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Is partial trace still cyclic?

  1. Sep 11, 2008 #1

    I know trace is usually cyclic, but is partial trace cyclic too? Why?


  2. jcsd
  3. Sep 11, 2008 #2
    Ok... so I know that it isn't cyclic now ... just by picking a random example, but if anyone knows the reason why it's not cyclic, and has a general proof as to why it's not, I'd be very grateful to hear it!

  4. Sep 12, 2008 #3


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    What's the definition of a partial trace?
  5. Sep 12, 2008 #4
    Normal trace is equivalent to the sum of the eigenvalues (or diagonal elements) of a matrix. Partial trace acts only on part of the system, so for a density matrix.. say it's a pure state but entangled,

    [tex]\rho_{AB}=\frac{1}{2}(|01\rangle +|10\rangle )(\langle 01|+\langle 10 |)[/tex]

    The partial trace over subsystem B gives the reduced density matrix [tex]\rho_A[/tex], so [tex]Tr_B(\rho_{AB})=\rho_A[/tex]


    [tex]\rho_A=_B\langle 0 |\rho_{AB}|0\rangle_B +_B\langle 1 |\rho_{AB}|1\rangle_B[/tex]
    [tex]\rho_A=|1\rangle \langle 1 | + |0\rangle \langle 0 | [/tex]

    My question is how do I prove that

    [tex]Tr_B (\rho \sigma) = Tr_B (\sigma \rho)[/tex]

    where [tex]\rho[/tex] and [tex]\sigma[/tex] are both density matrices of a system AB.

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