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Homework Help: Quantum mechanics: density matrix purification

  1. Jun 22, 2011 #1
    1. The problem statement, all variables and given/known data

    Given a matrix

    M(a) = (a -(1/4)i ; (1/4)i a)

    (semicolon separates rows)

    a) Determine a so that M(a) is a density matrix.
    b) Show that the system is in a mixed state.
    c) Purify M(a)

    3. The attempt at a solution

    a) from conditions for a density matrices

    1) M(a)=M(a)*
    2) tr(M(a))=1
    3) M(a)>=0

    form 1) a must be real
    from 2) a=1/2
    I'm not sure what 3) means, but if it means trace and determinant
    must be non-negative, than this is also fulfilled if a=1/2.

    b) I think that condition for system to be in mixed state is:
    M(a)^2 /= M(a)
    since this is true for M(a), system is in mixed state.

    c) Don't know how to solve this part, Maybe it has to do something with decomposing a matrix ?
  2. jcsd
  3. Jun 22, 2011 #2


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    M(a)≥0 means xM(a)x≥0 for all x.
  4. Jun 23, 2011 #3
    if I take x to be a vector with components (x y) then,
    x†M(a)x = a(xx† + yy†) + (1/4)i(xy† + x†y)

    this then means a(xx† + yy†)≥(1/4)i(xy† + x†y)
    but term on the left hand side is real and term on the right hand side may be complex,
    so how can I conclude when is x†M(a)x≥0 ?

    also is my answer to b) correct ?
    and if anyone would give me a little help on c)
  5. Jun 23, 2011 #4


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    First, you made a sign error. Second, note that x*y and xy* are conjugates.

    Your answer to (b) is correct.

    I'm not sure what they're asking for in (c).
    Last edited: Jun 23, 2011
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