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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Tomographic approach and distances between quantum states

A. V. Kusev, Olga V. Man’ko, Nicolaii V. Tcherniega

Published Mar 3, 2006• 2 citations
Publisher

Abstract

Notion of distance between quantum states is reviewed using Hellinger distance between probability distributions and classical formulations of quantum mechanics based on tomography scheme. The Hellinger distances between the states of damped oscillator (considered in the frame of Caldirola-Kanai model) with delta-kicked frequencies in the regime of weak and strong damping, between the photon number distribution functions in SRS process, and between spin tomograms are considered.

DOI
10.1117/12.675009

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Notion of distance between quantum states is reviewed using Hellinger distance between probability distributions and classical formulations of quantum mechanics based on tomography scheme. The Hellinger distances between the states of damped oscillator (considered in the frame of Caldirola-Kanai model) with delta-kicked frequencies in the regime of weak and strong damping, between the photon number distribution functions in SRS process, and between spin tomograms are considered.
Starting from the famous Pauli problem on the possibility of associating 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 between those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to non-negativity of probability density on phase space in the classical domain. The intersection of such sets is studied. The mathematical mechanism that allows us 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. The superposition rule and uncertainty relations in terms of probabilities as well as quantum basic equations like quantum evolution and energy spectra equations are given in an explicit form. A method to check experimentally the 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.
A review of the symplectic, optical tomography and photon number tomography is presented. Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied, and it is shown that the sets have common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vica versa, there exist tomograms of quantum states which are not admissible in classical mechanics. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively.
A review of the symplectic, optical tomography and photon number tomography is presented. Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied, and it is shown that the sets have common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vica versa, there exist tomograms of quantum states which are not admissible in classical mechanics. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively.
A review of the tomographic-probability representation of classical and quantum states is presented. The tomographic entropies and entropic uncertainty relations are discussed in connection with ambiguities in the interpretation of the state tomograms which are considered either as a set of the probability distributions of random variables depending on extra parameters or as a single joint probability distribution of these random variables and random parameters with specific properties of the marginals. Examples of optical tomograms of photon states, symplectic tomograms, and unitary spin tomograms of qudits are given. A new universal integral inequality for generic wave function is obtained on the base of tomographic entropic uncertainty relations.