Abstract.
One of the problems concerning entanglement witnesses (EWs) is the construction of them by a given set of operators. Here several multi-qubit EWs called stabilizer EWs are constructed by using the stabilizer operators of some given multi-qubit states such as GHZ, cluster and exceptional states. The general approach to manipulate the multi-qubit stabilizer EWs by exact(approximate) linear programming (LP) method is described and it is shown that the Clifford group play a crucial role in finding the hyper-planes encircling the feasible region. The optimality, decomposability and non-decomposability of constructed stabilizer EWs are discussed.
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References
M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)
The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation and Quantum Computation, edited by D. Bouwmeester, A. Ekert, A. Zeilinger (Springer, New York, 2000)
J. Preskill, The Theory of Quantum Information and Quantum Computation (California Inatitute of Technology, Pasadena, CA, 2000) http://www.theory.caltech.edu/poeole/preskill/ph229/
M. Horodecki, P. Horodecki, R. Horodecki, Springer Tr. Mod. Phys. 173, 151 (2001)
M. Lewenstein, D. Bruss, J.I. Cirac, B. Kraus, M. Kus, J. Samsonowicz, A. Sanpera, R. Tarrach, J. Mod. Opt. 47, 2841 (2000)
B.M. Terhal, Theor. Comput. Sci. 287, 313 (2002)
D. Bruss, J. Math. Phys. 43, 4237 (2002)
S.L. Woronowicz, Rep. Math. Phys. 10, 165 (1976)
M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223, 1 (1996)
G. Tóth, O. Gühne, Phys. Rev. A, 72, 022340 (2005)
M. Lewenstein, B. Kraus, J.I. Cirac, P. Horodecki, Phys. Rev. A 62, 052310 (2000); M. Lewenstein, B. Kraus, J.I. Cirac, P. Horodecki, Phys. Rev. A 63, 044304 (2001)
G. Tóth, O. Gühne, AIP Conf. Proc. 234, 234 (2004); Phys. Rev. Lett. 94, 060501 (2005)
O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, A. Sanpera, Phys. Rev. A 66, 062305 (2002); J. Mod. Opt. 50, 1079 (2003)
G. Tóth, Phys. Rev. A 69, 052327 (2004)
C. Brukner, V. Vedral, A. Zeilinger, Phys. Rev. A 73, 012110 (2006)
L.A. Wu, S. Bandyopadhyay, M.S. Sarandy, D.A. Lidar, Phys. Rev. A 72, 032309 (2005)
A.C. Doherty, P.A. Parrilo, F.M. Spedalieri, Phys. Rev. A 71, 032333 (2005)
R.O. Vianna, A.C. Doherty, Phys. Rev. A 74, 052306 (2006)
M.A. Jafarizadeh, M. Rezaee, S.K.A. Seyed Yagoobi, Phys. Rev. A 72, 062106 (2005)
M.A. Jafarizadeh, M. Rezaee, S. Ahadpour, Phys. Rev. A 74, 042335 (2006)
M.A. Jafarizadeh, G. Najarbashi, H. Habibian, Phys. Rev. A 75, 052326 (2007)
M.A. Jafarizadeh, R. Sufiani, e-print: arXiv:quant-ph/0703106 (2007)
S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004)
E.K.P. Chong, S.H. Żak, An Introduction to Optimization (John Wiley, NY, 2001)
D. Gottesman, Phys. Rev. A 54, 1862 (1996)
D. Gottesman, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1997
M. Van den Nest, J. Dehaene, B. De Moor, Phys. Rev. A 70, 032323 (2004)
R. Cleve, Phys. Rev. A 55, 4054 (1997)
J. Dehaene, M. Van den Nest, B. De Moor, F. Verstraete, Phys. Rev. A 67, 022310 (2003)
E. Hostens, J. Dehaene, B. De Moor, Phys. Rev. A 71, 042315 (2005)
R. Raussendorf, D.E. Browne, H.J. Briegel, Phys. Rev. A 68, 022312 (2003)
R.F. Werner, Phys. Rev. A 40, 4277 (1989)
W. Rudin, Functional Analysis (McGraw-Hill, Singapore, 1991)
W. Dür, J.I. Cirac, Phys. Rev. A 61, 042314 (2000)
F.G.S.L. Brandao, Phys. Rev. A 72, 022310 (2005)
A. Peres, Phys. Rev. Lett. 77, 1413 (1996)
O. Gühne, G. Tóth, P. Hyllus, H.J. Briegel, Phys. Rev. Lett. 95, 120405 (2005)
M. Van den Nest, J. Dehaene, B. De Moor, Phys. Rev. A 69, 022316 (2004)
M. Grassl, A. Klappenecker, M. Rötteler, e-print: arXiv:quant-ph/0703112v1 (2007)
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Jafarizadeh, M., Najarbashi, G., Akbari, Y. et al. Multi-qubit stabilizer and cluster entanglement witnesses. Eur. Phys. J. D 47, 233–255 (2008). https://doi.org/10.1140/epjd/e2008-00028-0
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DOI: https://doi.org/10.1140/epjd/e2008-00028-0