Spectral properties of entanglement witnesses

Abstract

Entanglement witnesses are observables which when measured, detect entanglement in a measured composed system. It is shown what kind of relations between eigenvectors of an observable should be fulfilled, to allow an observable to be an entanglement witness. Some restrictions on the signature of entaglement witnesses, based on an algebraic-geometrical theorem will be given. The set of entanglement witnesses is linearly isomorphic to the set of maps between matrix algebras which are positive, but not completely positive. A translation of the results to the language of positive maps is also given. The properties of entanglement witnesses and positive maps express as special cases of general theorems for $k$-Schmidt witnesses and $k$-positive maps. The results are therefore presented in a general framework.

arXiv commentpublished version, some proofs are more detailed, mistakes removed

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• Exposed positive maps: a sufficient condition
• A class of exposed indecomposable positive maps
• The Inverse Eigenvalue Problem for Entanglement Witnesses
• Geometric representation of two-qubit entanglement witnesses
• Inertias of entanglement witnesses
• Separability in terms of a single entanglement witness
• Spectral Properties of Symmetric Quantum States and Symmetric Entanglement Witnesses
• Norms and Cones in the Theory of Quantum Entanglement
• Optimal indecomposable witnesses without extremality as well as spanning property
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• The detection power of real entanglement witnesses under local unitary equivalence
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