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How can you test if a bipartite system is entangled?

  1. Mar 17, 2008 #1
    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?
     
  2. jcsd
  3. Mar 17, 2008 #2
    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.
     
  4. Mar 17, 2008 #3
    Thanks. But I couldn't figure out how to get the observables you mentioned in a real experiment.

     
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