Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    <\!0|\hat{H}|0\!> = E_0

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

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

    <\!0|\hat{A}|0\!> = A_0 = \text{const}

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

    i.e. you can observe oscillations of some physical magnitudes when the system is in the superposition state.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook