# Distinguishing between two quantum states

1. Jan 17, 2010

### 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.

thank you

2. Jan 20, 2010

### 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

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

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

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

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

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

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