On the meaning and interpretation of tomography in abstract Hilbert spaces

2005/10/20 by V. I. Man'ko, V. I. Manʹko, G. Marmo +4 · 3 citations
Computer Science · Physics and Astronomy · #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum optics and atomic interactions #quant-ph

paper · pdf · doi:10.1016/j.physleta.2005.10.063

published as Phys.Lett.A351:1-12,2006 · Submitted to Phys. Lett. A

arxiv created 2005/10/20 · openalex publication_date 2005/10/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The mechanism of describing quantum states by standard probability (tomographic one) instead of wave function or density matrix is elucidated. Quantum tomography is formulated in an abstract Hilbert space framework, by means of the identity decompositions in the Hilbert space of hermitian linear operators, with trace formula as scalar product of operators. Decompositions of 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.

Citations

• The Principles of Quantum Mechanics
• Reconstructing the density operator by using generalized field quadratures
• Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase
• Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams
• On the Quantum Correction For Thermodynamic Equilibrium

Cited by

• Bell's inequalities in the tomographic representation
• On the tomographic picture of quantum mechanics
• Qubit portraits of qudit states and quantum correlations