Nonlocal characteristics and argand diagram of two-qubit gates

TL;DR

This paper introduces a geometric approach using Argand diagrams and chord lengths to analyze the nonlocal properties of two-qubit gates, linking eigenvalue geometry to entanglement and gate classification.

Contribution

It presents a novel geometric framework connecting eigenvalue configurations to entanglement measures and classifies perfect entanglers within the Weyl chamber using convex hull analysis.

Findings

Chord lengths relate to entanglement measures.

Entangling power and gate typicality are expressed via chord lengths.

The Weyl chamber is partitioned into regions based on eigenvalue simplices.

Abstract

Nonlocal characteristics of a two-qubit gate are determined by its nonlocal part. The squared eigenvalues of the nonlocal part of a two-qubit gate exist on the unit circle in the complex plane. We show that two sets of chords, the chords connecting the squared eigenvalues with each other and those connecting a squared eigenvalue with the complex conjugate of others in the unit circle, can be used to describe the nonlocal characteristics of two-qubit gates. Lengths of both sets of chords are proportional to the amount of entanglement contained in certain pure states. The entangling power of a two-qubit gate can be expressed in terms of the squared lengths of the first set of chords. Similarly, we show that the gate typicality of a two-qubit gate can be expressed in terms of the squared lengths of the second set of chords and the linear entropy of a two-qubit gate can be expressed using the squared lengths of both sets of chords. Perfect entanglers are known to transform some product states into maximally entangled states. The convex hull of the squared eigenvalues of the nonlocal part of perfect entanglers contain the zero. We analyse the simplices containing the zero in the convex hull of the squared eigenvalues of the nonlocal part of perfect entanglers to construct a pair of orthonormal product states that can be transformed into maximally entangled states by the nonlocal part of perfect entanglers and divide the region of perfect entanglers in the Weyl chamber into three tetrahedral regions and eight bounding planes based on the uniqueness of the simplices containing the zero.