Discrimination of Bell states on unkown qubits

Question

Consider the scenario where Alice is given three qubits, and promised two of them are maximally entangled Bell state $\in_R \{\frac{|00+11>}{\sqrt{2}}, \frac{|00-11>}{\sqrt{2}}\}$ and one of them is BB84 state $\in_R \{ |0\rangle, |1\rangle, \frac{|0+1\rangle}{\sqrt{2}}, \frac{|0-1\rangle}{\sqrt{2}} \}$, but she does not which qubit is the BB84 state.

My questions are the follwing

1) Is there a (probabilistic) strategy for Alice to determine what bell state was given to her?

2) Suppose the set of Bell states to choose from was $\{\frac{|00+11>}{\sqrt{2}}, \frac{|00-11>}{\sqrt{2}}, \frac{|01+10>}{\sqrt{2}}, \frac{|01-10>}{\sqrt{2}}\}$. Would that make things different?

Alice only has one copy and has access to all the qubits (I am not talking about LOCC). We allow von Neumann measurements and don't care if the state discrimination is (non)destructive.

Answer 1

Let $\vert\psi_i\rangle$ be the 4 Bell states and $\vert\phi_k\rangle$ the 4 "BB84" states. Then, define a POVM measurement with elements $$ P_{ik\pi} = \tfrac{1}{12}\vert\psi_i\rangle\langle\psi_i\vert\otimes \vert\phi_k\rangle\otimes \langle\phi_k\vert\ , $$ where $\pi$ is a permutation of the three qubits (i.e., the three possible choices for the unentangled qubit, since the Bell part is symmetric under permutation).

Then, it is straightforward to check that $\{P_{ik\pi}\}$ forms a complete POVM, and whenever you obtain outcome $i,k,\pi$, you can be sure to have the corresponding state (while any other state will not deterministically yield this outcome).

Thus, this forms a probabilistic scheme for distinguishing these states. (Of course, there might be a better scheme for doing so, in particular for only 2 Bell states, but to assess that one would probably need a precise figure of merit.)