Non Abelian structures and the geometric phase of entangled qudits

TL;DR

This paper explores the topological and algebraic properties of two-qudit states under local unitary evolution, linking Lie algebra structures to geometric phases and fractional phases in quantum systems.

Contribution

It introduces a framework connecting su(d) Lie algebra elements with the geometric phase of entangled qudits, generalizing monopole-like formulas for multi-level systems.

Findings

Roots and weights parametrize cyclic evolutions and fractional phases.

Reformulation of the coset contribution to geometric phase.

Generalization of the monopole-like formula for qubits.

Abstract

In this work, we address some important topological and algebraic aspects of two-qudit states evolving under local unitary operations. The projective invariant subspaces and evolutions are connected with the common elements characterizing the su(d) Lie algebra and their representations. In particular, the roots and weights turn out to be natural quantities to parametrize cyclic evolutions and fractional phases. This framework is then used to recast the coset contribution to the geometric phase in a form that generalizes the usual monopole-like formula for a single qubit.