Constructing all entanglement witnesses from density matrices

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-Jamiołkowski 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.

5 pages. Added the computing $c_{{\rho }_q}^{\text{max}}$ in detail

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Cited by in corpus (6)

• Structural physical approximations and entanglement witnesses
• Entangled states in the role of witnesses
• The hierarchies of "witnesses" and the properties and characterization of entangled states as super entanglement witnesses
• Universal methods for extending any entanglement witness from the bipartite to the multipartite case
• Guaranteeing Completely Positive Quantum Evolution
• Characterization and properties of weakly optimal entanglement witnesses