How Does Distance Affect Entanglement Between Quantum Particles?

[friend]

How close must two particles get in order to become entangled? My guess would be that the polarizations of the vacuum due to the screening effect must overlap. Otherwise, they can't be said to be interacting. Is this right? Is the amount of entanglement proportional to the amount or strength of the overlap between the two screen? Do some pairs of particles not entangle because they don't interact?

 

[bhobba]

How close must two particles get in order to become entangled?

You are ascribing the property of distance that it doesn't have until observed to have it. It occurs due to the distance parameter in the Hamiltonian.

My guess would be that the polarizations of the vacuum due to the screening effect must overlap.

Its been explained before that is a wrong picture.

Otherwise, they can't be said to be interacting. Is this right? Is the amount of entanglement proportional to the amount or strength of the overlap between the two screen? Do some pairs of particles not entangle because they don't interact?

Entanglement is indeed an interaction. Or some particles can be created entangled.

Thanks
Bill

 

[friend]

Entanglement is indeed an interaction. Or some particles can be created entangled.

Does every interaction between particles create entanglement between those particles?
When can it be said that particles interact, when their wavefunctions overlap?

 

[bhobba]

Does every interaction between particles create entanglement between those particles?
When can it be said that particles interact, when their wavefunctions overlap?

No.

That depends on the Hamiltonian.

Thanks
Bill.

 

[friend]

All I need is one counterexample of two particles that do interact but do not get entangled. Is there such a thing? Thanks.

 

[DrChinese]

Entanglement, as bhobba, explains, is due to technical considerations. As a general rule, there is an interaction involved and the output particles must be indistinguishable.

However, this is not a strict requirement and there are circumstances in which entangled particles have never interacted. Hopefully this will help you to see that drawing a simple mental picture is only marginally beneficial.

http://arxiv.org/abs/1209.4191

 

[friend]

All I need is one counterexample of two particles that do interact but do not get entangled. Is there such a thing? Thanks.

According to this Wikipedia article on Entanglement, entanglement occurs when there are none-zero cross terms between states of two different Hilbert spaces. But aren't these non-zero cross terms the same as coupling constants between interacting fields? If so, it seems that interaction and entanglement always coincide. Or am I confusing quantum mechanical effects with QFT effects?

 

[friend]

What happens when the two entangled particles are far apart? Wouldn't the cross terms be zero since the particles are so far apart? The coefficients of the cross terms would still be non-zero. But the separate wave-functions from the different Hilbert spaces would have little if any overlap, and their product would be zero, right? So where is entanglement then if the cross terms are essentially zero?

 

[DrChinese]

What happens when the two entangled particles are far apart? Wouldn't the cross terms be zero since the particles are so far apart? The coefficients of the cross terms would still be non-zero. But the separate wave-functions from the different Hilbert spaces would have little if any overlap, and their product would be zero, right? So where is entanglement then if the cross terms are essentially zero?

I can't speak to the notation. But when there is multi-particle entanglement, there is no distance (or time) component. There is just one system. There are no separate wave functions in such case.

 

[friend]

I can't speak to the notation. But when there is multi-particle entanglement, there is no distance (or time) component. There is just one system. There are no separate wave functions in such case.

A person measuring an entangled particle on the other side of the universe may not know that it has an entangled partner. He sees only the particle behaving as if it has its own separate wave-function.

 

[DrChinese]

A person measuring an entangled particle on the other side of the universe may not know that it has an entangled partner. He sees only the particle behaving as if it has its own separate wave-function.

That is correct. It is an entangled system until an observation is performed on one of the entangled bases. Distance is not a factor in any manner in the observed results.

To put it another way: gravitational attraction of 2 objects varies with the inverse square of their separation. There is nothing like that with entanglement.

 

[friend]

It seems entanglement survives distance and velocity. Does it survive acceleration? Entanglement is usually described as bringing two particles together so that their wave functions mix and become entangled. Then measuring one affects how the other will be measured. This is said to be the case no matter how far apart the two entangled particles are. And there are cases where two photons can be entangled even though they are headed off in opposite directions. So entanglement survives great distance and zero velocity. And it also survives when the particles are traveling at the speed of light. So I suppose it survives any velocity between zero and "c".

But I have to wonder about acceleration. In order to accelerate any particle one has to give it energy, which requires an interaction with other particles. And wouldn't this interaction destroy the entanglement? So how much acceleration does it take to destroy entanglement, if any? Is there a measure of entanglement based on acceleration? And as far as that goes, how do they separate the two entangled particles to great distances apart after they were brought together to entangle them to begin with? If they are separated after being together, wouldn't that require acceleration? Or do they get entangled as they pass each other at constant velocity?

 

[DrChinese]

But I have to wonder about acceleration. In order to accelerate any particle one has to give it energy, which requires an interaction with other particles. And wouldn't this interaction destroy the entanglement? So how much acceleration does it take to destroy entanglement, if any?

Your premise is incorrect. All interactions do not destroy entanglement. There are lots of interactions in which entanglement survives. For example, entangled particles moving through fiber are being accelerated (as they change direction) and they remain entangled in spin and frequency. But during those interactions, entanglement may be lost on one or more bases.

The issue of acceleration due to gravity is complicated by the question of whether gravity is quantized. As a general rule, you can ignore that question (ignore gravitational considerations).

 

[jfizzix]

If two particles are interacting with one another, they can in principle become entangled, though that would depend on the nature of the interaction.
Similarly, interaction can destroy entanglement as well (the Schrödinger equation is time-symmetric).
It is even possible for two particles to interact in such a way that they entangle and disentangle over and over again.
This can be accomplished by say, a pair of spin-1/2 particles magnetically coupled to each other.

If two independent particles don't interact with each other, there is no way for them to become entangled, unless they can interact with a separate pair of already entangled particles.
Ultimately, the creation of the original entanglement requires interaction.

If two particles become entangled, and then no longer interact with each other or anything else, they stay entangled, even if they travel very far apart.
Indeed, the amount of entanglement would be exactly the same as it was just as the interaction stopped.

Interaction with a third party need not destroy entanglement (as Dr Chinese points out), but the more either particle becomes entangled with the "environment", the less the particles can be entangled with each other.
This is known as the monogamy of entanglement.

 

[friend]

Interaction with a third party need not destroy entanglement (as Dr Chinese points out), but the more either particle becomes entangled with the "environment", the less the particles can be entangled with each other.
This is known as the monogamy of entanglement.

Is there a measure of entanglement, or are particles either totally entangled or not? Does it depend on how deeply their wave functions have overlapped? How does one calculate the amount of entanglement? Is it measured by the entropy of entanglement? Thanks.

 

[jfizzix]

Is there a measure of entanglement, or are particles either totally entangled or not? Does it depend on how deeply their wave functions have overlapped? How does one calculate the amount of entanglement? Is it measured by the entropy of entanglement? Thanks.

On the subject of measuring entanglement, you may find this insights article I wrote helpful.
https://www.physicsforums.com/insights/measuring-quantum-entanglement/

 

[Nugatory]

According to this Wikipedia article on Entanglement, entanglement occurs when there are none-zero cross terms between states of two different Hilbert spaces. But aren't these non-zero cross terms the same as coupling constants between interacting fields?

They're pretty much completely unrelated. These lecture notes are a good explanation of what the "cross-terms" here are.

 

[friend]

Thanks, guys for the articles. That was interesting reading. What I got from all that is that entanglement exists when there are cross terms between the wave functions from different Hilbert spaces. What I'm still wondering is how that differs from interaction between particles. There are cross terms that serve as coupling constant in the Lagrangian for two interacting fields. Is that a form of entanglement? Or am I comparing apples with oranges? Do each of those coupled fields belong to its own Hilbert space? Or is it Fock space? It's confusing because a particle can be described with a wave function of a Hilbert space. But the same particle can also be described as an excitation of a field (in Fock space?).

 

[friend]

It's confusing because a particle can be described with a wave function of a Hilbert space. But the same particle can also be described as an excitation of a field (in Fock space?).

Perhaps it would help to think in terms of whether Hilbert spaces are included in Fock space. From the Wikipedia article on Fock space, it says,

"Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H,
{F_\nu }(H)\,\, = \,\,\overline { \oplus _{n = 0}^\infty {S_\nu }{H^{ \otimes n}}}
Here {{S_\nu }} is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic (\nu = +) or fermionic (\nu = -) statistics, and the overline represents the completion of the space."

So if you have two interacting fields in the Lagrangian multiplying each other with a coupling constant, does that represent two separate Fock spaces? Does that mean that the coupling constant between the fields gets passed on to a coupling between the two wave functions in the two Hilbert spaces? That would indicate that whatever interacts necessarily also gets entangled. I'm looking for advice on the math here. It would seem that if Fock space is made up of Hilbert space, then operations on Fock space get passed on (in some form) to the Hilbert space that makes up the Fock space. Any insight into the algebra of all this is appreciated. Thanks.