On the ring of local polynomial invariants for a pair of entangled qubits

TL;DR

This paper studies the polynomial invariants of two-qubit entanglement, describing a minimal basis and inequalities that characterize the entanglement properties through algebraic invariants.

Contribution

It introduces a special integrity basis for the ring of invariants and explicitly details the polynomial inequalities due to positivity constraints.

Findings

Identified a minimal set of primary and secondary invariants for the ring.

Derived explicit polynomial inequalities constraining the invariants.

Characterized the algebraic structure of invariants related to two-qubit entanglement.

Abstract

The entanglement characteristics of two qubits are encoded in the invariants of the adjoint action of SU(2) x SU(2) group on the space of density matrices defined as the space of positive semi-definite Hermitian matrices. The corresponding ring of polynomial invariants is studied. The special integrity basis for this ring is described and constraints on its elements due to the positive semi-definiteness of density matrices are given explicitly in the form of polynomial inequalities. The suggested basis is characterized by the property that only a minimal number of invariants, namely two primary invariants of degree 2, 3 and one secondary invariant of degree 4 appearing in the Hironaka decomposition of the ring are subject to the polynomial inequalities.