X States of the Same Spectrum and Entanglement as All Two-Qubit States

TL;DR

This paper introduces an explicit family of two-qubit X states that are entanglement-preserving unitary equivalent to all general states, providing a concrete solution to a previously conjectured and numerically supported idea.

Contribution

The paper proves the existence of explicit entanglement-preserving unitaries transforming general two-qubit states into X states, with a compact implicit solution and exact explicit forms.

Findings

Established explicit X states with same spectrum and entanglement as all two-qubit states.

Provided a compact implicit solution for the entanglement-preserving unitaries.

Derived an exact explicit form of the X-state family.

Abstract

We present an explicit family of two-qubit X states with entanglement-preserving unitary (EPU) equivalence to the set of general states; that is, for any spectrum-entanglement combination achievable by general states, this family contains an X state of the same spectrum and entanglement. This idea was originally conjectured by the author and supported with strong numerical evidence in arXiv:1310.7038. Then, in Ann. Phys. 351 (2014) 79, the authors proved the existence of such two-qubit unitary transformations, but found the parameters to be transcendental, eluding explicit solution. Here, by a different method, we prove the existence of such transformations, obtain a compact implicit solution for them, and provide an exact, explicit form of the desired X-state family.