Optimal estimation of qubit states with continuous time measurements

Communications in Mathematical Physics

Volumn 277, Issue 1, 2008, Pages 127-160

  Guţǎ, Mǎdǎlin ^a   Janssens, Bas ^a   Kahn, Jonas ^b  

^a UNIVERSITY OF NOTTINGHAM   (United Kingdom)

^b UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 36448995402     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-007-0357-5     Document Type: Article

References (58)

1. Artiles L., Gill R.D. and Guţǎ M. (2005). An invitation to quantum tomography. J. Royal Statist. Soc. B (Methodological) 67: 109-134
2. Bagan E., Baig M. and Muñoz-Tapia R. (2002). Optimal Scheme for Estimating a Pure Qubit State via Local Measurements. Phys. Rev. Lett. 89: 277904
3. Bagan E., Baig M., Muñoz-Tapia R. and Rodriguez A. (2004). Collective versus local measurements in a qubit mixed-state estimation. Phys. Rev. A 69: 010304(R)
4. Bagan E., Ballester M.A., Gill R.D., Monras A. and Muñoz-Tapia R. (2006). Optimal full estimation of qubit mixed states. Phys. Rev. A 73: 032301
5. Bagan E., Monras A. and Muñoz-Tapia R. (2005). Comprehensive analysis of quantum pure-state estimation for two-level system. Phys. Rev. A 71: 062318
6. Barndorff-Nielsen O.E., Gill R. and Jupp P.E. (2003). On quantum statistical inference (with discussion). J. R. Statist. Soc. B 65: 775-816
7. Barndorff-Nielsen O.E. and Gill R.D. (2000). Fisher information in quantum statistics. J. Phys. A 33: 1-10
8. Bouten L., Guţǎ M. and Maassen H. (2004). Stochastic Schrödinger equations. J. Phys. A 37: 3189-3209
9. Butucea C., Guţǎ M. and Artiles L. (2007). Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data. Ann. Stat 35(2): 465-494
10. Caves C.M. (1982). Quantum limits on noise in linear amplifiers. Phys. Rev. D 26: 1817-1839
11. Cirac J.I., Ekert A.K. and Macchiavello C. (1999). Optimal Purification of Single Qubits. Phys. Rev. Lett. 82: 4344
12. D'Ariano G.M., Leonhardt U. and Paul H. (1995). Homodyne detection of the density matrix of the radiation field. Phys. Rev. A 52: R1801-R1804
13. Dyson F.J. (1956). General Theory of Spin-Wave Interactions. Phys. Rev. 102: 1217-1230
14. Embacher F. and Narnhofer H. (2004). Strategies to measure a quantum state. Ann. of Phys. (N.Y.) 311: 220
15. Fisher D.G., Kienle S.H. and Freyberger M. (2000). Quantum-state estimation by self-learning measurements. Phys. Rev. A 61: 032306
16. Fujiwara A. (2006). Strong consistency and asymptotic efficiency for adaptive quantum estimation problems. J. Phys. A 39: 12489-12504
17. Fujiwara A. and Nagaoka H. (1995). Quantum Fisher metric and estimation for pure state models. Phys. Lett. A 201: 119-124
18. Gardiner C.W. and Zoller P. (2004). Quantum Noise. Springer, Berlin-Heidelberg-New York
19. Geremia J., Stockton J.K. and Mabuchi H. (2004). Real-Time Quantum Feedback Control of Atomic Spin-Squeezing. Science 304: 270-273
20. Gill, R.D.: Asymptotic information bounds in quantum statistics. http://arxiv.org/abs/math.ST/0512443, 2005, to appear in Annals of Statistics
21. Gill R.D. and Massar S. (2000). State estimation for large ensembles. Phys. Rev. A 61: 042312
22. Guţǎ M. and Kahn J. (2006). Local asymptotic normality for qubit states. Phys. Rev. A 73: 052108
23. Guţǎ, M., Jenčová, A.: Local asymptotic normality in quantum statistics, preprint quant-ph/0606213, to appear in Commun. Math. Phys
24. Hannemann T., Reiss D., Balzer C., Neuhauser W., Toschek P.E. and Wunderlich C. (2002). Self-learning estimation of quantum states. Phys. Rev. A 65: 050303(R)
25. Hayashi, M.: Presentations at MaPhySto and QUANTOP Workshop on Quantum Measurements and Quantum Stochastics, Aarhus, 2003, and Special Week on Quantum Statistics, Isaac Newton Institute for Mathematical Sciences, Cambridge, 2004
26. Hayashi M. (2002). Two quantum analogues of Fisher information from a large deviation viewpoint of quantum estimation. J. Phys. A: Math. Gen. 35: 7689-7727
27. Hayashi, M.: Quantum estimation and the quantum central limit theorem. Bull. Math. Soc. Japan 55, 368-391 (2003) (in Japanese; Translated into English in quant-ph/0608198)
28. Hayashi, M., editor: Asymptotic theory of quantum statistical inference: selected papers. River Edge, NJ: World Scientific, 2005
29. Hayashi, M., Mastumoto, K.: Asymptotic performance of optimal state estimation in quantum two level system. http://arxive.org/list/quant-ph/0411073, 2001
30. Hayashi, M., Matsumoto, K.: Statistical Model with Measurement Degree of Freedom and Quantum Physics. In: M. Hayashi, editor. Asymptotic theory of quantum statistical inference: selected papers, River Edge, NJ: World Scientific, 2005, pp. 162-170,
31. (English translation of a paper in Japanese published in Surikaiseki Kenkyusho Kokyuroku 35, 7689-7727 (2002))
32. Helstrom C.W. (1976). Quantum Detection and Estimation Theory. Academic Press, New York
33. Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. Amsterdam: North-Holland, 1982
34. Holtz R. and Hanus J. (1974). On coherent spin states. J. Phys. A 7: 37
35. Hudson R.L. and Parthasarathy K.R. (1984). Quantum Itô's formula and stochastic evolutions. Commun. Math. Phys. 93: 301-323
36. Janssens, B.: Unifying decoherence and the Heisenberg principle. http://arxiv.org/abs/quant-ph/0606093, 2006
37. Jones K.R. (1994). Fundamental limits upon the measurement of state vectors. Phys. Rev. A 50: 3682
38. Kahn, J., Guţǎ, M., Matsumoto, K.: Local asymptotic normality for d-dimensional quantum states. in preparation
39. Keyl M. and Werner R.F. (2001). Estimating the spectrum of a density operator. Phys. Rev. A 64: 052311
40. Latorre J.I., Pascual P. and Tarrach R. (1998). Minimal Optimal Generalized Quantum Measurements. Phys. Rev. Lett. 81: 1351
41. Le Cam L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer Verlag, New York
42. Leonhardt U., Munroe M., Kiss T., Richter T. and Raymer M.G. (1996). Sampling of photon statistics and density matrix using homodyne detection. Optics Commun 127: 144-160
43. Leonhardt U., Paul H. and D'Ariano G.M. (1995). Tomographic reconstruction of the density matrix via pattern functions. Phys. Rev. A 52: 4899-4907
44. Mack H., Fischer D.G. and Freyberger M. (2000). Enhanced quantum estimation via purification. Phys. Rev. A 62: 042301
45. Massar S. and Popescu S. (1995). Optimal Extraction of Information from Finite Quantum Ensembles. Phys. Rev. Lett. 74: 1259-1263
46. Matsumoto K. (2002). A new approach to the Cramer-Rao type bound of the pure state model. J. Phys. A 35(13): 3111-3123
47. Nagaoka, H.: On the parameter estimation problem for quantum statistical models. In: M. Hayashi, editor. Asymptotic Theory of Quantum Statistical Inference, River Edge, NJ: World Scientific (2005), pp 125-132
48. Schiller S., Breitenbach G., Pereira S.F., Müller T. and Mlynek J. (1996). Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation. Phys. Rev. Lett. 77: 2933-2936
49. Smithey D.T., Beck M., Raymer M.G. and Faridani A. (1993). Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. Phys. Rev. Lett. 70: 1244-1247
50. Vaart A. (1998). Asymptotic Statistics. Cambridge University Press, Cambridge
51. Wellner J. and Vaart A. (1996). Weak Convergence and Empirical Processes. Springer, New York
52. Vidal G., Latorre J.I., Pascual P. and Tarrach R. (1999). Optimal minimal measurements of mixed states. Phys. Rev. A 60: 126
53. Vogel K. and Risken H. (1989). Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40: 2847-2849
54. Wald A. (1943). Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations is Large. Trans. Amer. Math. Soc. 54: 426-482
55. Werner R.F. (1998). Optimal cloning of pure states. Phys. Rev. A 58: 1827-1832
56. Yuen H.P. and Lax M. (1973). Multiple-parameter quantum estimation and measurement of non-selfadjoint observables. IEEE Trans. Inform. Theory 19: 740
57. Zavatta A., Viciani S. and Bellini M. (2004). Quantum to classical transition with single-photon-added coherent states of light. Science 306: 660-662
58. Zyczkowski K. and Sommers H.J. (2005). Average fidelity between random quantum states. Phys. Rev. A 71: 032313