1. Title: Manipulating Multi-partite Entanglement Witnesses by using Linear Programming By: M.A. Jafarizadeh, G. Najarbashi, Y. Akbari, Hessam Habibian Department of Theoretical Physics and Astrophysics, Tabriz University, Tabriz 51664, Iran Iran International Conference on Quantum Information Kish Island, Iran 7-10 September 2007

2. Outline  Motivation  Entanglement Witnesses & Convex Optimization  Manipulating Entanglement Witnesses by Linear Programming  Multi-partite Entanglement Witnesses  Optimality, Decomposability & Positive maps  Conclusion

3. Motivation

6. 1. Entanglement Witnesses

7. Bounded entangled states(BES) which can be detected by Non-Decomposable EWs but not PPT criteria. NPPT PPT separable

8. 3. Manipulating EWs by LP method

13. 4. Multi-qudit Reduction type EWs Jafarizadeh, Najarbashi, Habibian: PRA_75_052326

19. 4.1 Optimality and Decomposability of

20. Decomposable ? Non-Decomposable

21. 4.2 Positive Maps

24. 5. Stabilizer EWs Jafarizadeh, Najarbashi, Akbari, Habibian, arXiv: 0704.2414v1

26. Nonlinear Convex Optimization Conclusion  The advantage of reducing manipulation of EWs to LP problem is to find optimal (Decomposable or Non-decomposable) EWs and it leads to detect larger areas of entanglement even when we solve our problem with approximated LP ones.  Studying Decomposable and Non-decomposable regions in multi-partite systems are so difficult which might be simplified by this method.  Multi-partite EWs open a new aspects to introduce many positive maps. Thanks for your attention