Grover oracle

For an implicit natural number $N$ and an explicit index $w_0$ ranging over $\{0,1,\dots,N-1\}$ (the type of $w_0$ being nonempty already forces $N\ge 1$, though this is not stated as a hypothesis, only implied by $w_0$'s existence), oracle w0 is defined as the continuous $\mathbb{C}$-linear map on $\mathbb{C}^N$ given by the identity map minus $2$ times the rank-one operator built from the standard basis vector $e_{w_0}\in\mathbb{C}^N$ (the vector with coordinate value $1$ at index $w_0$ and $0$ elsewhere), used as both the "output" and "input" vector of that rank-one operator. Concretely, the rank-one operator sends $z\mapsto \langle e_{w_0},z\rangle\, e_{w_0}$, and since $\langle e_{w_0},z\rangle=\overline{1}\cdot z_{w_0}=z_{w_0}$, this rank-one operator sends $z\mapsto z_{w_0}\,e_{w_0}$; hence for any $z\in\mathbb{C}^N$, oracle w0 sends $z$ to the vector whose $w_0$-th coordinate is $z_{w_0}-2z_{w_0}=-z_{w_0}$ and whose $i$-th coordinate for every $i\neq w_0$ equals $z_i$ unchanged — that is, the map negates exactly the $w_0$-th coordinate and leaves every other coordinate fixed.