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Homework Help: Pure state, mixed state and measurement

  1. Jun 7, 2014 #1
    Hello guys,

    1. The problem statement, all variables and given/known data

    the problem goes as follows:

    "Which measurement should you do on a statistical ensemble of qubits in order to distinguish between the pure state |Ψ>= cos(θ)|0> + sin(θ)|1> and the mixed state ρ=cos^2(θ)|0><0| + sin^2(θ)|1><1| "

    2. Relevant equations

    I am not even sure I grasp the atmosphere of the problem...

    3. The attempt at a solution

    I understand the basic differences between superposition and mixture, but still I can't work a solution inside my head.
    So... any help would be much appreciated.

    Thank you in advance!
    Last edited: Jun 7, 2014
  2. jcsd
  3. Jun 7, 2014 #2


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    I'm not sure about this, as I haven't worked much with density matrices, but wouldn't measuring the expectation value of the operator A=|0><1|+|1><0| give a different result for the pure and mixture states? Do you know how to calculate expectation values for mixed states?
  4. Jun 7, 2014 #3
    Ok I see your point! My concern now is whether operator A is a fictional-theoretical operator for the sake of the problem or must be a real one. Must it be self-adjoint? What is the characterization of operator A?
  5. Jun 7, 2014 #4


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    The operator I defined is hermitian, so in principle it represents a measurable quantity, but I'm not sure how to interpret its meaning physically.
  6. Jun 7, 2014 #5
    hilbert2 I thank you very very much!! The problem itself is based on conceptual understanding so I guess we should not really care whether operator A has physical meaning or not.

    Again, thank you very much!
  7. Jun 9, 2014 #6
    hilbert2's operator |0><1| + |1><0| is just [itex]\sigma_x[/itex], so maybe not so weird after all :smile:
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