Eur. Phys. J. D 55, 729-744 (2009)
https://doi.org/10.1140/epjd/e2009-00264-8

Constructing entanglement witness via real skew-symmetric operators

¹ Department of Theoretical Physics and Astrophysics, Tabriz University, 51664, Tabriz, Iran
² Institute for Studies in Theoretical Physics and Mathematics, 19395-1795, Tehran, Iran
³ Research Institute for Fundamental Sciences, 51664, Tabriz, Iran

Corresponding authors: ^a jafarizadeh@tabrizu.ac.ir - ^b behzadi@tabrizu.ac.ir

Received: 14 May 2009
Revised: 25 August 2009
Published online: 31 October 2009

Abstract

In this work, new types of EWs are introduced. They are constructed by using real skew-symmetric operators defined on a single party subsystem of a bipartite d⊗d system and a maximal entangled state in that system. A canonical form for these witnesses is proposed which is called canonical EW in corresponding to canonical real skew-symmetric operator. Also for each possible partition of the canonical real skew-symmetric operator corresponding EW is obtained. The method used for d⊗d case is extended to d1⊗d2 systems. It is shown that there exist Cd2d1 distinct possibilities to construct EWs for a given d1⊗ d2 Hilbert space. The optimality and nd-optimality problem is studied for each type of EWs. In each step, a large class of quantum PPT states is introduced. It is shown that among them there exist entangled PPT states which are detected by the constructed witnesses. Also the idea of canonical EWs is extended to obtain other EWs with greater PPT entanglement detection power.