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.

published version, some proofs are more detailed, mistakes removed

References in corpus (5)

• Geometry of quantum systems: density states and entanglement
• Spectral properties of entanglement witnesses
• The Jamiołkowski isomorphism and a conceptionally simple proof for the correspondence between vectors having Schmidt number $k$ and $k$-positive maps
• Differential Geometry of Bipartite Quantum States
• Non-decomposable quantum dynamical semigroups and bound entangled states

Cited by in corpus (11)

• Geometry of faithful entanglement
• Spectral properties of entanglement witnesses
• Geometric representation of two-qubit entanglement witnesses
• Inertias of entanglement witnesses
• Spectral Properties of Symmetric Quantum States and Symmetric Entanglement Witnesses
• Norms and Cones in the Theory of Quantum Entanglement
• Entanglement witness and nonlocality in confidence of measurement from multipartite quantum state discrimination
• Group-Theoretic Perspective on the PPT and Realignment Criteria in the Magic Simplex for Bipartite Qutrits
• The detection power of real entanglement witnesses under local unitary equivalence
• Effective criteria for entanglement witnesses in small dimensions
• Spectral characterizations of entanglement witnesses