We introduce a family of linear maps which are positive but not completely positive. We exhibit examples of nondecomposable maps and ${2}^{N}\ensuremath{\bigotimes}{2}^{N}$, $N\ensuremath{\geqslant}2$, bound entangled states---e.g., nondistillable bipartite states of $N+N$ qubits. Such states, as the standard Bell diagonal states, are diagonal in a maximally entangled basis and, apart remaining positive under partial transposition, are not detected by the realignment criterion.We introduce a family of linear maps which are positive but not completely positive. We exhibit examples of nondecomposable maps and ${2}^{N}\ensuremath{\bigotimes}{2}^{N}$, $N\ensuremath{\geqslant}2$, bound entangled states---e.g., nondistillable bipartite states of $N+N$ qubits. Such states, as the standard Bell diagonal states, are diagonal in a maximally entangled basis and, apart remaining positive under partial transposition, are not detected by the realignment criterion.

arXiv:quant-ph/0411098Physical Review AJan 31, 200632 citations

Separability criteria and bounds for entanglement measures

Heinz‐Peter Breuer

Employing a recently proposed separability criterion we develop analytical lower bounds for the concurrence and for the entanglement of formation of bipartite quantum systems. The separability criterion is based on a nondecomposable positive map which operates on state spaces with even dimension N >= 4, and leads to a class of nondecomposable optimal entanglement witnesses. It is shown that the bounds derived here complement and improve the existing bounds obtained from the criterion of positive partial transposition and from the realignment criterion.Employing a recently proposed separability criterion we develop analytical lower bounds for the concurrence and for the entanglement of formation of bipartite quantum systems. The separability criterion is based on a nondecomposable positive map which operates on state spaces with even dimension N >= 4, and leads to a class of nondecomposable optimal entanglement witnesses. It is shown that the bounds derived here complement and improve the existing bounds obtained from the criterion of positive partial transposition and from the realignment criterion.

arXiv:quant-ph/0606185Journal of Physics A Mathematical and GeneralSep 5, 200695 citations

Quantum Mappings and Characterization of Entangled Quantum States

Sergey N. Filippov

Journal of Mathematical SciencesAug 1, 201916 citations

Entanglement of mixed states in systems of n qubits

Renán Cabrera, W. E. Baylis

Bulletin of the American Physical SocietyJun 7, 20070 citations

N+N

qubits. Such states, as the standard Bell diagonal states, are diagonal in a maximally entangled basis and, apart remaining positive under partial transposition, are not detected by the realignment criterion.