How can you test if a bipartite system is entangled?

[ouacc]

say an ensemble consisting of N IDENTICAL bipartite systems. The system can be in either of the two situations.

A) Two particles in a bipartite system are entangled as |psi>= 1/sqrt(2) ( |00> + |11> )
so the density matrix is
rho1 = |psi> <psi| = 1/2 *{1 0 0 1} * {1; 0; 0; 1}
= 0.5* { 1 0 0 1; 0 0 0 0; 0 0 0 0; 1 0 0 1}

B) The bipartite system is in a mixed state. But the initial states of the two particles are the same.
they are either in |00> or |11> with probability 1/2 respectively.
So, the density matrix is
rho2= 0.5 {1 0 0 0; 0 0 0 0; 0 0 0 0; 0 0 0 1}

In summary, there are either N pairs of entangled particles or N pairs of independent but in-the-same-initial-state particles. Assume N is large enough.

How to design measurements on the ensemble, so that I can tell whether it is entangled?

 

[tomasko789]

say an ensemble consisting of N IDENTICAL bipartite systems. The system can be in either of the two situations.

A) Two particles in a bipartite system are entangled as |psi>= 1/sqrt(2) ( |00> + |11> )
so the density matrix is
rho1 = |psi> <psi| = 1/2 *{1 0 0 1} * {1; 0; 0; 1}
= 0.5* { 1 0 0 1; 0 0 0 0; 0 0 0 0; 1 0 0 1}

B) The bipartite system is in a mixed state. But the initial states of the two particles are the same.
they are either in |00> or |11> with probability 1/2 respectively.
So, the density matrix is
rho2= 0.5 {1 0 0 0; 0 0 0 0; 0 0 0 0; 0 0 0 1}

In summary, there are either N pairs of entangled particles or N pairs of independent but in-the-same-initial-state particles. Assume N is large enough.

How to design measurements on the ensemble, so that I can tell whether it is entangled?

So we have:
rho1 = 1/2*(|00>+|11>)(<00|+<11|)= shorter notation =1/2*|00+11><00+11|
rho2 = 1/2*(|00><00| + |11><11|)

Consider observable:
A = 1/2*|00+11><00+11| -1/2*|00-11><00-11|

Tr(A*rho1) = 1/2
Tr(A*rho2) = 0
=> we get different mean values, for more measurements we can distinguish between the two cases.
However, this is just a theoretical observable. If you mean the actual technical realisation of such a measurement on the system I have no idea how to do it.

 

[ouacc]

Thanks. But I couldn't figure out how to get the observables you mentioned in a real experiment.

So we have:
rho1 = 1/2*(|00>+|11>)(<00|+<11|)= shorter notation =1/2*|00+11><00+11|
rho2 = 1/2*(|00><00| + |11><11|)

Consider observable:
A = 1/2*|00+11><00+11| -1/2*|00-11><00-11|

Tr(A*rho1) = 1/2
Tr(A*rho2) = 0
=> we get different mean values, for more measurements we can distinguish between the two cases.
However, this is just a theoretical observable. If you mean the actual technical realisation of such a measurement on the system I have no idea how to do it.