Grover diffusion operator

For a natural number $N$ (explicit argument), diffusion N is defined as the continuous $\mathbb{C}$-linear map on $\mathbb{C}^N$ given by $2$ times the rank-one operator built from the vector $s:=$uniformSuperposition N (used as both arguments of the rank-one construction) minus the identity map. Concretely, the rank-one operator sends $z\mapsto\langle s,z\rangle\, s$, so for every $z\in\mathbb{C}^N$, diffusion N sends $z\mapsto 2\langle s,z\rangle\, s-z$, where $\langle s,z\rangle=\sum_{i=0}^{N-1}\overline{s_i}z_i=\tfrac{1}{\sqrt N}\sum_{i=0}^{N-1}z_i$ (each $s_i=1/\sqrt N$ being real, so it equals its own conjugate), again using the junk-value convention $\sqrt{0}=0$, $x/0=0$. In the degenerate case $N=0$ this makes $s$ the (unique, trivial) vector of $\mathbb{C}^0$ and the formula reduces to $z\mapsto -z$ on that zero-dimensional space, i.e. negation of the identity map.