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arXiv:quant-ph/0507078 · IMPERIAL COLLEGE PRESS eBooks

Homodyne Tomography and the Reconstruction of Quantum States of Light

Giacomo Mauro D’Ariano, Lorenzo Maccone, Massimiliano Federico Sacchi

Published Feb 1, 2007• 8 citations
arXivHTMLPDF

Abstract

Quantum tomography is a procedure to determine the quantum state of a physical system, or equivalently, to estimate the expectation value of any operator. It consists in appropriately averaging the outcomes of the measurement results of different observables, obtained on identical copies of the same system. Alternatively, it consists in maximizing an appropriate likelihood function defined on the same data. The procedure can be also used to completely characterize an unknown apparatus. Here we focus on the electromagnetic field, where the tomographic observables are obtained from homodyne detection.

DOI
10.1142/9781860948169_0008

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Quantum tomography is a procedure to determine the quantum state of a physical system, or equivalently, to estimate the expectation value of any operator. It consists in appropriately averaging the outcomes of the measurement results of different observables, obtained on identical copies of the same system. Alternatively, it consists in maximizing an appropriate likelihood function defined on the same data. The procedure can be also used to completely characterize an unknown apparatus. Here we focus on the electromagnetic field, where the tomographic observables are obtained from homodyne detection.
Tomography is the name under which all state reconstruction techniques are denoted,\none of the most recognized examples being medical tomography. Quantum\nstate tomography is a procedure to determine the quantum state of a physical system.\nBy performing homodyne measurements on resonance fluorescence from an\nartificial atom coupled to a one-dimensional transmission line, its quantum state\ncan be reconstructed. Resonance fluorescence is one of the simplest setups that\nresults in non-classical states of light. If these states are non-classical in the sense\nthat they have a negative Wigner function, they can be used as a computational\nresource for quantum computing.\nThere are many different approaches to quantum computing. Some, like gate\nbased quantum computing using discrete variables like qubits, have been extensively\nresearched, both theoretically and experimentally. There exists and alternative\napproach: continuous variable quantum computing. The continuous variables\nwe will be concerned with are the components of the electromagnetic field that\nconstitute the resonance fluorescence.\nThere are different parameters that affect the nature of the resonance fluorescence,\nfor example, the number of transmission lines the atom is coupled to, or\nthe strength of the driving field. In this work, we develop the tools necessary to\nnumerically simulate homodyne detection of resonance fluorescence for different\nsets of parameters, and reconstruct the quantum state as well as calculating the\nWigner negativity.
Summary form only given. Experimental techniques for measurement of the quantum state of light have been the subject of intensive investigation for some time. Tomographic techniques have been applied to experiments such as the homodyne measurement of the Wigner function of a single mode of light and of the density matrix of the polarization degrees of freedom of a pair of entangled photons. In this technique, the density matrix (or Wigner function) which characterizes the quantum state of the system being measured is found from a linear transformation of experimental data. There are a number of drawbacks to the method, principally in that the recovered state might not, because of experimental noise, correspond to a physical state. For example, density matrices for any quantum state must be Hermitian, positive semi-definite matrices with unit trace. The tomographically measured matrices often fail to be positive semi-definite. To avoid this problem, the maximum likelihood approach to the estimation of quantum states has been developed. We evaluate two different approaches to the maximum likelihood technique, namely defining the likelihood function f as a weighted sum of squared variances from measured data, and defining it in terms of information content. These are compared with results of standard tomographical schemes.
Summary We describe quantum tomography as an inverse statistical problem in which the quantum state of a light beam is the unknown parameter and the data are given by results of measurements performed on identical quantum systems. The state can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We present consistency results for pattern function projection estimators and for sieve maximum likelihood estimators for both the density matrix of the quantum state and its Wigner function. We illustrate the performance of the estimators on simulated data. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. rates of convergence, adaptation and studying other estimators; a main purpose of the paper is to bring these to the attention of the statistical community.
This chapter provides the relevant theory and techniques of quantum tomography. Quantum tomography aims to determine the state of the system by determining its density matrix by means of a series of measurements done on fairly a large number of identically prepared copies of the same system. Wolfgang Ernst Pauli problem asks the question whether measurements of probability densities of position and momentum of a particle determine its quantum state wave function. The chapter outlines the optical homodyne tomography technique of reconstruction of Wigner function. Polarized light wave is described by the commonly used Stokes parameters introduced by George Gabriel Stokes in 1852. Quantum process tomography (QPT) aims to give full information on the dynamics of a quantum state. QPT can be used to improve the design and control of quantum hardware. Direct characterization of quantum dynamics differs in many ways from standard quantum process tomography and ancilla-assisted process tomography.