What are some examples of imperfect entanglement in spin 1/2 systems?

[James MC]

Hi, I'm wondering what is meant by degrees of entanglement, and am looking for a simple concrete example. Here's a guess at an example, and then a more general definition based on it:
Examples of perfect entanglement would be the Bell singlet states. To use a position basis example, for two particles confined to one dimension we might have perfect entanglement:

c1(|x1>₁|x10>₂) + c2(|x2>₁|x11>₂)

Now for imperfect entanglement or "less" entanglement we might have:

c1|x1>₁(c2|x10>₂+c3|x11>₂) + c4|x2>₁(c2|x10>₂+c3|x12>₂)

So in this example I'm thinking that if you measure particle 1 and collapse it position x1, then you don't fully collapse particle 2; you only collapse it to a superposition of x10 and x11; similarly if you measure 1 and collapse it to x2, you don't fully collapse particle 2, you only collapse it to a superposition of x10 and x12.

So then general definition of degrees of entanglement would be "the more entangled the particles are (in basis B) the more that a collapse of one particle (in B) will lead to a more confined collapse (i.e. reducing superposition spread) of the other particle (in B).

Does my example make sense (not sure if the c#'s really work out here)?
Does the example illustrate the idea of degrees of entanglement?
Do you know of better simple illustrations of the idea?
What are some realistic illustrations of the idea?

Interested in your thoughts!

 

[naima]

look at the curve in entropy
you move between two unentangled states (null entropy) and in the middle there is a Bell state (entropy = 1) which is maximally entangled.

 

[James MC]

look at the curve in entropy
you move between two unentangled states (null entropy) and in the middle there is a Bell state (entropy = 1) which is maximally entangled.

Thanks but if I could understand your link then I would have little need to ask this question. Setting aside very general complex measures (e.g. using entropy) of entanglement degrees, I would just like to see a straightforward example of imperfect entanglement. For example I'm wondering if the simple one dimensional space example I offered is an example.

 

[kith]

The idea is the following. We measure an observable O₁ on particle 1 and an observable O₂ on particle 2. We then look at the correlations of the outcomes we get for pairs of observables {O₁, O₂}. If we can find a pair of observables with no correlation of outcomes at all, we have no entanglement. Else, if we can find a pair, where all the outcomes are perfectly (anti-)correlated, we have perfect entanglement. In between these extremes, we have imperfect entanglement (which can be quantified by the entropy of the reduced density matrices).

If possible, we often use the same observable for both particles. This covers many situations, but not the example of which you think that it describes imperfect entanglement. This state is also maximally entangled. The reason is that all your states are orthogonal to each other. You can easily construct an observable for particle 2 which yields perfect correlations.

Position states are a more advanced example because the Hilbert space is infinite-dimensional (and strictly speaking, the position states don't even live in a simple Hilbert space at all). Is is easier to talk about two spin 1/2 systems first.

Now try to construct a state with imperfect entanglement for two spin 1/2 systems.