# Using LOCC Operations to Distinguish Bell States

Example taken from http://www.quantiki.org/wiki/LOCC_operations:

Suppose I have two Bell states $|\phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ and $|\psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$. Two people, given one qubit each, can differentiate between these states in the following way. The first person measures their qubit, and they send this result to the second person. The second person then measures their qubit to figure out which Bell state they were given. For example, if the first person measured 0 and the second person measured 1, they must have the Bell state $|\psi^+\rangle$.

My question is, is there a way to distinguish between the "positive" and "negative" versions of the same Bell state? For instance, given $|\phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ and $|\phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$, can two people distinguish these in the same manner as above?

My intuition tells me no, since there's no way to distinguish between either state if, say, person 1 observes a 0 and person 2 observes a 0.

## 1 Answer

Yes, it is possible to distinguish between $|\phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle +|11\rangle)$ and $|\phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle -|11\rangle)$.

All that is needed is a local change of basis, on both sides, to states $$|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \;\; \text{and} \;\; |-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$$ This is equivalent to $$|0\rangle = \frac{1}{\sqrt{2}}(|+\rangle + |-\rangle) \;\; \text{and} \;\; |1\rangle = \frac{1}{\sqrt{2}}(|+\rangle - |-\rangle)$$ which means that $|\phi^{\pm}\rangle$ become $$|\phi^+\rangle = \frac{1}{2\sqrt{2}}\left[(|+\rangle + |-\rangle)(|+\rangle + |-\rangle) + (|+\rangle - |-\rangle)(|+\rangle - |-\rangle) \right] = \frac{1}{\sqrt{2}}(|++\rangle +|--\rangle)\\ |\phi^-\rangle = \frac{1}{2\sqrt{2}}\left[(|+\rangle + |-\rangle)(|+\rangle + |-\rangle) - (|+\rangle - |-\rangle)(|+\rangle - |-\rangle) \right] = \frac{1}{\sqrt{2}}(|+-\rangle +|-+\rangle)$$ Apply the procedure you described to the states $|\pm\rangle$ on each side and the result will tell you what is the entangled state.