  A complex conjugation of unitary quantum map is a second-order map (supermap)
that maps a unitary operator $U$ to its complex conjugate $U^*$. First, we
present a deterministic quantum protocol that universally implements the
complex conjugation supermap when we are given a blackbox quantum circuit,
guaranteed to implement some unitary operation, whose only known description is
its dimension. We then discuss the complex conjugation supermap in the context
of entanglement theory and derive a conjugation-based expression of the
$G$-concurrence. Finally, we present a physical process involving identical
fermions from which the complex conjugation protocol is derived as a simulation
of the process using qudits.
