Is entanglement in QFT independent of particle interactions?

[friend]

Is quantum entanglement described in the second quantization procedure of QFT? Or is entanglement only part of the first quantization procedure of quantum mechanics?

[Mentor's note - Edited to remove personal speculation]

 

[friend]

Is quantum entanglement described in the second quantization procedure of QFT? Or is entanglement only part of the first quantization procedure of quantum mechanics?

In the wikipedia.org article on entanglement, it describes entanglement in terms of inseparable wavefunctions, each of which has its own Hilbert space. Yet, wavefunctions and Hilbert spaces are entities of first quantization, Fock spaces are entities of QFT. It sounds from this wiki-article that entanglement is a first quantization phenomena. What phenomena would strictly be the result of QFT entanglement?

 

[Demystifier]

Hilbert space and entanglement are entities of both first and second quantization. Wave function (as one possible representation of a state in the Hilbert space) is often used in first quantization but rarely in second quantization.

 

[friend]

Hilbert space and entanglement are entities of both first and second quantization. Wave function (as one possible representation of a state in the Hilbert space) is often used in first quantization but rarely in second quantization.

It is said that first quantization along with Special Relativity automatically gives QFT. From this perspective, yes, everything about QM (including entanglement) is included in QFT. But I think there is something else going on with QFT that's not in QM. QFT has the added feature of making the fields obtained in QM into operators which act on the Fock space entities (what are they called, [wave functions]?). This is why is it called second quantization. As I recall, those operator fields of QFT are not even called wave functions anymore; they are operators that no longer give amplitudes. If this is the case, then I don't see how QFT can add wave functions in inseparable ways to describe entanglement. Your instruction would be appreciated.

 

[Demystifier]

It is said that first quantization along with Special Relativity automatically gives QFT. From this perspective, yes, everything about QM (including entanglement) is included in QFT. But I think there is something else going on with QFT that's not in QM. QFT has the added feature of making the fields obtained in QM into operators which act on the Fock space entities (what are they called, [wave functions]?). This is why is it called second quantization. As I recall, those operator fields of QFT are not even called wave functions anymore; they are operators that no longer give amplitudes. If this is the case, then I don't see how QFT can add wave functions in inseparable ways to describe entanglement. Your instruction would be appreciated.

For the relation between field operators and wave functions in QFT, see
http://lanl.arxiv.org/abs/quant-ph/0609163 [Found.Phys.37:1563-1611,2007]
Sec. 8, especially eqs. (74) and (76).

 

[friend]

For the relation between field operators and wave functions in QFT, see
http://lanl.arxiv.org/abs/quant-ph/0609163 [Found.Phys.37:1563-1611,2007]
Sec. 8, especially eqs. (74) and (76).

I like the last sentence of section 8, "Thus, instead of saying that QFT solves the problems of relativistic QM, it is more honest to say that it merely sweeps them under the carpet."

Perhaps the question can be put another way. Is entanglement a phenomena between particles only? Or can "fields" be entangled?

 

[sheaf]

Perhaps the question can be put another way. Is entanglement a phenomena between particles only? Or can "fields" be entangled?

Perhaps the correct question to ask is "can quantum field states be entangled?"
I would say yes, the vacuum state has been described as maximally entangled.

 

[king vitamin]

Entanglement in quantum field theory is currently an active and very interesting field of research. It is often quantified by the entanglement entropy (the von Neumann entropy of the reduced density matrix), but in principle you actually need the whole entanglement spectrum (the eigenvalues of the reduced density matrix) to obtain all info about entanglement, which is an even harder task. The problem is basically solved in 2D CFT due to the usual infinite-dimensional conformal symmetry, and there are many results in free field theories. See http://arxiv.org/abs/hep-th/0405152 for Calabrese and Cardy's original "replica" method for obtaining Renyi/entanglement entropies (as well as many of the original results). Most results for interacting theories are numerical (but I know of some very interesting analytic results). The newest edition of Fradkin's condensed matter textbook has a pretty good discussion of the current picture, which is still very incomplete.

 

[Demystifier]

Perhaps the correct question to ask is "can quantum field states be entangled?"
I would say yes, the vacuum state has been described as maximally entangled.

I agree with the first statement, but I don't think that vacuum is maximally entangled.

 

[Jazzdude]

I agree with the first statement, but I don't think that vacuum is maximally entangled.

Such a statement would be required to specify a decomposition into tensor factors anyway. There is not obvious or natural such decomposition for the vacuum, so the statement as it is appears to be meaningless to me.

 

[Demystifier]

Such a statement would be required to specify a decomposition into tensor factors anyway. There is not obvious or natural such decomposition for the vacuum, so the statement as it is appears to be meaningless to me.

In the case of bosonic fields, a natural basis is the basis of eigenstates of $\phi({\bf x})$. In this basis one can see explicitly that the vacuum (ground state of coupled harmonic oscillators) is entangled, but not maximally entangled. See e.g. the book
B. Hatfield, Quantum Field Theory of Point Particles and Fields

 

[Jazzdude]

In the case of bosonic fields, a natural basis is the basis of eigenstates of $\phi({\bf x})$. In this basis one can see explicitly that the vacuum (ground state of coupled harmonic oscillators) is entangled, but not maximally entangled. See e.g. the book
B. Hatfield, Quantum Field Theory of Point Particles and Fields

That's not exactly what I meant. I was making a statement about a natural factorisation, not a basis that is convenient to work with in your rest frame. Of course if you ask about spatial entanglement this is what you go with, but maybe that's already the wrong question.

 

[friend]

As I recall, a quantized field consists of a various number of particles with various different momentum to make up the field. I think I can understand that the particles of this field all have to be correlated in order to make up that particular field configuration. Is that what is meant to have an entangled field? What I'm not sure of is that any particular field configuration seems arbitrary. Or are you saying that one particular field configuration is somehow entangled with a different field configuration?

 

[Demystifier]

Or are you saying that one particular field configuration is somehow entangled with a different field configuration?

Field at position x is entangled with field at another position x'.

 

[friend]

Field at position x is entangled with field at another position x'.

Are you referring to only one field being involved, and only one particle of that field at x being entangled with only one other particle of that same field at a different point, x'? If so, then how is this different from the usual entanglement between two particles in first quantization?

 

[king vitamin]

As I recall, a quantized field consists of a various number of particles with various different momentum to make up the field. I think I can understand that the particles of this field all have to be correlated in order to make up that particular field configuration. Is that what is meant to have an entangled field? What I'm not sure of is that any particular field configuration seems arbitrary. Or are you saying that one particular field configuration is somehow entangled with a different field configuration?

You can have entanglement with any number of fields (many people just consider 1). The studies I talked about above were usually referring to the entanglement of the vacuum, in which there are no particles. One usually thinks about cutting space in half and asking how one half of space is entangled with the other half. It turns out that the leading contribution is proportional to the area between the two halves you consider.

I suppose you could look at excited field configurations too, but in any case you consider how the state in one region of space is entangled with another. You should be careful not to confuse the field (an operator) with the state, whose entropy you are interested in, but you usually study entanglement using density matrices which are computed via a path integral over fields.

 

[friend]

The studies I talked about above were usually referring to the entanglement of the vacuum, in which there are no particles. One usually thinks about cutting space in half and asking how one half of space is entangled with the other half. It turns out that the leading contribution is proportional to the area between the two halves you consider.

It sounds like you are saying that the properties of the vacuum field somehow correlate to the properties of the space it occupies. Can this be so without the properties of a field at a point being correlated with the properties of the corresponding point in space? Perhaps there is a differential version of the path integral for this correlation between vacuum fields and space.

 

[Demystifier]

Are you referring to only one field being involved, and only one particle of that field at x being entangled with only one other particle of that same field at a different point, x'? If so, then how is this different from the usual entanglement between two particles in first quantization?

I am referring to a field (I am not sure what do you mean by one field), and I do not refer to particles at all. This is different from first quantization precisely because I have entanglement without referring to particles.