= Solution
For any joint lower-register state, the <probability> is $P_0=(1+\langle S\rangle)/2$, where $S$ is its exchange operator. The Bell-state <eigenvalues> from part a therefore give
$$
\boxed{P_0(\Phi^+)=P_0(\Phi^-)=P_0(\Psi^+)=1,\qquad P_0(\Psi^-)=0.}
$$
For the <spin singlet state>, controlled exchange adds a minus sign to the control-1 branch, so the final <Hadamard gate> puts the ancilla in $|1\rangle$. For the three symmetric states the ancilla returns to $|0\rangle$.
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.
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