Entangled states in the role of witnesses

TL;DR

This paper introduces a novel framework where entangled states serve as witnesses for detecting entanglement and other quantum properties, providing new classifications, hierarchies, and insights into the structure of entangled states.

Contribution

It develops a hierarchy of entanglement witnesses, introduces the concepts of finer and optimal entangled states, and classifies entangled states into disjoint families centered on optimal states.

Findings

Entangled states can act as witnesses for entanglement detection.

A hierarchy of witnesses for classes of observables is established.

Entangled states are classified into disjoint families with optimal states at the core.

Abstract

Quantum entanglement lies at the heart of quantum mechanics and quantum information processing. In this work, we show a new framework where entangled states play the role of witnesses. We extend the notion of entanglement witnesses, developing a hierarchy of witnesses for classes of observables. This hierarchy captures the fact that entangled states act as witnesses for detecting entanglement witnesses and separable states act as witnesses for the set of non-block-positive Hermitian operators. Indeed, more hierarchies of witnesses exist. We introduce the concept of \emph{finer} and \emph{optimal} entangled states. These definitions not only give an unambiguous and non-numeric quantification of entanglement and a new perspective on edge states but also answer the open question of what the remainder of the best separable approximation of a density matrix. Furthermore, we classify all entangled states into disjoint families with optimal entangled states at its heart. This implies that we can focus only on the study of a typical family with optimal entangled states at its core when we investigate entangled states. Our framework also assembles many seemingly different findings with simple arguments that do not require lengthy calculations.