglement in the vacuum state.

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  • $\begingroup$ Thank you for taking the time to answer. Even though I have heard about the idea of vacuum being filled with particles could you please give me a mathematical argument for the same. In a free field theory the particle number is conserved and the number operator acting on the ground state gives zero. But, even in free field theories there is entanglement between different regions of spacetime, What can this entanglement be attributed to? $\endgroup$ – Rajath Krishna R May 18 '17 at 5:36
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    $\begingroup$ Also, is the idea of QFT being an effective description of the collective motion of underlying qubits a mathematically rigorously derived idea? If so, what does these qubits physically represent? $\endgroup$ – Rajath Krishna R May 18 '17 at 5:36
  • $\begingroup$ The entanglement is attributed to the fluctuating quantum field. Even if the particle number is zero, the quantum field still has a nonzero fluctuation. Consider the ground state of a simple harmonic oscillator, if you act the number operator $a^\dagger a$ to the ground state, the number is zero, there no particle, but the ground state wave function $\exp(-x^2)$ is not trivial, the position (as well as momentum) of the oscillator is still fluctuating. This is the vacuum fluctuation, and this is where the vacuum entanglement comes from. $\endgroup$ – Everett You May 19 '17 at 1:35
  • $\begingroup$ These qubits are the underlying fundamental degrees of freedom that makes up everything (including spacetime, matter, and forces). This is the idea of "It from Qubit", there is a whole summer school on the various aspect of this idea (perimeterinstitute.ca/conferences/it-qubit-summer-school) $\endgroup$ – Everett You May 19 '17 at 1:44
  • $\begingroup$ What does it mean for the quantum fields to be entangled? I went through some more literature on this and have edited the question which I believe has made the question a bit more precise. I request you to provide an answer in the context of the explicitly given entangled state in the question. $\endgroup$ – Rajath Krishna R May 19 '17 at 15:27

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