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Homework Help: Entanglement, projection operator and partial trace

  1. Jan 9, 2016 #1
    1. The problem statement, all variables and given/known data
    Consider the following experiment: Alice and Bob each blindly draw a marble from a vase that contains one black and one white marble. Let’s call the state of the write marble [itex]|0〉[/itex] and the state of the black marble [itex]|1〉[/itex].
    Consider what the state of Bob’s marble is when Alice finds a white marble
    2. Relevant equations

    3. The attempt at a solution
    So I found the mixed state of Bob and alice's particle to be:
    [tex] \rho=\frac{1}{2}|0,1\rangle \langle0,1|+\frac{1}{2}|1,0\rangle \langle1,0|[/tex]
    And i know that finding a white marble can be described in the following way:
    [tex] \rho^B=\frac{Tr_A(|0\rangle_A\langle0|\rho)}{Tr(|0\rangle_A\langle0|\rho)} [/tex]
    where [itex]Tr_A [/itex] is the partial trace w.r.t Alice's system.
    And just by reasoning i know the answer should be [itex]|1\rangle\langle1|[/itex] but im struggling to prove that by solving the above equation.
    Here's my attempt:
    [tex] \rho^B=\frac{Tr_A(|0\rangle_A\langle0|(|0,1\rangle \langle0,1|+|1,0\rangle \langle1,0|))}{Tr(|0\rangle_A\langle0|(|0,1\rangle \langle0,1|+|1,0\rangle \langle1,0|))} [/tex]
    [tex] \rho^B=\frac{Tr_A(|0\rangle_A\langle0|(|0\rangle \langle0| \otimes |1\rangle \langle1|+|1\rangle \langle1|\otimes|0\rangle \langle0|))}{Tr(|0\rangle_A\langle0|(|0\rangle \langle0| \otimes |1\rangle \langle1|+|1\rangle \langle1|\otimes|0\rangle \langle0|))} [/tex]
    But im not quite sure where to go from there, im a little inexperienced using Braket notation so any pointers would be greatly appreciated.
  2. jcsd
  3. Jan 14, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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