An introduction to the tomographic picture of quantum mechanics

TL;DR

This paper reviews the tomographic probability approach to quantum mechanics, explaining how quantum states can be represented by probability distributions and exploring their relation to classical states, with applications to entanglement and experimental verification.

Contribution

It provides a comprehensive pedagogical overview of quantum tomograms, their mathematical construction, and their connection to classical probabilities, including new insights into entanglement and uncertainty relations.

Findings

Connection between quantum and classical tomograms via inequalities.

Explicit forms of superposition and uncertainty relations in probabilities.

Method proposed for experimental verification of uncertainty relations using optical tomograms.

Abstract

Starting from the famous Pauli problem on the possibility to associate quantum states with probabilities, the formulation of quantum mechanics in which quantum states are described by fair probability distributions (tomograms, i.e. tomographic probabilities) is reviewed in a pedagogical style. The relation between the quantum state description and the classical state description is elucidated. The difference of those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to nonnegativity of probability density on phase space in the classical domain. Intersection of such sets is studied. The mathematical mechanism which allows to construct different kinds of tomographic probabilities like symplectic tomograms, spin tomograms, photon number tomograms, etc., is clarified and a connection with abstract Hilbert space properties is established. Superposition rule and uncertainty relations in terms of probabilities as well as quantum basic equation like quantum evolution and energy spectra equations are given in explicit form. A method to check experimentally uncertainty relations is suggested using optical tomograms. Entanglement phenomena and the connection with semigroups acting on simplexes are studied in detail for spin states in the case of two qubits. The star-product formalism is associated with the tomographic probability formulation of quantum mechanics.