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Bipartite state purity

  1. Nov 12, 2015 #1
    1. The problem statement, all variables and given/known data

    Consider the bipartite state:

    |q> = a/sqrt (2) (|1_A 1_B> +|0_A 0_B>)+sqrt((1-a^2)/2) (|0_A 1_B>+|1_A 0_B>)

    where a is between or equal to 0 and 1

    a) compute the state of the subsystem p_b

    b) compute the purity of p_b as a function of a

    c) for what values of a is the purity of p_b at a minimum/maximum

    d) Compute the entanglement entropy of the bipartite state, for what value of a is it at a min/max


    3. The attempt at a solution

    I have done a) and found p_b to be :

    p_b = |q_b><q_b|

    where |q_b> = (a+sqrt(1-a^2))/sqrt(2) (|0_B>+|1_B>)

    the computation for this is long and I don't want to replicate it here...

    b)

    I simply applied:

    p (0_B,0_B) = <0_B|p_b|0_b> for the 4 combinations of |0_b> and |1_b> and got the matrix, then taking the trace of this for the diagonal terms I get 1/2 and 1/2 so the purity was equal to 1 and thus not dependent on a

    c) It is unrelated

    d) I know that the equation is

    S (p_AB) = S (Tr_B p_AB)

    but I don't know how to proceed from here

    Any help such as how to proceed or checking my previous steps would be greatly appreciated!
     
  2. jcsd
  3. Nov 15, 2015 #2
    Could anybody help with this please :smile:
     
  4. Nov 16, 2015 #3

    DrClaude

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    Staff: Mentor

    How is purity defined?
     
  5. Nov 16, 2015 #4
    In my notes purity is defined as the square of the trace of a state's density operator, it is between 1/d and 1 where d is the dimension of the Hilbert space. P = 1 for pure states. The states with maximal classical uncertainty (maximally mixed states) have the minimum possible purity P = 1/d, we say that the state is 'maximally mixed'.
     
  6. Nov 17, 2015 #5

    DrClaude

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    You have a small mistake there. ##\mathrm{Tr}(\rho)## gives you the normalization of ##\rho##, so you will always get ##\mathrm{Tr}(\rho)^2 = 1##. To calculate the purity, you need to take the trace of the square of the density operator, ##\mathrm{Tr}(\rho^2)##.
     
  7. Nov 17, 2015 #6
    Ah, that makes it work!

    Alright so the only part I am confused about then is part d) and how to compute the entanglement entropy mathamatically
     
  8. Nov 17, 2015 #7

    DrClaude

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    Staff: Mentor

    Your missing an equation for calculating the entropy of a density matrix.
     
  9. Nov 17, 2015 #8
    Yes I know. That is what I am searching for, I do not have it in my notes and there is no textbook :(

    Do you know this equation?
     
  10. Nov 17, 2015 #9

    DrClaude

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    Staff: Mentor

  11. Nov 17, 2015 #10
    Thank you for the equation and the link!

    Hmm I am getting a negative entropy for some values of a

    How come you posted it as log 2 but it simply says log, presumably log 10 on the site?
     
  12. Nov 17, 2015 #11

    DrClaude

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    Staff: Mentor

    I put the 2 there to be explicit, as I guessed this would cause confusion. In this field, log always mean the logarithm in base 2.
     
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