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Journal of Physics A Mathematical and General

Tomographic representation of spin and quark states

A. B. Klimov, Olga V Man apos ko, Vladimir I Man ko, Yu.F. Smirnov, V.N. Tolstoy

Published Jul 12, 2002• 51 citations
Publisher

Abstract

We present a short review of the general principles of constructing tomograms of quantum states. We derive a general tomographic reconstruction formula for the quantum density operator of a system with a dynamical Lie group. In the reconstruction formula, the multiplicity of irreducible representation in Clebsch–Gordan decomposition is taken into account. Various approaches to spin tomography are discussed. An integral representation for the tomographic probability is found and a contraction of the spin tomogram to the photon-number tomography distribution is considered. The case of SU (3) tomography is discussed with the examples of quark states (related to the simplest triplet representations) and octet states.

DOI
10.1088/0305-4470/35/29/312

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We present a short review of the general principles of constructing tomograms of quantum states. We derive a general tomographic reconstruction formula for the quantum density operator of a system with a dynamical Lie group. In the reconstruction formula, the multiplicity of irreducible representation in Clebsch–Gordan decomposition is taken into account. Various approaches to spin tomography are discussed. An integral representation for the tomographic probability is found and a contraction of the spin tomogram to the photon-number tomography distribution is considered. The case of SU (3) tomography is discussed with the examples of quark states (related to the simplest triplet representations) and octet states.
We review the probabilistic representation of quantum mechanics within which states are described by the probability distribution rather than by the wavefunction and density matrix. Uncertainty relations have been obtained in the form of integral inequalities containing measurable optical tomograms of quantum states. Formulas for the transition probabilities and purity parameter have been derived in terms of the tomographic probability distributions. Inequalities for Shannon and Rényi entropies associated with quantum tomograms have been obtained. A scheme of the star product of tomograms has been developed.
It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrödinger–Dirac picture of quantum mechanics on Hilbert spaces. In this picture, states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture, which has evolved in the C ⋆ -algebraic approach to quantum mechanics, starts with the algebra of observables and introduces states as a derived concept. The equivalence between these two pictures amounts, essentially, to the Gelfand–Naimark–Segal construction. In this construction, the abstract C ⋆ -algebra is realized as an algebra of operators acting on a constructed Hilbert space. The representation that is defined may be reducible or irreducible, but in either case it allows us to identify a unitary group associated with the C ⋆ -algebra by means of its invertible elements. In this picture both states and observables are appropriate functions on the group; it also follows that quantum tomograms are strictly related with appropriate functions (positive-type) on the group. In this paper we present, using very simple examples, a tomographic description emerging from the set of ideas connected with the C ⋆ -algebra picture of quantum mechanics. In particular, we introduce the tomographic probability distributions for finite and compact groups, and formulate an autonomous criterion to recognize a given probability distribution as a tomogram of quantum state.
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.
In this job, we will present a theory called Quantum Tomography that is the natural extension of the theory of detection of signals in classical telecommunications to Quantum Mechanics. This theory mainly consists in the reconstruction of a quantum state of a system through a probability distribution measured directly in the laboratory, usually called Tomogram. This Thesis contains five chapters. In the first one, we will show the birth of this theory as an adaptation of homodyne and heterodyne classical detection to Quantum Mechanics. In the second, we will describe a tomographic description of Quantum Mechanics on C*-algebras by splitting the theory in two parts: a Generalized Sampling Theory and a Generalized Positive Transform. In the third, we will present the first numerical algorithm that solves the Clebsh-Gordan decomposition problem for any finite-dimensional unitary representation of any finite or compact Lie group. This algorithm receives the name of SMILY algorithm. In the fourth and fifth chapters, we will discuss tomography of systems parametrized with fields. In the fourth, we will reconstruct states of classical systems and in the fifth, we will do the same with quantum ones, but here in two different ways: the first one will be done by adapting the reconstruction formula of chapter four, and the second one by means of the axiomatic description of Wightamn-Streater of a Quantum Field Theory.
Entangled and separable states of a bipartite (multipartite) system are studied in the tomographic representation of quantum states. Properties of tomograms (joint probability distributions) corresponding to entangled states are discussed. The connection with star-product quantization is presented. U(N)-tomography and spin tomography as well as the relation of the tomograms to positive and completely positive maps are considered. The tomographic criterion of separability (necessary and sufficient condition) is formulated in terms of the equality of the specific function depending on unitary group parameters and positive map semigroup parameters to unity. Generalized Werner states are used as an example.