On the algebra of local unitary invariants of pure and mixed quantum states

TL;DR

This paper investigates the algebraic structure of local unitary invariants in quantum states, proposing conjectures about their freeness and generators, and provides explicit forms for invariants of degree two.

Contribution

It introduces conjectures linking the invariants' algebraic structure to free groups and derives explicit invariant forms for degree two states.

Findings

Conjecture that the inverse limit algebra is free with generators related to free group conjugacy classes.

Explicit expressions for degree-two invariants in pure and mixed states.

Demonstration of the equivalence of the conjectures through representation theory.

Abstract

We study the structure of the inverse limit of the graded algebras of local unitary invariant polynomials using its Hilbert series. For k subsystems, we conjecture that the inverse limit is a free algebra and the number of algebraically independent generators with homogenous degree 2m equals the number of conjugacy classes of index m subgroups in a free group on k-1 generators. Similarly, we conjecture that the inverse limit in the case of k-partite mixed state invariants is free and the number of algebraically independent generators with homogenous degree m equals the number of conjugacy classes of index m subgroups in a free group on k generators. The two conjectures are shown to be equivalent. To illustrate the equivalence, using the representation theory of the unitary groups, we obtain all invariants in the m=2 graded parts and express them in a simple form both in the case of mixed and pure states. The transformation between the two forms is also derived. Analogous invariants of higher degree are also introduced.