On the ring of local unitary invariants for mixed X-states of two qubits

TL;DR

This paper characterizes the structure of local unitary invariant polynomials for mixed X-states of two qubits, revealing a simplified algebraic description that aids in understanding their entangling properties.

Contribution

It establishes the invariant polynomial ring structure specifically for mixed X-states, connecting SU(2)xSU(2) invariants to SO(2)xSO(2) invariants with explicit generators.

Findings

Invariant ring for X-states is generated by five polynomials.

Injective ring homomorphism relates SU(2)xSU(2) and SO(2)xSO(2) invariants.

Provides algebraic tools for analyzing entanglement in X-states.

Abstract

Entangling properties of a mixed 2-qubit system can be described by the local homogeneous unitary invariant polynomials in elements of the density matrix. The structure of the corresponding invariant polynomial ring for the special subclass of states, the so-called mixed X-states, is established. It is shown that for the X-states there is an injective ring homomorphism of the quotient ring of SU(2)xSU(2) invariant polynomials modulo its syzygy ideal and the SO(2)xSO(2)-invariant ring freely generated by five homogeneous polynomials of degrees 1,1,1,2,2.