Class Of Entanglement Witnesses Research Articles

How many mutually unbiased bases are needed to detect bound entangled states?

From a practical perspective it is advantageous to develop methods that verify entanglement in quantum states with as few measurements as possible. In this paper we investigate the minimal number of mutually unbiased bases (MUBs) needed to detect bound entanglement in bipartite -dimensional states, i.e. entangled states that are positive under partial transposition. In particular, we show that a class of entanglement witnesses (EWs) composed of MUBs can detect bound entanglement if the number of measurements is greater than . This is a substantial improvement over other detection methods, requiring significantly fewer resources than either full quantum state tomography or measuring a complete set of d + 1 MUBs. Our approach is based on a partial characterisation of the (non-)decomposability of EWs. We show that non-decomposability is a universal property of MUBs, which holds regardless of the choice of complementary observables, and we find that both the number of measurements and the structure of the witness play an important role in the detection of bound entanglement.

Greenberger-Horne-Zeilinger andWentanglement witnesses for the noninteracting Fermi gas

The existence and nature of tripartite entanglement of a noninteracting Fermi gas (NIFG) is investigated. Three classes of parametrized entanglement witnesses (EWs) are introduced with the aim of detecting genuine tripartite entanglement in the three-body reduced density matrix and discriminating between the presence of the two types of genuine tripartite entanglement, $W\B$ and $\mathrm{GHZ}\W$ (the convex set of $B$ states is comprised of mixed states of product and biseparable states; that of $W$ states is comprised of mixed states of $B$ states and $W$-type pure entangled states; and the GHZ (Greenberger-Horne-Zeilinger) set contains generic mixtures of any kind for a tripartite system). By choosing appropriate EW operators, the problem of finding GHZ and $W$ EWs is reduced to linear programming. Specifically, we devise $W$ EWs based on a spin-chain model with periodic boundary conditions, and we construct a class of parametrized GHZ EWs by linearly combining projection operators corresponding to all the different state-vector types arising for a three-fermion system. A third class of EWs is provided by a GHZ stabilizer operator capable of distinguishing $W\B$ from $\mathrm{GHZ}\B$ entanglement, which is not possible with $W$ EWs. Implementing these classes of EWs, it is found that all states containing genuine tripartite entanglement are of $W$ type, and hence states containing $\mathrm{GHZ}\W$ genuine tripartite entanglement do not arise. Some genuine tripartite entangled states that have a positive partial transpose (PPT) with respect to some bipartition are detected. Finally, it is demonstrated that a NIFG does not exhibit ``pure'' $W\B$ genuine tripartite entanglement: three-party entanglement without any separable or biseparable admixture does not occur.

Manipulating multiqudit entanglement witnesses by using linear programming

A new class of entanglement witnesses (EWs) called reduction type entanglement witnesses is introduced, which can detect some multi-qudit entangeled states including PPT ones with Hilbert space of dimension $d_{_{1}}\otimes d_{_{2}}\otimes...\otimes d_{_{n}}$. The novelty of this work comes from the fact that the feasible regions turn out to be convex polygons, hence the manipulation of these EWs reduces to linear programming which can be solved \emph{exactly} by using simplex method. The decomposability and non-decomposability of these EWs are studied and it is shown that it has a close connection with eigenvalues and optimality of EWs. Also using the Jamio\l kowski isomorphism, the corresponding possible positive maps, including the generalized reduction maps of Ref. \cite{Hall1}, are obtained.

Entanglement witness derived from NMR superdense coding

It is shown that superdense coding (SDC) experiments by means of nuclear magnetic resonance (NMR) can show non-classical efficiency gain over classical communication only for nuclear spin polarization beyond a certain threshold, and this threshold coincides with that for non-separability of the density matrix. It is also claimed that transfer of two-bit information mediated by a single qubit in the previous NMR SDC experiments with low nuclear spin polarization is not ascribed to the non-classical effect induced by entanglement, but merely to a statistical effect in an ensemble system having a large number of molecules. Towards experimental detection of entanglement, a new class of entanglement witnesses is proposed, which is based on the measurement of nuclear spin magnetizations in the Bell basis and is suitable for actual NMR experiments.