On the meaning and interpretation of tomography in abstract Hilbert spaces.

The mechanism of describing quantum states by standard probabilities (tomograms) instead of wave function or density matrix is elucidated. Quantum tomography is formulated in an abstract Hilbert space framework, by means of decompositions of the identity in the Hilbert space of hermitian linear operators, with trace formula as scalar product of operators. Decompositions of the identity are considered with respect to over-complete families of projectors labeled by extra parameters and containing a measure, depending on these parameters. It plays the role of a Gram-Schmidt orthonormalization kernel. When the measure is equal to one, the decomposition of identity coincides with a positive operator-valued measure (POVM) decomposition. Examples of spin tomography, photon number tomography and symplectic tomography are reconsidered in this new framework.

[1]  V. Man'ko,et al.  Star-Product of Generalized Wigner-Weyl Symbols on SU(2) Group, Deformations, and Tomographic Probability Distribution , 2000 .

[2]  E. Sudarshan.  Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams , 1963 .

[3]  P. Dirac.  Principles of Quantum Mechanics , 1982 .

[4]  LETTER TO THE EDITOR: Reconstructing the density matrix of a spin s through Stern-Gerlach measurements: II , 1999, quant-ph/9903067.

[5]  Quantum Bi-Hamiltonian Systems , 2000, math-ph/0610011.

[6]  Vogel,et al.  Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. , 1989, Physical review. A, General physics.

[7]  Alternative commutation relations, star products and tomography , 2001, quant-ph/0112110.

[8]  Stefano Mancini,et al.  Density matrix from photon number tomography , 1996, quant-ph/9612004.

[9]  Olga V. Man'ko,et al.  Spin state tomography , 1997 .

[10]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[11]  C. Ross.  Found , 1869, The Dental register.

[12]  Y. Kano,et al.  A NEW PHASE-SPACE DISTRIBUTION FUNCTION IN THE STATISTICAL THEORY OF THE ELECTROMAGNETIC FIELD , 1965 .

[13]  Tobias J. Hagge,et al.  Physics , 1929, Nature.

[14]  Vladimir I. Man’ko,et al.  Squeeze tomography of quantum states , 2004 .

[15]  Classical-like description of quantum dynamics by means of symplectic tomography , 1996, quant-ph/9609026.

[16]  Vladimir I. Man’ko,et al.  Positive distribution description for spin states , 1997 .

[17]  R. G. Lerner,et al.  Encyclopedia of Physics , 1990 .

[18]  Wolfgang Pauli,et al.  Handbuch der Physik , 1904, Nature.

[19]  V. I. Man'ko,et al.  Reconstructing the density operator by using generalized field quadratures , 1996, quant-ph/9606034.

[20]  Banaszek,et al.  Direct probing of quantum phase space by photon counting. , 1996, Physical review letters.

[21]  Stefano Mancini,et al.  Wigner function and probability distribution for shifted and squeezed quadratures , 1995 .

[22]  V. I. Man'ko,et al.  Symplectic tomography as classical approach to quantum systems , 1996 .

[23]  Roger G. Newton,et al.  Measurability of the spin density matrix , 1968 .

[24]  E. Wigner.  On the quantum correction for thermodynamic equilibrium , 1932 .

[25]  M. De.  Handbuch der Physik , 1957 .

[26]  Vogel,et al.  Unbalanced homodyning for quantum state measurements. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[27]  Richard S. Millman,et al.  Symmetries in Science , 1987 .

[28]  R. Glauber.  Coherent and incoherent states of the radiation field , 1963 .

[29]  L. Landau.  Das Dämpfungsproblem in der Wellenmechanik , 1927 .

[30]  Entanglement Structure of the Adjoint Representation of the Unitary Group and Tomography of Quantum States , 2003, quant-ph/0310022.

[31]  P. Bertrand,et al.  A tomographic approach to Wigner's function , 1987 .