Time reversal and n-qubit canonical decompositions

Journal of Mathematical Physics

Volumn 46, Issue 6, 2005, Pages

  Bullock, Stephen S ^a   Brennen, Gavin K ^a   O'Leary, Dianne P ^b  

^a NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY   (United States)

^b University of Maryland ^*  (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 21844439738     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1900293     Document Type: Article

References (50)

1. Albertini, F. and D'Alessandro, D., " Model identification for spin networks., " Linear Algebr. Appl. 394, 237 (2005).
2. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., and Sorensen, D., LAPACK Users' Guide, 3rd ed. (SIAM, Philadelphia, 1999).
3. Barnum, H. and Linden, N., " Monotones and invariants for 'multi-particle quantum states., ' " J. Phys. A 34, 6787 (2001).
4. Bennett, C., DiVincenzo, D., Smolin, J., and Wootters, W., " Mixed-state entanglement and quantum error correction., " Phys. Rev. A 54, 3824 (1996).
5. Bremner, M., Dodd, J., Nielsen, M., and Bacon, D., " Fungible dynamics: There are only two types of entangling multi-qubit interactions., " Phys. Rev. A 69, 012313 (2004).
6. Brennen, G. K., " An observable measure of entanglement for pure states of multi-qubit systems., " Quantum Inf. Comput. 3, 619 (2003).
7. Brennen, G. K. and Bullock, S. S., " Stability of global entanglement in thermal states of spin chains., " Phys. Rev. A 70, 052303 (2004).
8. Bullock, S. and Brennen, G., " Canonical decompositions of n -qubit quantum computations and concurrence., " J. Math. Phys. 45, 2447 (2004).
9. Bullock, S. and Brennen, G., " Characterizing the entangling capacity of n -qubit computations., " Proc. SPIE 5436, 127 (2004) (see http:math.nist.govSBullock).
10. Bullock, S. and Markov, I., " An elementary two-qubit quantum computation in twenty-three elementary gates., " Phys. Rev. A 68, 012318 (2003).
11. Bullock, S. S., " Note on the Khaneja Glaser decomposition., " Quantum Inf. Comput. 5, 396 (2004).
12. Bunse-Gerstner, A., Byers, R., and Mehrmann, V., " A chart of numerical methods for structured eigenvalue problems., " SIAM J. Matrix Anal. Appl. 13, 419 (1992).
13. Childs, A., Haselgrove, H., and Nielsen, M., " Lower bounds on the complexity of simulating quantum gates., " Phys. Rev. A 68, 052311 (2003).
14. D'Alessandro, D., " Constructive controllability of one and two spin 12 particles., " Proceedings of the 2001 American Control Conference, Arlington, VA, June 2001 (see http:http://www.public.iastate.edu/~dalessQC3. html/).
15. Dongarra, J., Gabriel, J., Koelling, D., and Wilkinson, J., " The eigenvalue problem for Hermitian matrices with time reversal symmetry., " Linear Algebr. Appl. 60, 27 (1984).
16. Dür, W., Vidal, G., and Cirac, J. I., " Three qubits can be entangled in two inequivalent ways., " Phys. Rev. A 62, 062314 (2000).
17. Dyson, F., " Statistical theory of energy levels of complex system, I.., " J. Math. Phys. 3, 140 (1961).
18. Eisert, J. and Briegel, H., " The Schmidt measure as a tool for quantifying multi-particle entanglement., " Phys. Rev. A 64, 022306 (2001).
19. Gottfried, K., Quantum Mechanics (Benjamin, 1966).
20. Gour, G., " A family of concurrence monotones and its applications., " http:www.arxiv.orgabsquant-ph0410148.
21. Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces (American Mathematical Society, Providence, RI, 2001), vol. 34, graduate studies in mathematics, (corrected reprint of the 1978 original edition).
22. Hill, S. and Wootters, W., " Entanglement of a pair of quantum bits., " Phys. Rev. Lett. 78, 5022 (1997).
23. Horodecki, M., Horodecki, P., and Horodecki, R., " Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments., " Springer Tracts in Modern Physics (2001), quant-ph0109124.
24. Khaneja, N., Brockett, R., and Glaser, S. J., " Time optimal control in spin systems., " Phys. Rev. A 63, 032308 (2001).
25. Khaneja, N. and Glaser, S., " Cartan decomposition of SU (2n) and control of spin systems., " Chem. Phys. 267, 11 (2001) (see http:www.sciencedirect.comsciencejournal03010104).
26. Kramers, H. A., " Theorie generale de la rotation paramagnetique dans les cristaux., " Proc. R. Acad. Sci. Amsterdam 33, 959 (1930).
27. Kramers, H. A., Quantum Mechanics (Dover, Phoenix, 2004), ISBN 0486495337.
28. Kraus, B. and Cirac, J. I., " Optimal creation of entanglement using a two-qubit gate., " Phys. Rev. A 63, 062309 (2001).
29. Lewenstein, M., Kraus, B., Horodecki, P., and Cirac, I., " Characterization of separable states and entanglement witnesses., " Phys. Rev. A 63, 044304 (2001).
30. Lieb, E., Schultz, T., and Mattis, D., " Two soluble models of an antiferromagnetic chain., " Ann. Phys. 16, 407 (1961).
31. Makhlin, Y., " Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations., " Quantum Inf. Process. 1, 243-252 (2002).
32. Meyer, D. and Wallach, N., " Global entanglement in multi-partite systems., " J. Math. Phys. 43, 4273 (2002).
33. Miyake, A. and Wadati, M., " Multi-partite entanglement and hyperdeterminants., " Quantum Inf. Comput. 2, 540 (2002).
34. Munkres, J., Elements of Algebraic Topology (Addison-Wesley, New York, 1984).
35. Nakahara, N., Vartiainen, J., Kondo, Y., and Tanimura, K., " Warp-drive quantum computation., " http:www.arxiv.orgabsquant-ph0308006.
36. Nielsen, M. and Chuang, I., Quantum Information and Computation (Cambridge University Press, Cambridge, 2000).
37. Ramakrishna, V., Ober, R. J., Flores, K. L., and Rabitz, H., " Control of a coupled two-spin system without hard pulses., " Phys. Rev. A 65, 063405 (2002).
38. Rungta, P., Buzek, V., Caves, C., Hillery, M., and Milburn, G., " Universal state inversion and concurrence in arbitrary dimensions., " Phys. Rev. A 64, 042315 (2001).
39. Sakurai, J. J., Modern Quantum Mechanics, revised Ed. (Addison-Wesley, New York, 1985).
40. Scott, A. and Caves, C., " Entangling power of the quantum baker's map., " J. Phys. A 36, 9553-9576 (2003).
41. Shende, V., Bullock, S., and Markov, I., " Recognizing small-circuit structure in two-qubit operators., " Phys. Rev. A 70, 012310 (2004).
42. Uhlmann, A., " Fidelity and concurrence of conjugated states., " Phys. Rev. A 62, 032307 (2000).
43. Vatan, F. and Williams, C., " Optimal realization of an arbitrary two-Qubit quantum gate., " Phys. Rev. A 69, 032315 (2004).
44. Verstraete, F., Dehaene, J., De Moor, B., and Verschelde, H., " Four qubits can be entangled in nine different ways., " Phys. Rev. A 65, 052112 (2002).
45. Vidal, G., " Entanglement monotones., " J. Mod. Opt. 47, 355 (2000).
46. Vidal, G. and Dawson, C., " A universal quantum circuit for two-qubit transformations with three CNOT gates., " Phys. Rev. A 69, 010301 (2004).
47. Wigner, E., Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959).
48. Wong, A. and Christensen, N., " A potential multiparticle entanglement measure., " Phys. Rev. A 63, 044301 (2001).
49. Zhang, J., Vala, J., Sastry, S., and Whaley, K., " Geometric theory of nonlocal two-qubit operations., " Phys. Rev. A 67, 042313 (2003).
50. Some authors only use the term Cartan involution in the case that g is a noncompact Lie algebra. In their terminology, this definition of a Cartan involution on the Lie algebra of a compact group, e.g., su (N), is the image of a Cartan involution of a noncompact Lie algebra through symmetric duality (g=k⊕p) ↔ (gdual =k⊕ip).