Tomographic reconstruction of quantum states in N spatial dimensions
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, Ndimensional harmonic oscillator system by simple inverse Radon transforms. We assume a timeindependent 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 finitetime measurements. As further examples, we consider free particles in N dimensions with periodic boundary conditions and particles in an Ndimensional box, where we find a similar condition of incommensurability and finite recurrence time.
 Authors:
 QUANTOP, Danish National Research Foundation Center for Quantum Optics, Department of Physics and Astronomy, University of Aarhus, DK8000 Aarhus C (Denmark)
 Publication Date:
 OSTI Identifier:
 20787065
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevA.73.042105; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 74 ATOMIC AND MOLECULAR PHYSICS; BOUNDARY CONDITIONS; COORDINATES; ENERGY LEVELS; HAMILTONIANS; HARMONIC OSCILLATORS; HARMONIC POTENTIAL; PARTICLES; PERIODICITY; PHASE SHIFT; RADON; TIME MEASUREMENT; TOMOGRAPHY
Citation Formats
Mouritzen, Anders S., and Moelmer, Klaus. Tomographic reconstruction of quantum states in N spatial dimensions. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.
Mouritzen, Anders S., & Moelmer, Klaus. Tomographic reconstruction of quantum states in N spatial dimensions. United States. doi:10.1103/PHYSREVA.73.0.
Mouritzen, Anders S., and Moelmer, Klaus. Sat .
"Tomographic reconstruction of quantum states in N spatial dimensions". United States.
doi:10.1103/PHYSREVA.73.0.
@article{osti_20787065,
title = {Tomographic reconstruction of quantum states in N spatial dimensions},
author = {Mouritzen, Anders S. and Moelmer, Klaus},
abstractNote = {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, Ndimensional harmonic oscillator system by simple inverse Radon transforms. We assume a timeindependent 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 finitetime measurements. As further examples, we consider free particles in N dimensions with periodic boundary conditions and particles in an Ndimensional box, where we find a similar condition of incommensurability and finite recurrence time.},
doi = {10.1103/PHYSREVA.73.0},
journal = {Physical Review. A},
number = 4,
volume = 73,
place = {United States},
year = {Sat Apr 15 00:00:00 EDT 2006},
month = {Sat Apr 15 00:00:00 EDT 2006}
}

We propose a highefficiency scheme to tomographically reconstruct an unknown quantum state by using a series of quantum nondemolition (QND) measurements. The proposed QND measurements of the qubits are implemented by probing the stationary transmissions through a driven dispersively coupled resonator. It is shown that only one kind of QND measurement is sufficient to determine all the diagonal elements of the density matrix of the detected quantum state. The remaining nondiagonal elements can be similarly determined by transferring them to the diagonal locations after a series of unitary operations. Compared with the tomographic reconstructions based on the usual destructive projectivemore »

Quantum quenches in two spatial dimensions using chain array matrix product states
We describe a method for simulating the real time evolution of extended quantum systems in two dimensions (2D). The method combines the benefits of integrability and matrix product states in one dimension to avoid several issues that hinder other applications of tensor based methods in 2D. In particular, it can be extended to infinitely long cylinders. As an example application we present results for quantum quenches in the 2D quantum [(2+1)dimensional] Ising model. As a result, in quenches that cross a phase boundary we find that the return probability shows nonanalyticities in time.Cited by 9 
Tomographic reconstruction of quantum correlations in excited BoseEinstein condensates
We propose to use quantum tomography to characterize the state of a perturbed BoseEinstein condensate. We assume knowledge of the number of particles in the zerowavenumber mode and of density distributions in space at different times, and we treat the condensate in the Bogoliubov approximation. For states that can be treated with the GrossPitaevskii equation, we find that the reconstructed density operator gives excellent predictions of the second moments of the atomic creation and annihilation operators, including the onebody density matrix. Additional inclusion of the momentum distribution at one point of time enables somewhat reliable predictions to be made formore » 
Quantum tomographic cryptography with Bell diagonal states: Nonequivalence of classical and quantum distillation protocols
We present a generalized tomographic quantum key distribution protocol in which the two parties share a Bell diagonal mixed state of two qubits. We show that if an eavesdropper performs a coherent measurement on many quantum ancilla states simultaneously, classical methods of secure key distillation are less effective than quantum entanglement distillation protocols. We also show that certain classes of Bell diagonal states are resistant to any attempt at incoherent eavesdropping.