An algebraically independent generating set of the algebra of local unitary invariants

TL;DR

This paper proves that the algebra of local unitary invariants in quantum systems is free and provides an explicit algebraically independent generating set, linking invariants to subgroup conjugacy classes.

Contribution

It establishes the freeness of the inverse limit of invariant polynomial algebras and constructs an explicit generating set with a novel connection to subgroup conjugacy classes.

Findings

The algebra of local unitary invariants is free.

The generating set's size relates to conjugacy classes of subgroups.

Provides an explicit algebraically independent generating set.

Abstract

We show that the inverse limit of the graded algebras of local unitary invariant polynomials of finite dimensional k-partite quantum systems is free, and give an algebraically independent generating set. The number of degree 2d invariants in the generating set is equal to the number of conjugacy classes of index d subgroups of a free group on k-1 generators.