State reconstruction formulae for the S-distributions and quadratures

Reports on Mathematical Physics

Volumn 66, Issue 1, 2010, Pages 55-84

  Kiukas, Jukka ^a   Pellonpää, Juha Pekka ^b   Schultz, Jussi ^b  

^a LEIBNIZ UNIVERSITY HANNOVER   (Germany)

^b UNIVERSITY OF TURKU   (Finland)

Author keywords

Informational completeness; Positive operator measures; Quasi probability distributions; Rotated quadrature distributions; State reconstruction

Indexed keywords

EID: 79955807806     PISSN: 00344877     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0034-4877(10)80024-2     Document Type: Article

References (41)

1. Welsch D.-G., Vogel W., Opatrný T. Homodyne detection and quantum state reconstruction. Progress in Optics 1999, Vol. XXXIX:63-211. Elsevier, Amsterdam. E. Wolf (Ed.).
2. Leonhardt U. Measuring the Quantum State of Light 1997, Cambridge University Press.
3. Vogel K., Risken H. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 1989, 40:2847-2849.
4. Bertrand J., Bertrand P. A tomographic approach to Wigner's function. Found. Phys 1987, 17:397-405.
5. Smithey D.T., Beck M., Raymer M.G., Faridina A. 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. 1993, 70:1244-1247.
6. Leibfried D., Meekhof D.M., King B.E., Monroe C., Itano W.M., Wineland D.J. Experimental determination of the motional quantum state of a trapped atom. Phys. Rev. Lett. 1996, 77:4281-4285.
7. Lougovski P., Solano E., Zhang Z.M., Walther H., Mack H., Schleich W.P. Fresnel representation of the Wigner function: an operational approach. Phys. Rev. Lett. 2003, 91:010401.
8. D'Ariano G.M., Macchiavello C., Paris M.G.A. Detection of the density matrix through optical homodyne tomography without filtered back projection. Phys. Rev. A 1994, 50:4298-4302.
9. Leonhardt U., Paul H., D'Ariano G.M. Tomographic reconstruction of the density matrix via pattern functions. Phys. Rev. A 1995, 52:4899-4907.
10. Leonhardt U., Munroe M., Kiss T., Richter Th., Raymer M.G. Sampling of photon statistics and density matrix using homodyne detection. Opt. Commun. 1996, 127:144-160.
11. Kühn H., Welsch D.-G., Vogel W. Determination of density matrices from field distributions and quasiprobabilities. J. Mod. Opt. 1994, 41:1607-1613.
12. Mancini S., Tombesi P., Man'ko V.I. Density matrix from photon number tomography. Europhys. Lett. 1997, 37:79-83.
13. De Nicola S., Fedele R., Man'ko M.A., Man'ko V.I. Fresnel tomography: a novel approach to the wave function reconstruction based on Fresnel representation of tomograms. Theoret. and Math. Phys. 2005, 144:1206-1213.
14. Paris M.G.A. Quantum state measurement by realistic heterodyne detection. Phys. Rev. A 1996, 53:2658-2663.
15. D'Ariano G.M., Mancini S., Tombesi P., Man'ko V.I. Reconstructing the density operator by using generalized field quadratures. Quant. Semiclass. Opt. 1996, 8:1017-1027.
16. Quantum State Estimation. Lect. Notes Phys. 2004, 649. Springer, Berlin. M.G.A. Paris, J. Řehácěk (Eds.).
17. Leonhardt U., Munroe M. Number of phases required to determine a quantum state in optical homodyne tomography. Phys. Rev. A 1996, 54:3682-3684.
18. Prugovečki E. Information-theoretical aspects of quantum measurement. Int. J. Theor. Phys. 1977, 16:321-331.
19. Busch P., Lahti P. The determination of the past and the future of a physical system in quantum mechanics. Found. Phys. 1989, 19:633-678.
20. Cooke R.G. Infinite Matrices and Sequence Spaces 1955, Dover Publications, Inc., New York.
21. Kiukas J., Lahti P. On the moment limit of quantum observables, with an application to the balanced homodyne detection. J. Mod. Opt. 2008, 55:1175-1198.
22. Kiukas J., Lahti P., Pellonpää J.-P. A proof for the informational completeness of the rotated quadrature observables. J. Phys. A Math. Theor. 2008, 41:175260.
23. Cassinelli G., D'Ariano G.M., De Vito E., Levrero A. Group theoretical quantum tomography. J. Math. Phys. 2000, 41:7940-7951.
24. Handbook of Mathematical Functions 1972, National Bureau of Standards, Applied Mathematics Series - 55, Tenth printing with corrections. M. Abramowitz, I.A. Stegun (Eds.).
25. Spanier J., Oldham K.B. An Atlas of Functions 1987, Hemisphere Publishing Company.
26. Armstrong B.H. Spectrum line profiles: the Voigt function. J. Quant. Spectrosc. Radiat. Transfer 1967, 7:61-88.
27. Milone L.A., Milone A.A.E. Evaluation of Dawson's function. Astrophys. Space Sci. 1988, 189-191.
28. McCabe J.H. A continued fraction expansion, with a truncated error estimate, for Dawson's integral. Math. Comp. 1974, 28:811-816.
29. Schreier F. The Voigt function and complex error function: a comparison of computational methods. J. Quant. Spectrosc. Radiat. Transfer. 1992, 48:743-762.
30. Riordan J. Combinatorial Identities 1968, Wiley, New York.
31. Cahill K.E., Glauber R.J. Ordered expansions in boson amplitude operators. Phys. Rev. 1969, 177:1857-1881.
32. Cahill K.E., Glauber R.J. Density operators and quasiprobability distributions. Phys. Rev. 1969, 177:1882-1902.
33. Leonhardt U., Paul H. Realistic optical homodyne measurements and quasiprobability distibutions. Phys. Rev. A 1993, 48:4598-4604.
34. D'Ariano G.M., Macchiavello C., Paris M.G.A. Optimized phase detection. Phys. Lett. A 1995, 198:286-294.
35. Kiukas J., Lahti P. A note on the measurement of phase space observables with an eight-port homodyne detector. J. Mod. Opt. 2008, 55:1891-1898.
36. Lahti P., Pellonpää J.-P. Continuous variable tomographic measurements. Phys. Lett. A 2009, 373:3435-3438.
37. Pellonpää J.-P. Quantum tomography, phase space observables, and generalized Markov kernels. J. Phys. A Math. Theor. 2009, 42:465303.
38. Werner R. Quantum harmonic analysis on phase space. J. Math. Phys. 1984, 25:1404-1411.
39. Gradshteyn I.S., Ryzhik I.M. Table of Integrals, Series, and Products 1980, Academic Press, Inc., Orlando, Corrected and Enlarged Edition.
40. Duran A.J. A bound on the Laguerre polynomials. Studia Math. 1991, 100:169-181.
41. P. Lahti, J.-P. Pellonpää and J. Schultz: Realistic eight-port homodyne detection and covariant phase space observables, J. Mod. Opt., in press, arXiv:1004.5006.