Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Kernel of star-product for spin tomograms

Octavio Castaños, Ram n L pez-Pe a, Margarita A Man ko, Vladimir I Man ko

Published Apr 8, 2003• 36 citations
Publisher

Abstract

The spin-state tomogram is presented as the modulus squared of the matrix element of the SU (2)-group irreducible representation. For the spin system, we established explicitly a realization of the set of operators defining for spin tomograms the star-product formalism in terms of irreducible tensors. On the set of spin tomograms, the delta-function and ‘standard’ and Moyal-like kernels of the tomogram star-product are calculated explicitly in terms of Clebsch–Gordan and Racah coefficients and matrix elements of the SU (2) irreducible representation. In the limit of infinite spin, the spin tomogram is shown to become the tomogram of the harmonic-oscillator state.

DOI
10.1088/0305-4470/36/16/316

Related papers

Star product, discrete Wigner functions, and spin-system tomograms

Петер Адам, V. A. Andreev, Аурелиан Исар +2 more

We develop the star-product formalism for spin states and consider different methods for constructing operator systems forming sets of dequantizers and quantizers, establishing a relation between them. We study the physical meaning of the operator symbols related to them. Quantum tomograms can also serve as operator symbols. We show that the possibility to express discrete Wigner functions in terms of measurable quantities follows because these functions can be related to quantum tomograms. We investigate the physical meaning of tomograms and spin-system tomogram symbols, which they acquire in the framework of the star-product formalism. We study the structure of the sum kernels, which can be used to express the operator symbols, calculated using different sets of dequantizers and also arising in calculating the star product of operator symbols, in terms of one another.

Theoretical and Mathematical Physics•Mar 1, 2016•10 citations

SPIN TOMOGRAPHY AND STAR-PRODUCT KERNEL FOR QUBITS AND QUTRITS

Vladimir I. Man’ko

Using the irreducible tensor-operator technique, we establish the relation between different forms of spin tomograms. Quantizer and dequantizer operators are presented in simple explicit forms and are specified for the low-spin states. The kernel of the star-product is evaluated for qubits and qutrits and its connection with a generic formula is found.

•Aug 5, 2013•15 citations

Tomographic representation of spin and quark states

A. B. Klimov, Olga V Man apos ko, Vladimir I Man ko +2 more

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.

Journal of Physics A Mathematical and General•Jul 12, 2002•51 citations

TOMOGRAPHIC MAP WITHIN THE FRAMEWORK OF STAR-PRODUCT QUANTIZATION

Olga V. Man’ko, Vladimir I. Man’ko, Giuseppe Marmo

Tomograms introduced for the description of quantum states in terms of probability distributions are shown to be related to a standard star-product quantization with appropriate kernels. Examples of symplectic tomograms and spin tomograms are presented.

arXiv:quant-ph/0112112•International Symposium on Quantum Theory and Symmetries (QTS)•Jun 1, 2002•11 citations

A Spinor Approach to the SU(2) Clebsch-Gordan Coefficients*

Ling-Di Yang, Famin Chen

Abstract We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.

Communications in Theoretical Physics•Jun 1, 2019•0 citations

Moyal-like form of the star product for generalizedSU(2) Stratonovich-Weyl symbols

A. B. Klimov, P.B. Espinoza

The star product is the basic tool used in the phase-space formulation of quantum mechanics. We find a differential form of the star product for a class of \mbox{$s$-parametrized} SU (2) Stratonovich-Weyl symbols which appear in the phase-space representation of spin-like systems. The limit of large spin is considered and the asymptotic form of the differential operator defining the star product is obtained.

Journal of Physics A Mathematical and General•Sep 24, 2002•39 citations
The spin-state tomogram is presented as the modulus squared of the matrix element of the SU (2)-group irreducible representation. For the spin system, we established explicitly a realization of the set of operators defining for spin tomograms the star-product formalism in terms of irreducible tensors. On the set of spin tomograms, the delta-function and ‘standard’ and Moyal-like kernels of the tomogram star-product are calculated explicitly in terms of Clebsch–Gordan and Racah coefficients and matrix elements of the SU (2) irreducible representation. In the limit of infinite spin, the spin tomogram is shown to become the tomogram of the harmonic-oscillator state.
We develop the star-product formalism for spin states and consider different methods for constructing operator systems forming sets of dequantizers and quantizers, establishing a relation between them. We study the physical meaning of the operator symbols related to them. Quantum tomograms can also serve as operator symbols. We show that the possibility to express discrete Wigner functions in terms of measurable quantities follows because these functions can be related to quantum tomograms. We investigate the physical meaning of tomograms and spin-system tomogram symbols, which they acquire in the framework of the star-product formalism. We study the structure of the sum kernels, which can be used to express the operator symbols, calculated using different sets of dequantizers and also arising in calculating the star product of operator symbols, in terms of one another.
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.
Abstract We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.
The star product is the basic tool used in the phase-space formulation of quantum mechanics. We find a differential form of the star product for a class of \mbox{sss-parametrized} SU (2) Stratonovich-Weyl symbols which appear in the phase-space representation of spin-like systems. The limit of large spin is considered and the asymptotic form of the differential operator defining the star product is obtained.