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Open Systems & Information Dynamics

Entanglement Witnesses for a Class of Bipartite States of n × n Qubits

Fabio Benatti, Mahya Karbalaii

Published Jun 1, 2013• 1 citation
Publisher

Abstract

We characterize the positive maps detecting the entangled bipartite states of n × n qubits that are diagonal with respect to the orthonormal basis constructed by tensor products of Pauli matrices acting on the totally symmetric state. We then discuss the case n = 2 for a class of states completely determined by the geometric patterns of subsets of a 16-point lattice.

DOI
10.1142/s1230161213500054

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We characterize the positive maps detecting the entangled bipartite states of n × n qubits that are diagonal with respect to the orthonormal basis constructed by tensor products of Pauli matrices acting on the totally symmetric state. We then discuss the case n = 2 for a class of states completely determined by the geometric patterns of subsets of a 16-point lattice.
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+NN+NN+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 characterize a convex subset of entanglement witnesses for two qutrits. Equivalently, we provide a characterization of the set of positive maps in the matrix algebra of 3 × 3 complex matrices. It turns out that boundary of this set displays elegant representation in terms of SO(2) rotations. We conjecture that maps parametrized by rotations are optimal, i.e. they provide the strongest tool for detecting quantum entanglement. As a byproduct we found a new class of decomposable entanglement witnesses parametrized by improper rotations from the orthogonal group O(2).