An introduction to the tomographic picture of quantum mechanics

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.

Probability representation and quantumness tests for qudits and two-mode light states

“…Maximum mean value of quantumness witness against number of levels N = 2j + 1 (dots). The solid line is an asymptote of this dependence and is predicted by (16). Negativity of all the values allows one to detect quantumness of any qudit state with an arbirary spin j.…”

“…The approach, called the tomographic-probability representation of quantum states, was developed, e.g., in [14,15] and reviewed recently in [16]. This approach turned out to be convenient [17] to study the quantumness test [5,6] given for qubits.…”

Torus as phase space: Weyl quantization, dequantization, and Wigner formalism

“…We define the Wigner transform of F in the representation (θ, N ) as the distribution on T 2 defined by 24) where the Fourier series converges in the sense of distribution, (since V θ,N (F ) is uniformly bounded).…”

Canonical transforms, quantumness and probability representation of quantum mechanics

“…The symplectic tomogram for the Moshinsky shutter problem can also be obtained as a solution to the tomographic evolution equation (12); for t 1 = t 2 = t, it reads [15] w(X, µ, ν, t) = 1 2|µ|…”