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Homework Help: Differentiation Quantum Phasis

  1. Jan 14, 2009 #1
    1. The problem statement, all variables and given/known data
    Given two quantum preparations
    [itex] \frac{1}{\sqrt{2}} \left( |0\rangle + | 1 \rangle \right) [/itex]
    [itex] \frac{1}{\sqrt{2}} \left( |0\rangle - | 1 \rangle \right) [/itex]
    Give a measurement that will distinguish between these two preparations with high probability.

    3. The attempt at a solution
    I'm thinking that there might be some other measurement basis with which I can apply in order to get a high probability of determining which is which, but I can't think of it.
  2. jcsd
  3. Jan 14, 2009 #2
    Do you know what 0 and 1 means? are they eigenstates of say angular momentum, Lz, states of harmonic oscillator? or something else?

    If you don't know what 0 1 are (except different energy eigenstates), you will just have to arbitrarily construct a Hermitian operator whose eigenstates are the ones above and call that a measurement. I'm quite sure this is not what the question wants.
  4. Jan 14, 2009 #3
    The original question is how to differentiate between the following states in a 2-dimensional Hilbert space:

    [tex] \frac{1}{\sqrt2} \left( | 0 \rangle + e^{3i\pi/4} | 1 \rangle \right) [/tex]
    [tex] \frac{1}{\sqrt2} \left( | 0 \rangle + e^{7i\pi/4} | 1 \rangle \right) [/tex]

    and the hint suggested that I use a [itex] \pi/4 [/itex] shifter [itex] | 0 \rangle\langle 0 | + e^{i\pi/4} |1 \rangle \langle 1| [/itex].
  5. Jan 15, 2009 #4


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    Hint: is a projection operator Hermitian?
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