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

arXiv:quant-ph/0511173 · Physical Review A

Tomographic reconstruction of quantum states in N spatial dimensions

Anders S. Mouritzen, Klaus Mølmer

Published Apr 14, 2006• 13 citations
arXivHTMLPDF

Abstract

Most quantum tomographic methods for massive particles using destructive measurement can only be used for motion in one dimension. We show how to infer the quantum state of a nonrelativistic, $N$-dimensional harmonic oscillator system by simple inverse Radon transforms. We assume a time-independent Hamiltonian for which the dynamics along each coordinate always occurs simultaneously with the others and measurements on all coordinates are performed simultaneously. This is unlike light systems where different modes can be delayed and phase shifted with respect to each other. A requirement of the procedure is that the angular frequencies of the $N$ harmonic potentials be incommensurable. We discuss the information available if the requirement of incommensurability is not fulfilled and also under what conditions the state can be reconstructed from finite-time measurements. As further examples, we consider free particles in $N$ dimensions with periodic boundary conditions and particles in an $N$-dimensional box, where we find a similar condition of incommensurability and finite recurrence time.

DOI
10.1103/physreva.73.042105

Related papers

Quantized Rotational Motion in a Plane

Jochen Autschbach

Abstract The angular momentum for the simplified case of a particle rotating in a fixed plane is treated. The ‘perimeter model’ is the analogue of the one-dimensional particle in a box (PiaB), with the particle moving on a circle with fixed radius. This requires cyclic – or periodic – boundary conditions. It is shown that the quantum perimeter model results can be obtained by re-interpreting the coordinate of the linear PiaB and by considering the periodic boundary conditions. The eigenvalue pattern leads to a 4n+2 Huckel rule. Next, the chapter discusses hindered rotations, such as the rotation of a methyl group around a C-C bond. The solutions to the hindered rotation problem combine features of the harmonic oscillator at low energies, with features of the perimeter model at high energies.

Oxford University Press eBooks•Dec 21, 2020•0 citations

Tomography in Abstract Hilbert Spaces

Vladimir I. Man’ko, Giuseppe Marmo, Alberto Simoni +1 more

The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.

arXiv:quant-ph/0604039•Open Systems & Information Dynamics•Aug 31, 2006•26 citations

Effective Schrodinger equation for one-dimensional systems with rapidly oscillating boundary conditions

N. P. Tretyakov, Павел Евгеньевич Голосов, Anton Kochanov

The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.

International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT)•Nov 1, 2018•1 citation

Reconstructing quantum states and expectations via dynamical tomography

Marco Peruzzo, Tommaso Grigoletto, Francesco Ticozzi

When the dynamics of a quantum system of interest is known, an informationally complete set of observables is not needed for state reconstruction via tomographic techniques: letting the system evolve before performing the measurement allows one to effectively extend the available ways to probe the system. This idea leads to dynamical quantum tomography, whose feasibility we characterize for general quantum dynamics using Krylov-based methods. Specializing in Markovian ones, we also provide deterministic tests and randomized ones to effectively assess parametric dynamics. The limits of the methods are explored by comparing unitary and open dynamics when a single observable is available, and the set of observables whose expectation can be reconstructed from the available ones characterized. The framework is illustrated with applications to a spin chain (with or without dissipation) and an electron-nuclear system.

arXiv:2509.24636•Physical Review A•Feb 13, 2026•0 citations

Phase-space tomography of the truncated harmonic oscillator states

Marta C. González, Celso L. Ladera

Phase-Space Tomography has been recently applied, in complete analogy to classical optics tomography, for the reconstruction of quantum states. The quantum systems thus far considered move in potentials for which only the coordinates of phase space are modified by simple affine transformations--shearing and rotation--, the state Wigner quasi-probability distribution remaining invariant. In this work we consider the case of the truncated harmonic oscillator, a more general, and potentially useful, potential, and show that state reconstruction is still feasible.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 19, 1999•0 citations

Quantum system driven by rapidly varying periodic perturbation

Tasko P. Grozdanov, M. J. Raković

A quantum system driven by a high-frequency periodic perturbation is studied. By using the asymptotic expansions in terms of inverse powers of the driving frequency, a class of unitary time-dependent canonical transformations is defined which renders the transformed Hamiltonians time independent. One representative of those Hamiltonians is the quasienergy operator, which is explicitly derived up to the fourth order. The classical limit of the theory and the possibility of separating the mean-motion Hamiltonian is discussed. It is shown that this separation can consistently be carried out to higher orders only in the case of a uniform external force. The application of the method to unperturbed systems with regular confining and singular potentials is discussed by considering the examples of harmonic oscillator, particle in the double-well potential, and particle in the Coulomb potential.

Physical Review A•Aug 1, 1988•65 citations
Most quantum tomographic methods for massive particles using destructive measurement can only be used for motion in one dimension. We show how to infer the quantum state of a nonrelativistic, NNN-dimensional harmonic oscillator system by simple inverse Radon transforms. We assume a time-independent Hamiltonian for which the dynamics along each coordinate always occurs simultaneously with the others and measurements on all coordinates are performed simultaneously. This is unlike light systems where different modes can be delayed and phase shifted with respect to each other. A requirement of the procedure is that the angular frequencies of the NNN harmonic potentials be incommensurable. We discuss the information available if the requirement of incommensurability is not fulfilled and also under what conditions the state can be reconstructed from finite-time measurements. As further examples, we consider free particles in NNN dimensions with periodic boundary conditions and particles in an NNN-dimensional box, where we find a similar condition of incommensurability and finite recurrence time.
Abstract The angular momentum for the simplified case of a particle rotating in a fixed plane is treated. The ‘perimeter model’ is the analogue of the one-dimensional particle in a box (PiaB), with the particle moving on a circle with fixed radius. This requires cyclic – or periodic – boundary conditions. It is shown that the quantum perimeter model results can be obtained by re-interpreting the coordinate of the linear PiaB and by considering the periodic boundary conditions. The eigenvalue pattern leads to a 4n+2 Huckel rule. Next, the chapter discusses hindered rotations, such as the rotation of a methyl group around a C-C bond. The solutions to the hindered rotation problem combine features of the harmonic oscillator at low energies, with features of the perimeter model at high energies.
The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.
The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.
When the dynamics of a quantum system of interest is known, an informationally complete set of observables is not needed for state reconstruction via tomographic techniques: letting the system evolve before performing the measurement allows one to effectively extend the available ways to probe the system. This idea leads to dynamical quantum tomography, whose feasibility we characterize for general quantum dynamics using Krylov-based methods. Specializing in Markovian ones, we also provide deterministic tests and randomized ones to effectively assess parametric dynamics. The limits of the methods are explored by comparing unitary and open dynamics when a single observable is available, and the set of observables whose expectation can be reconstructed from the available ones characterized. The framework is illustrated with applications to a spin chain (with or without dissipation) and an electron-nuclear system.
Phase-Space Tomography has been recently applied, in complete analogy to classical optics tomography, for the reconstruction of quantum states. The quantum systems thus far considered move in potentials for which only the coordinates of phase space are modified by simple affine transformations--shearing and rotation--, the state Wigner quasi-probability distribution remaining invariant. In this work we consider the case of the truncated harmonic oscillator, a more general, and potentially useful, potential, and show that state reconstruction is still feasible.
A quantum system driven by a high-frequency periodic perturbation is studied. By using the asymptotic expansions in terms of inverse powers of the driving frequency, a class of unitary time-dependent canonical transformations is defined which renders the transformed Hamiltonians time independent. One representative of those Hamiltonians is the quasienergy operator, which is explicitly derived up to the fourth order. The classical limit of the theory and the possibility of separating the mean-motion Hamiltonian is discussed. It is shown that this separation can consistently be carried out to higher orders only in the case of a uniform external force. The application of the method to unperturbed systems with regular confining and singular potentials is discussed by considering the examples of harmonic oscillator, particle in the double-well potential, and particle in the Coulomb potential.