Constructing all entanglement witnesses from density matrices

TL;DR

This paper presents a universal method to construct entanglement witnesses for any entangled state using density matrices, trace inequalities, and the Choi-Jamio{}kowski isomorphism, providing both theoretical and operational conditions.

Contribution

It introduces a general form of entanglement witnesses derived from density matrices and the Choi-Jamio{}kowski isomorphism, with necessary and sufficient conditions for their use.

Findings

A general procedure for constructing entanglement witnesses for all entangled states.

Derivation of the form $W=\rho - c_{\rho} I$ as a universal entanglement witness.

Operational and theoretical conditions for the validity of these witnesses.

Abstract

We demonstrate a general procedure to construct entanglement witnesses for any entangled state. This procedure is based on the trace inequality and a general form of entanglement witnesses, which is in the form $W=\rho-c_{\rho} I$, where $\rho$ is a density matrix, $c_{\rho}$ is a non-negative number related to $\rho$, and $I$ is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio{\l}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=\rho-c_{\rho}I$ in operation, as well as in theory.