Nonlocal characteristics of two-qubit gates from Argand diagram geometry

Abstract We show that the geometrical features of the unimodular squared eigenvalues of the nonlocal part of a two-qubit gate in the complex plane can be used to study the nonlocal characteristics of the two-qubit gate. It has been shown that the entangling power of a two-qubit gate can be expressed in terms of the mean squared length of the chords connecting the squared eigenvalues of the nonlocal part with each other. In this paper, we provide similar expressions for the gate typicality and the linear entropy of two-qubit gates. The expression of gate typicality involves the squared lengths of three chords connecting a squared eigenvalue of the nonlocal part with the complex conjugate of the other three squared eigenvalues. The expression of linear entropy is obtained from the expressions of entangling power and gate typicality. We also express them in terms of the linear entropies of certain two-qubit states. In addition, we show that the normalized entangling power of a two-qubit gate can be provided exact description in the Argand diagram. Finally, we analyze the simplices enclosing the origin in the convex hull of the squared eigenvalues of the nonlocal part of the perfect entangling two-qubit gates to construct a pair of orthonormal product states that can be transformed into maximally entangled states, and divide the region of perfect entanglers in the Weyl chamber into different regions based on the uniqueness of the simplices enclosing the origin.