Are Bell states distinguishable through LOCC?

Question

Define $|\psi^{00}\rangle = \frac{1}{\sqrt2}(|00\rangle + |11\rangle)$ and $|\psi^{01}\rangle = \frac{1}{\sqrt2}(|00\rangle - |11\rangle)$, and consider the state $$ |0\rangle\langle 0|^C\otimes |\psi^{00}\rangle\langle\psi^{00}|^{AB} + |1\rangle\langle 1|^C\otimes |\psi^{01}\rangle\langle\psi^{01}|^{AB}, $$ 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 $|\psi^{10}\rangle = \frac{1}{\sqrt2}(|01\rangle + |10\rangle)$, 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.

Answer 1

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.

Answer 2

We can distinguish these states using SingleQubit Unitaries and Measurements.

Lets the 2 states be $|\psi^{00}\rangle = \frac{1}{\sqrt2}(|00\rangle + |11\rangle)$ and $|\psi^{01}\rangle = \frac{1}{\sqrt2}(|00\rangle - |11\rangle)$. 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 $$I \otimes H|\psi^{00}\rangle = \frac{1}{\sqrt2}(|0\rangle \otimes |+\rangle + |1\rangle \otimes |-\rangle) = \frac{1}{\sqrt2}(|0\rangle|+\rangle + |1\rangle|-\rangle) = \frac{1}{\sqrt2}(|+\rangle|0\rangle + |-\rangle|1\rangle)$$

$$I \otimes H|\psi^{01}\rangle = \frac{1}{\sqrt2}(|0\rangle \otimes |+\rangle - |1\rangle \otimes |-\rangle) = \frac{1}{\sqrt2}(|0\rangle|+\rangle - |1\rangle|-\rangle) = \frac{1}{\sqrt2}(|-\rangle|0\rangle + |+\rangle|1\rangle)$$

Now Bob performs a measurement on his qubit. If the initial state was $|\psi^{00}\rangle$ then the result of the measurement could be $|0\rangle$ or $|1\rangle$ 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 $|0\rangle$ or $|1\rangle$ 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.

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