Distinguishing between two quantum states

[opsvival]

i'm given either |0> or cos\phi|0> + sin\phi|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

 

[Maxim Zh]

It depends on what measurement you can carry out.

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

<br /> &lt;\!0|\hat{H}|0\!&gt; = E_0<br />

<br /> (\cos\phi&lt;\!0| + \sin\phi&lt;\!1|)\hat{H}(\cos\phi|0\!&gt; + \sin\phi|1\!&gt;) =<br /> E_0 \cos^2\!\phi \,+ E_1 \sin^2\!\phi<br />

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

<br /> &lt;\!0|\hat{A}|0\!&gt; = A_0 = \text{const}<br />

<br /> (\cos\phi&lt;\!0|e^{i\omega_0 t} + \sin\phi&lt;\!1|e^{i\omega_1 t})<br /> \hat{A}<br /> (\cos\phi|0\!&gt;e^{-i\omega_0 t} + \sin\phi|1\!&gt;e^{-i\omega_1 t}) = A_1(t)<br />

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