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arXiv:1405.2021 · Physical Review A

Universal methods for extending any entanglement witness from the bipartite to the multipartite case

Bang-Hai Wang, Hai-Ru Xu, Simone Severini B

Published Aug 13, 2014• 3 citations
arXivHTMLPDF

Abstract

Any bipartite entanglement witness $W$ can be written as $W={c}_{\ensuremath{\sigma}}I\ensuremath{-}\ensuremath{\sigma}$, where $\ensuremath{\sigma}$ is a quantum state, $I$ is the identity matrix, and ${c}_{\ensuremath{\sigma}}$ is a non-negative number. We present a general method to extend the given entanglement witness to multipartite cases via purification, partial purification, and direct tensor of the quantum state $\ensuremath{\sigma}$. Our methods extend $\ensuremath{\sigma}$ but leave the parameter ${c}_{\ensuremath{\sigma}}$ untouched. This is very valuable since the parameter is generally not easy to compute.

DOI
10.1103/physreva.90.022312

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Any bipartite entanglement witness WWW can be written as W=c\ensuremathσI\ensuremath−\ensuremathσW={c}_{\ensuremath{\sigma}}I\ensuremath{-}\ensuremath{\sigma}W=c\ensuremathσ​I\ensuremath−\ensuremathσ, where \ensuremathσ\ensuremath{\sigma}\ensuremathσ is a quantum state, III is the identity matrix, and c\ensuremathσ{c}_{\ensuremath{\sigma}}c\ensuremathσ​ is a non-negative number. We present a general method to extend the given entanglement witness to multipartite cases via purification, partial purification, and direct tensor of the quantum state \ensuremathσ\ensuremath{\sigma}\ensuremathσ. Our methods extend \ensuremathσ\ensuremath{\sigma}\ensuremathσ but leave the parameter c\ensuremathσ{c}_{\ensuremath{\sigma}}c\ensuremathσ​ untouched. This is very valuable since the parameter is generally not easy to compute.
We demonstrate a general procedure to construct entanglement witnesses for any entangled state. This procedure is based on the trace inequality and a general form of entanglement witnesses, which is in the form $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$, where $\ensuremath{\rho}$ is a density matrix, ${c}_{\ensuremath{\rho}}$ is a non-negative number related to $\ensuremath{\rho}$, and III is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio\l{}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$ in operation, as well as in theory.
We derive a general framework that lifts any set of bipartite to multipartite entanglement witnesses and we show how positive maps can naturally be incorporated into this framework. We show that some previous approaches for multipartite entanglement detection are intimately connected to the witnesses derived from partial transposition and that such criteria can easily be outperformed in higher dimensions by nondecomposable maps. As an exemplary case we present a witness that is capable of detecting genuine multipartite entanglement in bound entangled states.
Entanglement in multipartite systems is a key resource for quantum information and communication protocols, making its verification in complex systems a necessity.Because an exact calculation of arbitrary entanglement probes is impossible, we derive and implement a numerical method to construct multipartite witnesses to uncover entanglement in arbitrary systems.Our technique is based on a substantial generalization of the power iteration-an essential tool for computing eigenvalues-and it is a solver for the separability eigenvalue equations, enabling the general formulation of optimal entanglement witnesses.Beyond our rigorous derivation and direct implementation of this method, we apply our approach to several examples of complexly quantum-correlated states and benchmark its general performance.Consequently, we provide a generally applicable numerical tool for the identification of multipartite entanglement.
Entanglement is an essential resource in quantum information science [1] and its presence in any quantum system can be experimentally detected through entanglement witness operators [2]. In particular, measuring a negative expectation value of a witness with high statistical confidence provides a necessary and sufficient condition to confirm the generation of a genuine multipartite [3] and/or d-level entangled state [4]. In recent years, the experimental generation of complex quantum states has intensified the need for witnesses that are capable of detecting such systems and are experimentally optimal at the same time. This means that the witness should require the least measurement effort (in terms of number and complexity of the measurement settings), include only projections on single qudits, while at the same time possessing a high noise tolerance (Fig. 1a). However, "experimentally-friendly" witnesses capable of accomplishing these tasks have not been derived yet.
We construct linear and nonlinear entanglement witnesses, by tensoring and partial tracing existing states and witnesses. We show that little shared quantum resources allow one to employ decomposable witnesses to obtain larger ones detecting locally undistillable states and find analytic witnesses for multipartite entangled states that are undistillable across all bipartitions. Crucially, we show how to efficiently optimize multicopy witnesses to desired states, which is a current challenge. As an example of both theoretical and experimental interest, existing trace polynomial witnesses are significantly improved while preserving their symmetries and implementability with randomized measurements. In addition to detecting entanglement, we also find new witnesses detecting k-copy distillability. A recipe for the single-copy case is shown to be effective for generic and Werner states.