= Solution
Choose orthonormal, flux-normalized incoming and outgoing channels and include all channels of this lossless scattering problem. Conservation of <probability> requires $\|\beta\|^2=\|\alpha\|^2$ for every input <vector>. With $\beta=S\alpha$,
$$
\alpha^\dagger(S^\dagger S-I)\alpha=0\quad\text{for every }\alpha,
\qquad\boxed{S^\dagger S=I.}
$$
For a square channel <matrix> this also gives $SS^\dagger=I$. A submatrix obtained by omitting loss channels need not be unitary, but that is not the stated two-channel model.
The given real <Hadamard matrix> is symmetric, unitary and its own inverse. Hence
$$
a_1^\dagger=\frac{b_1^\dagger+b_2^\dagger}{\sqrt2},\qquad
a_2^\dagger=\frac{b_1^\dagger-b_2^\dagger}{\sqrt2}.
$$
This same transformation preserves the <canonical commutation relations> for <bosons> and anticommutation relations for <fermions>. To extract <probabilities> one must use normalized <number states>, including their factorial factors, even when the supplied input expression is unnormalized.
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