Quantum entanglement and geometry of determinantal varieties

Journal of Mathematical Physics

Volumn 47, Issue 5, 2006, Pages

  Chen, Hao ^a  

^a FUDAN UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33744725888     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2194629     Document Type: Article

References (44)

1. Abhyankar, S., " Combinatorics of Young tableaux, determinantal varieties and the calculation of Hilbert functions (in French)., " Rend. Semin. Mat. Torino 42, 65-88 (1984).
2. Acin, A., Bruss, D., Lewenstein, M., and Sanpera, A., " Classification mixed three-qubit states., " Phys. Rev. Lett. 87, 040401 (2001).
3. Arbarello, E., Cornalba, M., Griffiths, P. A., and Harris, J., Geometry of Algebraic Curves, Volume I (Springer-Verlag, Berlin, 1985), Vol. I, Chap., Determinantal varieties.
4. Bennett, C. H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., and Wootters, W. K., " Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels., " Phys. Rev. Lett. 70, 1895 (1993).
5. Bennett, C. H., Cirac, J. I., Leifer, M. S., Leung, D. W., Linden, N., Popescu, S., and Vidal, G., " Optimal simulation of two-qubit Hamiltonians using general local operations., " Phys. Rev. A 66, 012305 (2002).
6. Bennett, C. H., DiVincenzo, D. P., Mor, T., Shor, P. W., Smolin, J. A., and Terhal, T. M., " Unextendible product bases and bound entanglement., " Phys. Rev. Lett. 82, 5385-5388 (1999).
7. Bennett, C. H., Popescu, S., Rohrlich, D., Smolin, J. A., and Thapliyal, A., " Exact and asymptotic measure of multipartite pure-state entanglement., " Phys. Rev. A 63, 012307 (2000).
8. Bennett, C. H., and Shor, P. W., " Quantum information theory., " IEEE Trans. Inf. Theory 44 (6), 2724-2742 (1998).
9. Brieskorn, E., and Knorrer, H., Plane Algebraic Curves (Birkhauser, Boston, 1981).
10. Brylinski, J-L., and Brylinski, R., " Invariant polynomial functions on k qudits., " in Mathematics of Quantum Computation, edited by, R. Brylinski, and, G. Chen, (CRC Press, Boca Raton, FL, 2002), pp. 277-286.
11. Brylinski, J-L., and Brylinski, R., " Universal quantum gates., " in Mathematics of Quantum Computation, edited by, R. Brylinski, and, G. Chen, (CRC Press, Boca Raton, FL, 2002), pp. 101-116.
12. Carteret, H. A., Higuchi, A., and Sudbery, A., " Multipartite generalization of the Schmidt decompositions., " J. Math. Phys. 41, 7932-7939 (2000).
13. Chen, H., " Necessary conditions for efficient simulation of Hamiltonians using local unitary operations., " Quantum Inf. Comput. 3, 249-257 (2003).
14. Chen, H., " Schmidt numbers of low rank bipartite mixed states., " Phys. Rev. A 67, 062301 (2003).
15. De Concini, C., Eisenbud, D., and Procesi, C., " Young diagrams and determinantal varieties., " Invent. Math. 56, 129-165 (1980).
16. Einstein, A., Podolsky, B., and Rosen, N., " Can quantum mechanical description of physical reality be considered complete., " Phys. Rev. 47, 777 (1935).
17. Eisenbud, D., " Linear sections of determinantal varieties., " Am. J. Math. 110, 541-575 (1988).
18. Eisenbud, D., Koh, J., and Stillman, M., " Determinantal equations for curves of high degree., " Am. J. Math. 110, 513-539 (1988).
19. Ekert, A., " Quantum cryptography based on Bell's theorem., " Phys. Rev. Lett. 67, 661 (1991).
20. Ekert, A., and Jozsa, R., " Quantum algorithms: Entanglement enhanced information processing., " Philos. Trans. R. Soc. London, Ser. A 356, no. 1743, 1762-1782 (1998).
21. Greenberger, D. M., Horne, M., and Zeilinger, A., in Bell's Theorem, Quantum Theory, and Conceptions of the Universe, edited by, M. Kafatos, (Kluwer Dordrecht, 1989).
22. Griffiths, P. A., " Infinitesimal variations of Hodges structure III: Determinantal varieties and the infinitesimal invariants of normal functions., " Compos. Math. 50, 267-324 (1983).
23. Harris, J., Algebraic Geometry, A First Course, Gradute Texts in Mathematics, Vol. 133 (Springer-Verlag, Berlin, 1992), especially Lecture 9 Determinantal Varieties.
24. Horodecki, P., " Separability criterion and inseparable mixed states with positive partial transpose., " Phys. Lett. A 232, 333 (1997).
25. Horodecki, M., Horodecki, P., and Horodecki, R., " Separability of mixed states: Necessary and sufficient conditions., " Phys. Lett. A 223, 8 (1996).
26. Horodecki, M., Horodecki, P., and Horodecki, R., " Mixed state entanglement and distillation: is there "bound" entanglement in nature., " Phys. Rev. Lett. 80, 5239 (1998).
27. Horodecki, M., Horodecki, P., and Horodecki, R., " Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps., " Phys. Lett. A 283, 1 (2001).
28. Horodecki, M., Horodecki, P., and Horodecki, R., " Mixed state entanglement and quantum communication., " in Quantum Information-Basic Concepts and Experiments, edited by, G. Alber, and, M. Wiener, (Springer, Berlin, 2001).
29. Horodecki, P., Lewenstein, M., Vidal, G., and Cirac, J. I., " Operational criterion and constructive checks for the separability of low rank density matrices., " Phys. Rev. A 62, 032310 (2000).
30. Kauffman, L. H., and Lomonaco S. J., Jr., " Quantum entanglement and topological entanglement., " New J. Phys. 4, 73 (2002).
31. Kiraus, B., Cirac, J. I., Karnas, S., and Lewenstein, M., " Separability in 2×N composite quantum systems., " Phys. Rev. A 61, 062302 (2000).
32. Lewenstein, M., Bruss, D., Cirac, J. I., Krus, B., Kus, M., Samsonowitz, J., Sanpera, A., and Tarrach, R., " Separability and distillablity in composite quantum systems-a premier., " J. Mod. Opt. 47, 2481 (2000).
33. Linden, N., and Popescu, S., " On multi-particle entanglement., " Fortschr. Phys. 46, 567 (1998).
34. Linden, N., Popescu, S., and Sudbery, A., " Non-local parameters of multiparticle density matrices., " Phys. Rev. Lett. 83, 243-247 (1999).
35. Meyer, D., and Wallach, N. R., " Global entanglement in multiparticle systems., " J. Math. Phys. 43, 4273-4278 (2002).
36. Nielsen, M. A., " Conditions for a class of entanglement transformations., " Phys. Rev. Lett. 83, 436 (1999).
37. Nielsen, M. A., and Kempe, J., " Separable states are more disordered globally than locally., " Phys. Rev. Lett. 86, 5184 (2001).
38. Peres, A., " Separability criterion for density matrices., " Phys. Rev. Lett. 77, 1413-1415 (1996).
39. Preskill, J., Physics 229: Advanced Mathematical Methods of Physics-Quantum Computation and Information (California Institute of Technology, Pasadena, CA, 1998), http://www.theory. caltech.edu/people/preskill/ph229/
40. Smolin, J. A., " Four-party unlockable bound-entangled states., " Phys. Rev. A 63, 032306 (2001).
41. Terhal, B. M., " Detecting quantum entanglements., " Theor. Comput. Sci. 287 (1), 313-335 (2002).
42. Terhal, B., and Horodecki, P., " Schmidt number for density matrices., " Phys. Rev. A 61, 040301 (2000).
43. Wallach, N. R., An Unentangled Gleason's Theorem, Contemp. Math. Vol. 305 (AMS, Providence, RI, 2002).
44. Werner, R. F., and Wolf, M. M., " Bell's inequality for states with positive partial transpose., " Phys. Rev. A 61, 062102 (2000).