On the stratifications of 2-qubits X-state space

TL;DR

This paper analyzes the structure of 2-qubit X-states by classifying them into different orbit types under global and local unitary group actions, providing equations and inequalities that define each class.

Contribution

It introduces a detailed stratification of 2-qubit X-states based on group actions, including explicit equations and inequalities for each class.

Findings

Partition of X-states into orbit types under global unitary actions

Partition of X-states into orbit types under local unitary actions

Explicit algebraic conditions defining each stratification component

Abstract

The 7-dimensional family $\mathfrak{P}_X$ of so-called mixed X-states of 2-qubits is considered. Two types of stratification of 2-qubits $X-$state space, i.e., partitions of $\mathfrak{P}_X\,$ into orbit types with respect to the adjoint group actions, one of the global unitary group $G_X \subset SU(4)$ and another one under the action of the local unitary group $LG_X \subset G_X$, is described. The equations and inequalities in the invariants of the corresponding groups, determining each stratification component, are given.