Complex conjugation ($z\mapsto \overline{z}$)

Complex conjugation sends a complex number $z=a+ib$ to its complex conjugate $\overline{z}=a-ib$. It is an involution and a field automorphism of $\mathbb{C}$, with fixed field $\mathbb{R}$. Applied componentwise to a complex vector, it produces the conjugated vector appearing in a Choi state for a rank-one Kraus operator.

This operation reflects a holomorphic function across the real axis while preserving holomorphic dependence on the new argument. A transform of real data with real kernel coefficients satisfies $F^{♯}=F$. In a polygonal global relation, complex rotation factors are conjugated as well; an explicit prefactor $i$ changes sign.