Photon-number reference for a Hadamard gate ($r_k=1-1/k,s_k=\max (k-2,0)/k$)

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 $k\ge 2$, 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 $|xz|\le 1/2$. 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.