= Photon-number reference for a Hadamard gate
{title2=$r_k=1-1/k,\quad s_k=\max(k-2,0)/k$}
A flat reference over $k$ adjacent <Fock states> has overlap $r_k$ with a one-photon shift and $s_k$ with a two-photon shift. These overlaps, obtained by counting shared number states, determine the reduced <qubit> channel of a number-conserving Hadamard interaction. For $k\ge2$, its squared fidelity with the desired pure output is $1-(1+xz)/(2k)$ for input <Bloch vector> $(x,y,z)$. Its worst-case infidelity is $3/(4k)$ because $|xz|\le1/2$. At $k=1$ the two-shift overlap is zero, not $-1$, and the fidelity is $1/2+(y^2-xz)/4$. Broad coherent number support, rather than merely a large fixed <photon> number, suppresses distinguishability of the outgoing field states.
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