Conditions for entanglement to exist

[neorich]

Hello,

I am interested in the conditions necessary for entangled states to be created. Unfortunately I only have access to introductory QM texts, and they talk about about how entanglement can exists between particles etc, but no mention of the creation of them (and the conditions required for this).

I have also read somwhere that for the entanglement to exists, the particles need to be created by the same source, and conserve angular momentum.

Is this correct and all that is required, or if not, what conditions are required to create entangled states?

Thanks for your help

neorich

 

[lalbatros]

Any interaction between two unentangled (sub-)systems makes (generally) them entangled to a certain extend.
I think this is applies also to what you have in mind.
Maybe you could explain you point of view a little more and check if it is compatible with what I said.

 

[neorich]

Thanks for your reply,

One of the examples given to explain an effect of entanglement is to do with the polarisations of two photons emitted from a source (travelling in opposite directions, although clearly this is just used to emphasize the point), and when you determine the polarisation of one of them, the state of polarisation of the other comes into existence.

But surely this doesn't happen for ANY two photons created anywhere, and independantly, does it? So what conditions are necessary for this to be the case? In other words, what conditions are necessary for two photons (or any particles) to become entangled?

Also, you mentioned about two unentangled sub systems becoming entangled (to a certain extent) when an interaction takes place, can you expand on this?

Thanks

Regards

neorich

 

[lalbatros]

In an unentangled system, measurement probabilities on one part A is independent of probabilities on part B.
But, if the part A and B have an interaction, after some time, this independence disappears.
This is easily seen from the solution of the coupled time-dependent Schrödinger equation:

i h df(a,b)/dt = (Ha + Ha + V) f(a,b)

where V represents the interaction.

If at an initial time f(a,b)=g(a)*h(b), the system is unentangled, and products of probabilities apply for the whole system.

If V=0, no interaction, the absence of entanglement will continue indefinitively.
However, if V =/= 0, the system will -generally- become entangled and the probabilities on each part of the system will not be independen anymore.

However, in special case and for special times the system may come back to an unentangled system. That's what I think more or less intuitively because the interaction can lead to an oscillatory behaviour of the entanglement.

It would be interresting to define a quantity that would represent the entanglement. It would be zero for unentangled systems and different of zero for entangled systems. I don't know if such a measure of entanglement should have an upper limit for some kind of "maximal entanglement".

With such a measure defined, one could solve the Schrödinger equation and see the evolution of the entanglement measure. Maybe we could see some oscillation.

What do you think?

I have often asked about "entanglement measures" but had few answers.
I found some litterature on that but was not satisfied (maybe no suitable for my hobby time!).
Could such an "entanglement measure" not be defined from elementary probability theory?

 

[neorich]

Thanks lalbatros,

It is interesting you talk about an oscillating entanglement behaviour, I had not really thought about this, only varying degrees of entanglement dependent on the conditions of the system, and (generally) decreasing with time.

It certainly would be useful to have a quantity we could deal with, but to use it in the Schrödinger equation, would this indicate that it would have some relationship to the energy of the system?

Are there any books you could recommend which deal with entanglement, either qualitatively or quantitatively (or preferably both).

Thanks again

Regards

neorich

 

[lalbatros]

By far the most "normal" behaviour is a decrease of entanglement.
It is difficult to keep a system entangled since interactions with the outside world will entangle the system with the world, which amount to blurr the initial entanglement.

There are certainly many books and many papers delaing with the subject.
I have not really found something like a review paper or a textbook dealing with the subject, but I will look for it. Maybe I have some luck ...