Characterizing the geometrical edges of nonlocal two-qubit gates

Abstract

Nonlocal two-qubit gates are geometrically represented by tetrahedron known as Weyl chamber within which perfect entanglers form a polyhedron. We identify that all edges of the Weyl chamber and polyhedron are formed by single parametric gates. Nonlocal attributes of these edges are characterized using entangling power and local invariants. In particular, ${\text{SWAP}}^{\ensuremath{-}\ensuremath{\alpha}}$ family of gates with $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1$ constitutes one edge of the Weyl chamber with ${\text{SWAP}}^{\ensuremath{-}1/2}$ being the only perfect entangler. Finally, optimal constructions of controlled-NOT using ${\text{SWAP}}^{\ensuremath{-}1/2}$ gate and gates belong to three edges of the polyhedron are presented.