Character of Locally Inequivalent Classes of States and Entropy of Entanglement

TL;DR

This paper characterizes pure bipartite quantum states with identical entanglement, showing they are either locally unitarily connected or incomparable, and discusses the structure of inequivalent classes based on Schmidt coefficients.

Contribution

It establishes the physical nature of states with equal entanglement and introduces criteria for their local equivalence or incomparability based on Schmidt rank and coefficients.

Findings

Infinite inequivalent classes of states with same entanglement exist for Schmidt rank ≥ 3.

States with same entanglement but higher Schmidt rank may differ in at least three Schmidt coefficients.

States with same entanglement in higher ranks can be either locally connected or incomparable.

Abstract

In this letter we have established the physical character of pure bipartite states with the same amount of entanglement in the same Schmidt rank that either they are local unitarily connected or they are incomparable. There exist infinite number of deterministically locally inequivalent classes of pure bipartite states in the same Schmidt rank (starting from three) having same amount of entanglement. Further, if there exists incomparable states with same entanglement in higher Schmidt ranks (greater than three), then they should differ in at least three Schmidt coefficients.