Entanglement universality of two-qubit X-states

TL;DR

This paper proves that every two-qubit state has an X-state counterpart with the same spectrum and entanglement, providing a method to transform between them while preserving key entanglement measures.

Contribution

It introduces a parametric unitary transformation that converts any two-qubit state into an X-state with identical entanglement properties.

Findings

Every two-qubit state has an X-state counterpart with the same spectrum and entanglement.

A semi-analytic method to set transformation parameters to preserve entanglement measures.

Explicit construction of X-states corresponding to the concurrence-purity diagram.

Abstract

We demonstrate that for every two-qubit state there is a X-counterpart, i.e., a corresponding two-qubit X-state of same spectrum and entanglement, as measured by concurrence, negativity or relative entropy of entanglement. By parametrizing the set of two-qubit X-states and a family of unitary transformations that preserve the sparse structure of a two-qubit X-state density matrix, we obtain the parametric form of a unitary transformation that converts arbitrary two-qubit states into their X-counterparts. Moreover, we provide a semi-analytic prescription on how to set the parameters of this unitary transformation in order to preserve concurrence or negativity. We also explicitly construct a set of X-state density matrices, parametrized by their purity and concurrence, whose elements are in one-to-one correspondence with the points of the concurrence versus purity (CP) diagram for generic two-qubit states.