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.
- 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 $|\psi^{01}\rangle$