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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, wepresent a deterministic quantum protocol that universally implements thecomplex conjugation supermap when we are given a blackbox quantum circuit,guaranteed to implement some unitary operation, whose only known description isits dimension. We then discuss the complex conjugation supermap in the contextof entanglement theory and derive a conjugation-based expression of the$G$-concurrence. Finally, we present a physical process involving identicalfermions from which the complex conjugation protocol is derived as a simulationof the process using qudits.

Complex conjugation supermap of unitary quantum maps and its universal implementation protocol