Inertias of entanglement witnesses

TL;DR

This paper investigates the inertias of entanglement witnesses, providing a comprehensive classification for qubit-qudit systems, and introduces criteria for separability and entanglement detection based on inertia properties.

Contribution

It systematically classifies inertias of entanglement witnesses in qubit-qudit systems and links inertia properties to separability criteria and entanglement classification.

Findings

All inertias for EWs in qubit-qudit systems are exhaustively classified.

A method to generate new inertias from existing ones is developed.

Inertias of EWs constructed by X-states are explicitly characterized.

Abstract

Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We study the inertias of EWs, i.e., the triplet of the numbers of negative, zero, and positive eigenvalues respectively. We focus on the EWs constructed by the partial transposition of states with non-positive partial transposes. We provide a method to generate more inertias from a given inertia by the relevance between inertias. Based on that we exhaust all the inertias for EWs in each qubit-qudit system. We apply our results to propose a separability criterion in terms of the rank of the partial transpose of state. We also connect our results to tripartite genuinely entangled states and the classification of states with non-positive partial transposes. Additionally, the inertias of EWs constructed by X-states are clarified.