Are Bell states distinguishable through LOCC?

Question

Define $|psi^{00}rangle = frac{1}{sqrt2}(|00rangle + |11rangle)$ and $|psi^{01}rangle = frac{1}{sqrt2}(|00rangle - |11rangle)$, and consider the state
$$ |0ranglelangle 0|^Cotimes |psi^{00}ranglelanglepsi^{00}|^{AB} + |1ranglelangle 1|^Cotimes |psi^{01}ranglelanglepsi^{01}|^{AB}, $$
where the subsystems are distributed among three parties Alice, Bob and Charlie. Is it possible for Alice and Bob to extract Charlies bit through LOCC?
Essentially, I am asking if Bell states are locally distinguishable. If we consider $|psi^{00}rangle$ and $|psi^{10}rangle = frac{1}{sqrt2}(|01rangle + |10rangle)$, it is easy to see they are indeed locally distinguishable, but I have not been able to come up with a measurement that reveals the phase.
Any help is appreciated.

DaftWullie · Accepted Answer

If you look at these states in the $X$ basis, they are
$$
|++rangle+|--rangle,qquad |+-rangle+|-+rangle.
$$
Thus, by both measuring in the $X$ basis and computing the parity of the answers, you can tell $C$'s bit value.

vasjain · Answer

We can distinguish these states using SingleQubit Unitaries and Measurements.
Lets the 2 states be $|psi^{00}rangle = frac{1}{sqrt2}(|00rangle + |11rangle)$ and $|psi^{01}rangle = frac{1}{sqrt2}(|00rangle - |11rangle)$. Let Alice be in possession of the 1st qubit and Bob be in possession of the 2nd qubit.
Bob applies a $H$ gate on his qubit. This would result in the following states
$$I otimes H|psi^{00}rangle = frac{1}{sqrt2}(|0rangle otimes |+rangle + |1rangle otimes |-rangle) 
= frac{1}{sqrt2}(|0rangle|+rangle + |1rangle|-rangle) 
= frac{1}{sqrt2}(|+rangle|0rangle + |-rangle|1rangle)$$
$$I otimes H|psi^{01}rangle = frac{1}{sqrt2}(|0rangle otimes |+rangle - |1rangle otimes |-rangle) 
= frac{1}{sqrt2}(|0rangle|+rangle - |1rangle|-rangle) 
= frac{1}{sqrt2}(|-rangle|0rangle + |+rangle|1rangle)$$
Now Bob performs a measurement on his qubit.
If the initial state was $|psi^{00}rangle$ then the result of the measurement could be $|0rangle$ or $|1rangle$ and Alice's qubit would be $|+rangle$ or $|-rangle$ respectively.
Similarly, if the initial state was $|psi^{01}rangle$ then the result of the measurement could be $|0rangle$ or $|1rangle$ and Alice's qubit would be $|-rangle$ or $|+rangle$ state respectively.
Now Alice can apply the $H$ gate on her qubit and then perform a Measurement on it.
If Bob's and Alice's results were $(0,0)$ or $(1,1)$ then the initial state was $|psi^{00}rangle$. On the other hand if Bob's and Alice's results were $(0,1)$ or $(1,0)$ then the initial state was $|psi^{01}rangle$.
Hence we can now formulate an Algorithm for distinguishing the phase.

Bob applies $H$ gate on his qubit.
Bob performs a measurement and sends the result of the measurement to Alice.
Alice applies $H$ gate on her qubit.
Alice performs a measurement and sends the result of the measurement to Bob.
Now if their results were the same then the state was $|psi^{00}rangle$ else it was $|psi^{01}rangle$

Are Bell states distinguishable through LOCC?

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