ua nt - p h / 06 09 05 0 v 2 1 1 Se p 20 06 Bound Entangled States With Negative Partial Transpose Exist !

We prove the existence of bound entangled states with negative partial transpose (NPT) in any d×d(d ≥ 3) Hilbert space with simple assumptions on Schmidt rank two states. We have assumed that the Schmidt rank two states should satisfy some bounds. Obviously the class of NPT bound entangled states belong to the class of conjectured to be bound entangled states by Divincenzo et.al [Phys. Rev. A, 61, 062312(2000)] and by Dür et.al [Phys. Rev. A, 61, 062313(2000)]. PACS number(s): 03.67.Hk, 03.65.Ud. The basic issue on the classification of mixed state entanglement at least on the level of bipartite systems solely depends upon whether there exist bound entangled states or not. The existence of PPT-bound (PPT means positive partial transpose) entangled states [1] and also the existence of NPT N−copy undistillable states [2, 3] for every positive integer N naturally indicates there may exist NPT-bound entangled states. In this work we are able to show the existence of NPT-bound entangled states with some assumptions on Schmidt rank two states. We first briefly describe the issue and the importance of the problem. In recent years it is found that quantum entanglement is an useful resource in performing several tasks in quantum information theory and quantum communication [4]. Maximally entangled states shared between two parties are essential ingredients in this respect [5]. Now due to the interaction with [email protected] [email protected], [email protected]