Completely positive covariant two-qubit quantum processes and optimal quantum NOT operations for entangled qubit pairs

The structure is investigated of all completely positive quantum operations that transform pure two-qubit input states of a given degree of entanglement in a covariant way. Special cases thereof are quantum NOT operations which transform entangled pure two-qubit input states of a given degree of entanglement into orthogonal states in an optimal way. Based on our general analysis all covariant optimal two-qubit quantum NOT operations are determined. In particular, it is demonstrated that only in the case of maximally entangled input states can these quantum NOT operations be performed perfectly.

Figure 1

The parameter space of points $(V,X,Y)$ for which the covariant process ${\Pi }_{V,X,Y}$ is completely positive forms the tetrahedron $\overline{ABCD}$.

Figure 2

The minimum error (18) and the errors of the three relevant UNOT processes and their dependence on the degree of entanglement $\alpha$. The solid line represents the optimal minimum error. The dashed line $U_{SEP}$ corresponds to an independent application of two one-qubit covariant UNOT operations $u^1$ to each qubit from the entangled pair. The dash-dotted line $U_{ME}$ corresponds to the ideal covariant UNOT map for maximally entangled states. The dotted line represents the minimum achievable error for an unknown two-qubit pure state if its degree of entanglement is unknown.