The Complexity of Relating Quantum Channels to Master Equations

Communications in Mathematical Physics

Volumn 310, Issue 2, 2012, Pages 383-418

  Cubitt, Toby S ^a,b   Eisert, Jens ^c,d   Wolf, Michael M ^e,f  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

^b UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

^c FREIE UNIVERSITÄT BERLIN   (Germany)

^d UNIVERSITY OF POTSDAM   (Germany)

^e UNIVERSITY OF COPENHAGEN   (Denmark)

^f TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84857234886     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-011-1402-y     Document Type: Article

References (52)

1. Nielsen M. A., Chuang I. L.: Quantum Computation and Quantum Information. Cambridge Univ. Press, Cambridge (2000).
2. Carmichael H. J.: Statistical Methods in Quantum Optics, Volume 1. Berlin-Heidelberg-New York: Springer, 2003.
3. Weiss, U.: Quantum dissipative systems. Series in Modern Condensed Matter Physics. River Edge, NJ: World Scientific, 1999.
4. Lindblad G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119 (1976).
5. Gorini V., Kossakowski A., Sudarshan E. C. G.: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17, 821 (1976).
6. Holevo A. S.: Statistical structure of quantum theory. Berlin-Heidelberg-New, York: Springer (2001).
7. Wolf M. M., Eisert J., Cubitt T. S., Cirac J. I.: Assessing non-Markovian quantum dynamics. Phys. Rev. Lett. 101, 150402 (2008).
8. Boulant, N., Havel, T. F., Pravia, M. A., Cory, D. G.: Robust method for estimating the Lindblad operators of a dissipative quantum process from measurements of the density operator at multiple time points. Phys. Rev. A 67 042322 (2003).
9. Boulant N., Emerson J., Havel T. F., Cory D. G.: Incoherent noise and quantum information processing. J. Chem. Phys. 121(7), 2955 (2004).
10. Howard M. et al.: Quantum process tomography and linblad estimation of a solid-state qubit. New J. Phys 8, 33 (2006).
11. Weinstein Y. S. et al.: Quantum process tomography of the quantum fourier transform. J. Chem. Phys 121, 6117 (2004).
12. Lidar D. A., Bihary Z., Whaley K. B.: From completely positive maps to the quantum Markovian semigroup master equation. Chem. Phys 268, 35 (2001).
13. Papadimitriou, C. H.: Computational Complexity. Reading, MA: Addison Wesley (1993).
14. Garey, M. R., Johnson D. S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Sanfrancisco, CA: W. H. Freeman (1979).
15. Norris J. R.: Markov Chains. Cambridge Univ. Press, Cambridge (1997).
16. Elfving, G.: Zur theorie der Markoffschen ketten. Acta Soc. Sei. Fennicae. 2(8) (1937).
17. Kingman J. F. C.: The imbedding problem for finite Markov chains. Z. Wahr 1, 14 (1962).
18. Kingman J. F. C., Williams D.: The combinatorial structure of non-homogeneous Markov chains. Z. Wahr 26, 77 (1973).
19. Fuglede B.: On the imbedding problem for stochastic and doubly stochastic matrices. Probab. Th. Rel. Fields, 80, 241 (1988).
20. Mukherjea A.: The role of nonnegative idempotent matrices in certain problems in probability. In: Charles R. Johnson, ed., Matrix theory and applications. Providence, RI: Amer. Math. Soc., 1990.
21. Cubitt, T. S.: The embedding problem for stochastic matrices is NP-hard. Manuscript in preparation.
22. Nielsen M. A., Knill E., Laflamme R.: Complete quantum teleportation using nuclear magnetic resonance. Nature 396, 52 (1998).
23. Vandersypen L. M. K., Chuang I. L.: NMR techniques for quantum control and computation. Rev. Mod. Phys 76, 1037 (2004).
24. Emerson J. et al.: Symmetrized characterization of noisy quantum processes. Science, 317, 1893 (2007).
25. Riebe M. et al.: Process tomography of ion trap quantum gates. Phys. Rev. Lett 97, 220407 (2006).
26. Monz T. et al.: Realization of the quantum Toffoli gate with trapped ions. Phys. Rev. Lett. 102, 040501 (2009).
27. O'Brien J. L. et al.: Quantum process tomography of a controlled-not gate. Phys. Rev. Lett. 93, 080502 (2004).
28. Lundeen J. S. et al.: Tomography of quantum detectors. Nature Phys 5, 27 (2009).
29. Howard M. et al.: Quantum process tomography and Linblad estimation of a solid-state qubit. New J. Phys 8, 33 (2006).
30. Choi M. D.: Completely positive linear maps on complex matrices. Lin. Alg. Appl 10, 285 (1975).
31. Jamiolkowski A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3, 275 (1972).
32. Wolf M. M., Cirac J. I.: Dividing quantum channels. Commun. Math. Phys. 279, 147 (2008).
33. Denisov L. V.: Infinitely divisible markov mappings in quantum probability theory. Th. Prob. Appl 33, 392 (1988).
34. Gurvits, L.: Classical deterministic complexity of Edmonds problem and quantum entanglement. In: Proceedings of the thirty-fifth ACM symposium on Theory of computing. New York: ACM Press, 2003, pp. 10-19.
35. Ioannou L. M.: Computational complexity of the quantum separability problem. Quant. Inf. Comp. 7(4), 335-370 (2007).
36. Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38(1), 49-95 (1996).
37. Horn R. A., Johnson C. R.: Topics in Matrix Analysis. Cambridge Univ. Press, Cambridge (1994).
38. Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer, Second edition, 1976.
39. Weilenmann J.: Continuity properties of fractional powers, of the logarithm, and of holomorphic semigroups. J. Func. Anal. 27, 1-20 (1978).
40. Moler C., van Loan C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3-49 (2003).
41. Schaefer, T. J.: The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC'78), New York: ACM Press, 1978, p. 216.
42. Stewart G. W.: Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Rev. 15(4), 727-764 (1973).
43. Golub, G. H., van Loan, C. F.: Matrix Computations. Baltimore, MD: Johns Hopkins University Press, Third edition, 1996.
44. Horn R. A., Johnson C. R.: Matrix Analysis. Cambridge Univ. Press, Cambridge (1990).
45. Schrijver A.: Theory of Linear and Integer Programming. Wiley, New York (1986).
46. Porkolab, L., Khachiyan, L.: Computing integral points in convex semi-algebraic sets. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS '97), Discataway, NJ: IEEE 1997, p. 162.
47. Porkolab, L.: On the Complexity of Real and Integer Semidefinite Programming. PhD thesis, Rutgers, (1996).
48. Cuthbert J. R.: The logarithm function for finite-state Markov semi-groups. J. London Math. Soc. 6(2), 524 (1973).
49. Johansen S.: Some results on the imbedding problem for finite markov chains. J. London Math. Soc. 8(2), 345 (1974).
50. Carette, P.: Characterizations of embeddable 3x3 stochastic matrices with a negative eigenvalue. New York J. Math. 1, 120 (1995).
51. Cuthbert J. R.: On uniqueness of the logarithm for Markov semi-groups. J. London Math. Soc. 4(2), 623 (1972).
52. Davies E. B.: Embeddable Markov matrices. Electronic J. Prob 15, 1474 (2010).