Abstract
This chapter is devoted to a study of informational completeness of a measurement outcome statistics and the related problem of state reconstruction. Special attention is given to the continuous variable case. After introducing the key concepts and the basic results, including a short discussion of the qubit case, the Pauli problem and the two basic ways of overcoming it are studied. Finally, the problem of state reconstruction on the basis of informational completeness is investigated.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Pauli, W.: General Principles of Quantum Mechanics. Springer, Berlin (1980). Translated from the German by P. Achuthan and K. Venkatesan, With an introduction by Charles P. Enz
Reichenbach, H.: Philosophic Foundations of Quantum Mechanics. University of California Press, California (1944)
Prugovečki, E.: Information-theoretical aspects of quantum measurement. Int. J. Theor. Phys. 16(5), 321–331 (1977)
Corbett, J.V., Hurst, C.A.: Are wave functions uniquely determined by their position and momentum distributions? J. Austral. Math. Soc. Ser. B 20(2), 182–201 (1977/78)
Kiukas, J., Schultz, J.: Informationally complete sets of gaussian measurements. J. Phys. A 46(48), 485303 (2013)
Helgason, S.: The Radon Transform. Progress in Mathematics, vol. 5, 2nd edn. Birkhäuser, Boston (1999)
Ali, S.T., Prugovečki, E.: Classical and quantum statistical mechanics in a common Liouville space. Phys. A 89(3), 501–521 (1977)
Werner, R.: Quantum harmonic analysis on phase space. J. Math. Phys. 25(5), 1404–1411 (1984)
Kiukas, J., Lahti, P., Schultz, J., Werner, R.F.: Characterization of informational completeness for covariant phase space observables. J. Math. Phys. 53(10), 102103 (2012)
Busch, P., Lahti, P.J.: The determination of the past and the future of a physical system in quantum mechanics. Found. Phys. 19(6), 633–678 (1989)
Healy Jr., D.M., Schroeck Jr., F.E.: On informational completeness of covariant localization observables and Wigner coefficients. J. Math. Phys. 36(1), 453–507 (1995)
D’Ariano, G.M., Perinotti, P., Sacchi, M.F.: Informationally complete measurements and group representation. J. Opt. B 6(6), S487–S491 (2004)
Wünsche, A., Bužek, V.: Reconstruction of quantum states from propensities. Quantum Semiclass. Opt. 9(4), 631–653 (1997)
Kiukas, J., Pellonpää, J.-P., Schultz, J.: State reconstruction formulae for the \(S\)-distributions and quadratures. Rep. Math. Phys. 66(1), 55–84 (2010)
Kiukas, J., Pellonpää, J.-P., Schultz, J.: Density matrix reconstruction from displaced photon number distributions. J. Phys. A 43(9), 095303, 18 (2010)
Schultz, J.: State reconstruction via infinite matrix inversion. Phys. Scripta T140(1), 014060 (2010)
Leonhardt, U., Paul, H., D’Ariano, G.M.: Tomographic reconstruction of the density matrix via pattern functions. Phys. Rev. A 52, 4899–4907 (1995)
Cassinelli, G., D’Ariano, G.M., De Vito, E., Levrero, A.: Group theoretical quantum tomography. J. Math. Phys. 41(12), 7940–7951 (2000)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)
Spanier, J., Oldham, K.B.: An Atlas of Functions. Hemisphere Publishing Company, New York (1987)
Riordan, J.: Combinatorial Identities. Wiley, New York (1968)
Cahill, K.E., Glauber, R.J.: Ordered expansions in boson amplitude operators. Phys. Rev. 177, 1857–1881 (1969)
Cahill, K.E., Glauber, R.J.: Density operators and quasiprobability distributions. Phys. Rev. 177, 1882–1902 (1969)
Vogel, K., Risken, H.: Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40, 2847–2849 (1989)
Smithey, D.T., Beck, M., Raymer, M.G., Faridani, 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. 70, 1244–1247 (1993)
Leonhardt, U., Paul, H.: Realistic optical homodyne measurements and quasiprobability distributions. Phys. Rev. A 48, 4598–4604 (1993)
D’Ariano, G.M., Macchiavello, C., Paris, M.G.: Optimized phase detection. Phys. Lett. A 198(4), 286–294 (1995)
Pellonpää, J.-P.: Quantum tomography, phase-space observables and generalized Markov kernels. J. Phys. A 42(46), 465303, 18 (2009)
Heinosaari, T., Mazzarella, L., Wolf, M.M.: Quantum tomography under prior information. Comm. Math. Phys. 318(2), 355–374 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Busch, P., Lahti, P., Pellonpää, JP., Ylinen, K. (2016). State Reconstruction. In: Quantum Measurement. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-43389-9_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-43389-9_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-43387-5
Online ISBN: 978-3-319-43389-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)