×

zbMATH — the first resource for mathematics

On the tomographic picture of quantum mechanics. (English) Zbl 1237.81043
Summary: We formulate necessary and sufficient conditions for a symplectic tomogram of a quantum state to determine the density state. We establish a connection between the (re)construction by means of symplectic tomograms with the construction by means of Naimark positive definite functions on the Weyl-Heisenberg group. This connection is used to formulate properties which guarantee that tomographic probabilities describe quantum states in the probability representation of quantum mechanics.

MSC:
81P50 Quantum state estimation, approximate cloning
53D05 Symplectic manifolds, general
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Mancini, S.; Man’ko, V.I.; Tombesi, P., Phys. lett. A, 213, 1, (1996)
[2] Mancini, S.; Man’ko, V.I.; Tombesi, P., Found. phys., 27, 801, (1997)
[3] Man’ko, V.I.; Marmo, G.; Simoni, A.; Stern, A.; Sudarshan, E.C.G.; Ventriglia, F., Phys. lett. A, 351, 1, (2006)
[4] Ibort, A.; Man’ko, V.I.; Marmo, G.; Simoni, A.; Ventriglia, F., Phys. scr., 79, 065013, (2009)
[5] Esposito, G.; Marmo, G.; Sudarshan, E.C.G., From classical to quantum mechanics, (2004), Cambridge University Press Cambridge
[6] von Neumann, J., Nach. ges. wiss. Göttingen, 11, 245, (1927)
[7] Wigner, E., Phys. rev., 40, 749, (1932)
[8] Radon, J., Ber. verh. Sachs. akad., 69, 262, (1917)
[9] Narcowich, F.J.; O’Connell, R.F., Phys. rev. A, 34, 1, (1986)
[10] Ibort, A.; Man’ko, V.I.; Marmo, G.; Simoni, A.; Ventriglia, F., J. phys. A: math. theor., 42, 155302, (2009) · Zbl 1161.81393
[11] Naimark, M.A., Normed algebras, (1972), Wolters-Noordhoff Publ. Groningen · Zbl 0073.08902
[12] D’Ariano, G.M.; Mancini, S.; Man’ko, V.I.; Tombesi, P., J. opt. B: quantum semiclass. opt., 8, 1017, (1996)
[13] Man’ko, V.I.; Marmo, G.; Simoni, A.; Sudarshan, E.C.G.; Ventriglia, F., Rep. math. phys., 61, 337, (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.