The algebra of local unitary invariants of identical particles

TL;DR

This paper studies the algebraic structure of local unitary invariants for quantum systems with identical particles, revealing they form free algebras with generators described by graphs, which serve as polynomial entanglement measures.

Contribution

It introduces a new combinatorial framework for understanding the invariants of quantum states with identical particles, showing these algebras are free and providing explicit generators.

Findings

Algebras of invariants are free in the infinite-dimensional limit.

Generators are described by graph-based combinatorial structures.

These generators serve as minimal polynomial entanglement measures.

Abstract

We investigate the properties of the inverse limit of the algebras of local unitary invariant polynomials of quantum systems containing various types of fermionic and/or bosonic particles as the dimensions of the single particle state spaces tend to infinity. We show that the resulting algebras are free and present a combinatorial description of an algebraically independent generating set in terms of graphs. These generating sets can be interpreted as minimal sets of polynomial entanglement measures distinguishing between states showing different nonclassical behaviour.