Optimal manipulations with qubits: Universal-NOT gate

Quantum information processing is usually associated with a specific computational basis. Nevertheless, for a deeper understanding of the fundamental principles of quantum computing, it is essential to investigate what is the fidelity of universal operations that are basis independent; i.e., the task is to perform optimally a specific operation on a qubit or a quantum register that is in an unknown state. In this paper we introduce the universal-NOT gate that takes as an input a qubit in an arbitrary state |Ψ〉 and generates an output that is as close as possible to the orthogonal state |Ψ^⊥〉.