Operator-Schmidt decomposition and the geometrical edges of two-qubit gates

TL;DR

This paper explores the operator-Schmidt decomposition of two-qubit gates, revealing geometric structures and characterizing edges of the entanglement polyhedron using Schmidt strength, with implications for understanding maximally entangled gates.

Contribution

It investigates the Schmidt decomposition's geometric aspects, characterizes controlled unitaries, and analyzes entanglement properties of two-qubit gates within the tetrahedral and polyhedral structures.

Findings

Schmidt number 2 gates are controlled unitaries.

One tetrahedral edge has maximum Schmidt strength.

Maximally entangled gates lie on a specific edge.

Abstract

Nonlocal two-qubit quantum gates are represented by canonical decomposition or equivalently by operator-Schmidt decomposition. The former decomposition results in geometrical representation such that all the two-qubit gates form tetrahedron within which perfect entanglers form a polyhedron. On the other hand, it is known from the later decomposition that Schmidt number of nonlocal gates can be either 2 or 4. In this work, some aspects of later decomposition are investigated. It is shown that two gates differing by local operations possess same set of Schmidt coefficients. Employing geometrical method, it is established that Schmidt number 2 corresponds to controlled unitary gates. Further, all the edges of tetrahedron and polyhedron are characterized using Schmidt strength, a measure of operator entanglement. It is found that one edge of the tetrahedron possesses the maximum Schmidt strength, implying that all the gates in the edge are maximally entangled.