 Solution

For any joint lower-register state, the probability is $P_0=(1+⟨S⟩)/2$, where $S$ is its exchange operator. The Bell-state eigenvalues from part a therefore giveFor the spin singlet state, controlled exchange adds a minus sign to the control-1 branch, so the final Hadamard gate puts the ancilla in $|1⟩$. For the three symmetric states the ancilla returns to $|0⟩$.

The product-state overlap formula cannot be applied to arbitrary entangled lower inputs. All four Bell states have identical one-qubit reduced states $I/2$, yet the spin singlet state result differs maximally. The network measures exchange symmetry of the joint state, rather than fidelity of quantum states inferred from its two marginals.