  Entanglement detection problem is one of the important problem in quantum
information theory. Gurvit showed that this problem is NP complete and thus
this may be the possible reason that only one criterion is not sufficient to
detect all entangled states. There are some powerful entanglement detection
criterion such as partial transposition criterion, realignment criterion but it
may not be possible to implement them successfully in the experiment. This
situation can be avoided if the entanglement is detected through the
construction of witness operator method. In this work, we take an analytical
approach to construct a witness operator. To achieve this task, we first
construct a linear map using partial transposition and realignment operation.
Then we find some conditions on the parameters of the map for which the map
represent a positive map. Further, we have constructed a Choi matrix
corresponding to the map and have shown that it is not completely positive. We
then construct an entanglement witness operator, which is based on the linear
combination of the function of Choi matrix and the identity matrix and it can
detect both NPTES and PPTES. Finally, we prove its efficiency by detecting
several bipartite bound entangled states which were previously undetected by
some well-known separability criteria. We also compared the detection power of
our witness operator with three well-known powerful entanglement detection
criteria, namely, dV criterion, CCNR criterion and the separability criteria
based on correlation tensor (CT) proposed by Sarbicki et. al. and find that our
witness operator detect more entangled states than these criterion.

Quantum Physics