© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1204.5231 · Physica Scripta

A pedagogical presentation of aC⋆-algebraic approach to quantum tomography

Alberto Ibort, Vladimir I. Man’ko, Giuseppe Marmo, Alberto Simoni, F. Ventriglia

Published Nov 22, 2011• 22 citations

Abstract

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.

DOI: 10.1088/0031-8949/84/06/065006

Related papers

An introduction to the tomographic picture of quantum mechanics

Alberto Ibort, Vladimir I. Man’ko, Giuseppe Marmo +2 more

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.

arXiv:0904.4439•Physica Scripta•Jun 1, 2009•262 citations

Groupoids and the tomographic picture of quantum mechanics

Alberto Ibort, Vladimir I. Man’ko, G Marmo +3 more

The existing relation between the tomographic description of quantum states and the convolution algebra of certain discrete groupoids represented on Hilbert spaces will be discussed. The realizations of groupoid algebras based on qudit, photon-number (Fock) states and symplectic tomography quantizers and dequantizers will be constructed. Conditions for identifying the convolution product of groupoid functions and the star--product arising from a quantization--dequantization scheme will be given. A tomographic approach to construct quasi--distributions out of suitable immersions of groupoids into Hilbert spaces will be formulated and, finally, intertwining kernels for such generalized symplectic tomograms will be evaluated explicitly.

arXiv:1309.2782•Physica Scripta•Oct 8, 2013•18 citations

On the tomographic description of quantum systems: theory and applications

Alberto López-Yela

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.

arXiv:1512.08248•Dec 27, 2015•1 citation

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

The probability representation as a new formulation of quantum mechanics

Margarita A. Man’ko, Vladimir I. Man’ko

We present a new formulation of conventional quantum mechanics, in which the notion of a quantum state is identified via a fair probability distribution of the position measured in a reference frame of the phase space with rotated axes. In this formulation, the quantum evolution equation as well as the equation for finding energy levels are expressed as linear equations for the probability distributions that determine the quantum states. We also give the integral transforms relating the probability distribution (called the tomographic-probability distribution or the state tomogram) to the density matrix and the Wigner function and discuss their connection with the Radon transform. Qudit states are considered and the invertible map of the state density operators onto the probability vectors is discussed. The tomographic entropies and entropic uncertainty relations are reviewed. We demonstrate the uncertainty relations for the position and momentum and the entropic uncertainty relations in the tomographic-probability representation, which is suitable for an experimental check of the uncertainty relations.

Journal of Physics Conference Series•Aug 24, 2012•3 citations

A probabilistic approach to quantum mechanics based on ‘tomograms’

Michele Caponigro, Stefano Mancini, Vladimir I. Man’ko

Abstract It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical‐like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.

arXiv:quant-ph/0605144•Fortschritte der Physik•May 11, 2006•8 citations

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.

The existing relation between the tomographic description of quantum states and the convolution algebra of certain discrete groupoids represented on Hilbert spaces will be discussed. The realizations of groupoid algebras based on qudit, photon-number (Fock) states and symplectic tomography quantizers and dequantizers will be constructed. Conditions for identifying the convolution product of groupoid functions and the star--product arising from a quantization--dequantization scheme will be given. A tomographic approach to construct quasi--distributions out of suitable immersions of groupoids into Hilbert spaces will be formulated and, finally, intertwining kernels for such generalized symplectic tomograms will be evaluated explicitly.

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.

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 present a new formulation of conventional quantum mechanics, in which the notion of a quantum state is identified via a fair probability distribution of the position measured in a reference frame of the phase space with rotated axes. In this formulation, the quantum evolution equation as well as the equation for finding energy levels are expressed as linear equations for the probability distributions that determine the quantum states. We also give the integral transforms relating the probability distribution (called the tomographic-probability distribution or the state tomogram) to the density matrix and the Wigner function and discuss their connection with the Radon transform. Qudit states are considered and the invertible map of the state density operators onto the probability vectors is discussed. The tomographic entropies and entropic uncertainty relations are reviewed. We demonstrate the uncertainty relations for the position and momentum and the entropic uncertainty relations in the tomographic-probability representation, which is suitable for an experimental check of the uncertainty relations.

Abstract It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical‐like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.