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

## TL;DR

This paper introduces a universal quantum protocol for implementing complex conjugation of unitary maps, explores its implications in entanglement theory, and demonstrates a physical process involving fermions for simulation.

## Contribution

It provides the first deterministic, dimension-only dependent protocol for complex conjugation supermaps and connects this to entanglement measures and fermionic systems.

## Key findings

- Universal protocol for complex conjugation supermap with blackbox circuits
- Expression of G-concurrence based on conjugation
- Physical fermionic process simulating the conjugation protocol

## Abstract

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.