On General Measure of Entanglement

[krete]

On General Measure of Entanglement


Hi, sirs
My major is computer science. While I currently need to deal with some physical problems. Thank you very much for your help.

It is well-known that Von Neumann entropy is a measure of entanglement for bipartite pure states. However, I think that the effectiveness of this measure depends on the existence of the Schmidt decomposition for bipartite pure states. If we want to measure multipartite entanglement, it seems that the interpretation of Schmidt decomposition can not be expanded directly. Might you kindly tell me some messures on multipartite entanglement? Thanks a lot!

 

[soarce]

Hi krete,

As far I know, this is still an open question. When we have multi-particle systems a lot of possibilities to be entangled: for example, one particle with other two particles, two particles with other two particles ... and so on. Although there is some work on systems with three and four particle, it is not an easy task to extrapolate the results to many-particle systems in general.

 

[krete]

Dear soarce,
Thank you very much for your kind suggestions.
I have another question. I think that the quantum entanglements and classically statistical dependence is equivalent if measurements have been finished. To be specific, if we repeatedly measure an ensemble of a pure state |psi> = a|00> + b|01> + c|10> + d|11>, where |a|^2+|b|^2+|c|^2+|d|^2=1 and count the events of |00>, |01>, |10> and |11>. We will find the following relation holds:
Pr(ij)=Pr(i)*Pr(j), i,j=0,1, if and only if |psi>=(x0|0>+x1|1>)*(y0|0>+y1|1>) (* stands for tensor product)
It my intuition right? Thanks.

 

[soarce]

Dear soarce,
I think that the quantum entanglements and classically statistical dependence is equivalent if measurements have been finished. To be specific, if we repeatedly measure an ensemble of a pure state |psi> = a|00> + b|01> + c|10> + d|11>, where |a|^2+|b|^2+|c|^2+|d|^2=1 and count the events of |00>, |01>, |10> and |11>.

I am not sure if I undestrood what are you saying. Do you try to establish some procedure to measure the amount entanglement, or just to detect it (without saying anything about its amount) ?

You have an unknown state |psi> and you perform a series of measurements Pr(ij), Pr(i), Pr(j), where i,j=0,1. Then you check the relation:

We will find the following relation holds:
Pr(ij)=Pr(i)*Pr(j), i,j=0,1, if and only if |psi>=(x0|0>+x1|1>)*(y0|0>+y1|1>) (* stands for tensor product)

(This is true, one can check it using density matrix formalism.)

If this relation holds then the state is not entanglet.


Remark:
i) When measuring Pr(i) ( or Pr(j) ) one has to project the other state onto some linear combination of |0> and |1>, just to trace out its outcome.

ii) If you are interested in studying composite systems I recommend you the book of A. Peres, "Quantum Theory: Concepts and Methods". It is not an easy one, but provides you the basics of correlated quantum systems.

 

[soarce]

Just a short remark on the relation you wrote

Pr(ij)=Pr(i)*Pr(j) for all i,j=0,1, if and only if |psi>=(x0|0>+x1|1>)*(y0|0>+y1|1>)

The lefthand side must be verified for all combinationas of i and j; whatever observable we measure we will find that the subsystem are unentagled.

 

[krete]

I am not sure if I undestrood what are you saying. Do you try to establish some procedure to measure the amount entanglement, or just to detect it (without saying anything about its amount) ?

I just want to detect it.

If you are interested in studying composite systems I recommend you the book of A. Peres, "Quantum Theory: Concepts and Methods". It is not an easy one, but provides you the basics of correlated quantum systems.

Thanks very much for your suggestion. I will try to find the book in the library.

The lefthand side must be verified for all combinationas of i and j; whatever observable we measure we will find that the subsystem are unentagled.

Sure, you are exactly correct.


Actually, I have to deal with the problem measuring pure k-partite entanglements (I admit this problem might not be well-defined since the meaning of 'pure' is not very clear. Roughly speaking, pure k-partite entanglements can not be reduced to lower-order entanglements). I find a method of classical statistics, which might solve this problem. But it seems that we should demonstrate some equivalence between entanglements and classical dependences before we can apply a classical method to describe entanglements.

 

[soarce]

I came across this references on multiparticle entanglement:

http://www.iop.org/EJ/abstract/0305-4470/34/35/310 (from 2001, rather old)

http://www3.interscience.wiley.com/cgi-bin/summary/117903263/SUMMARY?CRETRY=1&SRETRY=0

If you search "detecting multiparticle entanglement" by google, you will find a lot of references.

 

[krete]

I came across this references on multiparticle entanglement:

http://www.iop.org/EJ/abstract/0305-4470/34/35/310 (from 2001, rather old)

http://www3.interscience.wiley.com/cgi-bin/summary/117903263/SUMMARY?CRETRY=1&SRETRY=0

If you search "detecting multiparticle entanglement" by google, you will find a lot of references.

got it, thanks a lot:)!