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

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.