Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 10D / b

Yes. The states $|{\Phi }^{+}⟩$ and $|{\Phi }^{-}⟩$ are orthogonal, sinceA joint projective measurement in the Bell basis therefore distinguishes them with certainty. Concretely, a controlled-NOT followed by a Hadamard gate converts the Bell basis into the computational basis before measurement.

No. Taking the partial trace over $B$ givesEvery measurement on $A$ alone therefore has identical outcome probabilities in the two cases. This is the local indistinguishability of Bell-state phase.

Yes. Alice and Bob both measure in the Hadamard basis $|+⟩,|-⟩$. The identitiesshow that their outcomes agree for $|{\Phi }^{+}⟩$ and differ for $|{\Phi }^{-}⟩$. After exchanging one classical bit they know which state they had. This is the LOCC discrimination of two Bell states.