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

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

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

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.

 

[Entropia]

I would first like to clarify that distinguishing between two quantum states is a fundamental problem in quantum mechanics, and it is a topic that is still being actively researched and studied. It is not a simple task, and there is no one definitive method for doing so. However, I will provide some guidance and suggestions to help you in your task.

Firstly, it is important to understand that a quantum state represents the state of a quantum system, and it is described by a wave function. In your case, the two states |0> and cos\phi|0> + sin\phi|1> represent two possible states of a quantum system, and the fair coin toss is determining which state is chosen.

To distinguish between these two states, you will need to perform a measurement on the system. A measurement in quantum mechanics involves applying a physical observable (such as position, momentum, energy, etc.) to the system and obtaining a corresponding measurement outcome. The outcome of the measurement will depend on the state of the system, and this is how we can determine which state we are given.

To perform a measurement, we need to prepare the system in a specific state. In your case, you are given either |0> or cos\phi|0> + sin\phi|1>, but you do not know which one. So, the first step would be to prepare the system in one of these states. This can be done by applying a unitary operation to the system. A unitary operation is a mathematical operation that preserves the norm of the wave function, and it is used to transform one state into another. In your case, you can construct a unitary operation by using a rotation matrix, which will rotate the state vector in a specific direction.

Once you have prepared the system in a specific state, you can then perform the measurement. The measurement outcome will depend on the state of the system, and it will give you information about which state you are given. If the measurement outcome corresponds to |0>, then you can conclude that you were given the state |0>. If the measurement outcome corresponds to cos\phi|0> + sin\phi|1>, then you can conclude that you were given the state cos\phi|0> + sin\phi|1>. However, if the measurement outcome does not correspond to either of these states, then it is possible that the state was a superposition of both states, and further analysis would