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Distinguishing between two quantum states

  1. Jan 17, 2010 #1
    i'm given either |0> or cos[tex]\phi[/tex]|0> + sin[tex]\phi[/tex]|1> by a fair coin toss.
    and I don't know which state i'm given.

    i need guess which state was chosen.

    i think the method is to do a unitary operation on the states, and do the measurement,
    but I'm not sure how to construct a unitary, and I'm still not clear what this creating unitary and doing the measure is doing.

    need help please.

    thank you
     
  2. jcsd
  3. Jan 20, 2010 #2
    It depends on what measurement you can carry out.

    If |0> and |1> are eigenstates of the Hamiltonian with energies [tex]E_0[/tex] and [tex]E_1[/tex] ([tex]E_0 \neq E_1[/tex]) and you can measure the average energy of the system, then you will get

    [tex]
    <\!0|\hat{H}|0\!> = E_0
    [/tex]

    [tex]
    (\cos\phi<\!0| + \sin\phi<\!1|)\hat{H}(\cos\phi|0\!> + \sin\phi|1\!>) =
    E_0 \cos^2\!\phi \,+ E_1 \sin^2\!\phi
    [/tex]

    For any operator [tex]\hat{A}[/tex] which does not commute with the Hamiltonian:

    [tex]
    <\!0|\hat{A}|0\!> = A_0 = \text{const}
    [/tex]

    [tex]
    (\cos\phi<\!0|e^{i\omega_0 t} + \sin\phi<\!1|e^{i\omega_1 t})
    \hat{A}
    (\cos\phi|0\!>e^{-i\omega_0 t} + \sin\phi|1\!>e^{-i\omega_1 t}) = A_1(t)
    [/tex]

    i.e. you can observe oscillations of some physical magnitudes when the system is in the superposition state.
     
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