Universal-NOT gate

The action of a NOT gate on a classical bit results in a change of its valu e from a 0 to a 1 and vice versa. The action of the classical NOT gate is i n principle perfect because with fidelity equal to unity it complements the value of a bit. The action of the quantum NOT gate in a computational basi s \0] and \1] is very similar to the action of the classical NOT gate. Howe ver, a more general quantum mechanical operation which corresponds to a cla ssical NOT gate would take a qubit in an arbitrary state \Psi] and produce a qubit in the state \Psi(perpendicular to)] orthogonal to \Psi]. This oper ation is anti-unitary and therefore, cannot be realized exactly. So how wel l we can do? We find a unitary transformation acting on an input qubit and some auxiliary qubits, which represent degrees of freedom of the quantum NO T gate itself, which approximately realizes the NOT operation on the state of the original qubit. We call this 'device' a universal-NOT gate because t he size of the error it produces is independent of the input state, We show that an optimal U-NOT gate which has as its input N identical qubits and p roduces M outputs achieves a fidelity of F = (N + 1)/ (N + 2), which is equ al to the fidelity of estimation of the input qubits. We also show that whe n a priori information about the state of the input qubit is available, the fidelity of a quantum NOT gate can be much better than the fidelity of est imation.