The cause of entanglement / coupling

[Talisman]

The "cause" of entanglement / coupling

Hi all, two questions. I don't have a physics background (but have a decent mathematical background, and have dabbled in QM).

* What is the difference between entanglement and coupling? Wikipedia says they're "almost" the same thing, except for the distance at which they occur. Mathematically, I guess they're the same (i.e., represented by a non-separable joint state)?

* What are various possible causes of entanglement? Examples that come to mind:
1) Pair production -- byproduct of conservation laws.
2) Entanglement with measurement apparatus -- intuitively, that the device (if it's accurate) must indicate the result corresponding to the eigenstate it found the particle in.

Are there other canonical examples?

 

[Hawkwind]

* What is the difference between entanglement and coupling? Wikipedia says they're "almost" the same thing, except for the distance at which they occur.

Hi,

I wouldn't say so, I think that these are very different "mechanisms": coupling is the consequence of an interaction. Think for instance of coupled harmonical oscillators.

Entanglement, on the other hand, ist not based on an interaction. The "classical" example is indeed a system of 2 components, which are entangled as the consequence of a conservation law (e.g. conservation of total angular momentum).


Interactions between 2 coupled obects propagate with the speed of the interaction (e.g. "c" ) and are always accompanied by an exchange of momentum an energy.

Entanglement results in a specific forecast for a set of measurements on a statistical ensemble of identical experiments; the prediction is of the kind: if you measure "1" on object A, then you will measure "0" on object B - a correlation. No exchange of energy, momentum or information of any kind between the components is necessary to establish this correlation.

 

[tom.stoer]

Entanglement is a purely mathematical property which can be formulated generically on a Hilbert space. Let's assume we have a two-particle Hibert space and we have two state vectors. An entangled state looks like

$$|\psi\rangle = |A\rangle|B\rangle+|B\rangle|A\rangle$$

That means it is a linear combinbation of a vector where the first particle is in state A and the second particle is in state B with a vector where the first particle is in state B and the second particle is in state A.

This property of the system consisting of the two particles in completely agnostic for any interaction between the two particles and applies even if there is no such interaction, i.e. for free particles.

 

[Hawkwind]

Entanglement is a purely mathematical property which can be formulated generically on a Hilbert space.

I wonder what do you want to indicate by the phrase "purely mathematical property" ?
Entanglement is physics: it is the nonlocal character of quantum mechanics.

 

[tom.stoer]

I wonder what do you want to indicate by the phrase "purely mathematical property" ?
Entanglement is physics: it is the nonlocal character of quantum mechanics.

I fully agree. I simply mean that you need no "physical ingredients" like fields, interactions, Hamiltonian and all that stuff.

 

[jambaugh]

Entanglement is "simply" correlation, i.e. a previously established relationship between observables of two systems. In this sense it is no different from classical correlation i.e. if an asteroid of mass M breaks into two pieces the mass of one is M minus the mass of the other.

The peculiarity we find in quantum correlation wherein we give it a special name is that you can know more about the correlation between systems than you know about each individual system itself. In particular you can correlate all variables including variables which cannot be simultaneously observed. e.g. correlate both momentum and position for a pair of particles.

 

[Talisman]

Entanglement is a purely mathematical property which can be formulated generically on a Hilbert space. Let's assume we have a two-particle Hibert space and we have two state vectors. An entangled state looks like

$$|\psi\rangle = |A\rangle|B\rangle+|B\rangle|A\rangle$$

That means it is a linear combinbation of a vector where the first particle is in state A and the second particle is in state B with a vector where the first particle is in state B and the second particle is in state A.

This property of the system consisting of the two particles in completely agnostic for any interaction between the two particles and applies even if there is no such interaction, i.e. for free particles.

Hmm... so I'm totally comfortable with the mathematical properties of entangled states, but it sounds like coupling is unrelated. It could be that the http://en.wikipedia.org/wiki/Quantum_coupling" is misleading when it says that it is "a state similar to entanglement." In any case, I guess it's not as relevant to my question as I thought.

For part 2, I am looking for situations that are known to lead to entanglement. States prepared in the laboratory are interesting (e.g., SPDC), as are those governed by basic laws (as in conservation, in spontaneous pair production), as well as more "implicit" cases (e.g., a measuring device is only accurate by definition if its pointer states are entangled with the eigenstates of the observable we seek to measure).

Thanks for the answers so far!

 

[Zarqon]

I would say causing entanglement boils down to one thing: indistinguishability.

An example, consider two atoms close to each other and a single photon coming in. If you are somehow able to detect that the photon was indeed absorbed, but the atoms are much closer to each other than the spread of the photon wavepacket, then you get the entangled state |ge>+|eg>, with g, e meaning ground and excited state respectively, because you cannot tell in which atom the photon was absorbed, i.e. they are indistinguishable in this aspect.

Another example is spontaneous parametric down conversion, as you mentioned. In this case you have two non-linear crystals after each other, but at different angles. In this case, just as in the previous example, the trick is to make sure they are not separated more than the photon wavepacket, such that it's not possible to tell in which of the two crystals the down conversion took place, i.e. the two crystals are indistinguishable in this regard. Because of the different angles of the crystals this leads to photons coming out of the two-crystals setup being entangled in polarization.

 

[tom.stoer]

... but it sounds like coupling is unrelated. It could be that the http://en.wikipedia.org/wiki/Quantum_coupling" is misleading when it says that it is "a state similar to entanglement."

I don't know; the text is not very clear b/c they say that

... it is a state similar to quantum entanglement but whereas quantum entanglement can take place over long distances quantum coupling is restricted to quantum scales.

and

While the individual particles may fluctuate their values, the quantum states of the two qubits remain locked in relation to each other, via Coulomb force.

The first statement seems to mean that this is essentially quantum entangement w/o the need for any coupling, whereas the second statement says that there is a Coulomb force. It's confusing.

Perhaps it is somehow related to the exchange interaction http://en.wikipedia.org/wiki/Exchange_interaction which contains both a potential and an exchange term. In that sense it could be an additional effect caused by both entanglement and the presence of an interaction.

 

[Hawkwind]

Perhaps it is somehow related to the exchange interaction http://en.wikipedia.org/wiki/Exchange_interaction which contains both a potential and an exchange term. In that sense it could be an additional effect caused by both entanglement and the presence of an interaction.

But the discussion there relates to magnetic moments: entanglement is a far more general effect, which concerns electrically neutral components as well (for instance photons).

I don't see so much similarity between quantum entanglement and couplings by physical interactions.

Nevertheless, it's true that quantum statistics can result in effective interactions (exchange interaction, degeneracy pressure).

 

[tom.stoer]

I know that entanglement has nothing to do with interactions (see post #3).

I only tried to find out what Wikipedia means by "quantum coupling" which seems to need some kind of interaction. Unfortunately the Wiki article does not cite any reference from which this could become clear.

 

[haael]

The problem with the entanglement/superposition is that it requires new philosophical paradigm. Namely it requires the "OR" logic, the idea that logical alternative may be physically realized.

The classical physics supports only the "AND" logic. Suppose there is a box and 2 object: an apple and a banana. Classical physics allows the following states:
* An apple is in the box.
* A banana is in the box.
* An apple AND a banana are in the box.

The quantum physics introduces the new state:
* An apple OR a banana is in the box.

Of course, in the classical physics we also can assert the last sentence, but it is assumed that "in reality" only the 3 former are physical.

Let's define:
A = "An apple is in the box."
B = "A banana is in the box."
P(...) = "It is physically realized, that ..."

Then in classical physics:
A or B => P(A) or P(B)

We may not know if an apple or a banana is in the box, but God knows the Truth, that "in reality" either an apple, a banana or both are in the box.

In quantum physics it is no longer the case. The sentence "A or B" may indeed be physically realized. It is the Truth, the "reality".
A or B => P(A or B)

Most interpretations of QM try to translate this somehow into "AND" logic. The reality in this view is still classical, no physically realized alternatives, at the cost of nonlocality, superluminal communication and so on.

 

[tom.stoer]

One should add that "A OR B" here really means "A or B" and not "EITHER A OR B".

 

[jambaugh]

I would say causing entanglement boils down to one thing: indistinguishability.

I don't think that is so much a criterion but a convenient method.

Causing entanglement is simply a matter of measuring (or preparing) some correlative observable. For example determining that the total spin of two half spin particles is 0 automatically makes their spins entangled. They needn't be identical.

Interaction does play a part in entanglement but only in the preparation stage.

 

[The_Duck]

* What are various possible causes of entanglement? Examples that come to mind:
1) Pair production -- byproduct of conservation laws.
2) Entanglement with measurement apparatus -- intuitively, that the device (if it's accurate) must indicate the result corresponding to the eigenstate it found the particle in.

Are there other canonical examples?

I think pretty much any nontrivial quantum mechanical system is going to exhibit entanglement. E.g. shoot a wave packet at another wave packet. After the collision the particle states are entangled because while there are many possible outgoing trajectories for each particle, there is a strong correlation between the outgoing trajectory of particle A and the outgoing trajectory of particle B. For example, inasmuch as the initial state had a definite total energy and momentum, measuring the energy and momentum of one outgoing particle tells you the energy and momentum of the other by conservation.

 

[Talisman]

I think pretty much any nontrivial quantum mechanical system is going to exhibit entanglement. E.g. shoot a wave packet at another wave packet. After the collision the particle states are entangled because while there are many possible outgoing trajectories for each particle, there is a strong correlation between the outgoing trajectory of particle A and the outgoing trajectory of particle B. For example, inasmuch as the initial state had a definite total energy and momentum, measuring the energy and momentum of one outgoing particle tells you the energy and momentum of the other by conservation.

Thanks! In this case, conservation of energy and momentum are the "mechanism" again. I have no trouble with the math or "philosophy" of entanglement, but my physics is lacking. I suppose a remedial course would do me some good :)

 

[The_Duck]

In this case, conservation of energy and momentum are the "mechanism" again.

I'm going to say something stupid if I talk too long, but: I'd say that here the mechanism of entanglement is the interaction between the particles, which happens to conserve energy and momentum. This fact of conservation just makes it particularly easy to state certain correlations between the outgoing particles. Even if the interaction took place in some potential that varied in time in space, so that energy and momentum weren't conserved, there would still be correlations between the outgoing particles, which is entanglement.

 

[jambaugh]

Heisenberg's uncertainty principle comes into play here in that correlation observations do not commute with factor observables so one cannot simultaneously observe the stronger quantum correlations and simultaneously observe and thus predict independent observables with respect to each factor system. [By factor system I mean e.g. either half of an entangled pair since the composite of the two is described by a tensor product of the descriptions of the parts. ]

An entangled pair of systems is a pair of systems for which a correlation has been observed so that by the generalized HUP one cannot know how either factor system will behave w.r.t. any of its observables. Just as you cannot simultaneously observe say position and momentum, you cannot simultaneously observe correlation of both in a pair of particles and either position or momentum for either half of the pair.

I think the main quantum peculiarity is that although we cannot simultaneously measure e.g. position and momentum of a single system we can observe a correlation of both in a composite system by virtue of the fact that say the sum of momenta is independent and commuting with the differences in positions. Hence we can observe simultaneously that the sum of momenta is 0 and the difference in positions is 0.

Note that with entanglement, entropy is no longer additive but subadditive. The (VonNeumann) entropy of the composite is less than the sum of the entropies of the parts. I find this very helpful in understanding both quantum entropy and entanglement as this subadditivity is intimately connected with entanglement. I believe you can define the entropy of a system to be the degree to which it is entangled with the environment.