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Kluwer Academic Publishers eBooks

Quantum Tomography: Renormalization of Incompatible Observations

Z. Hradil, Robert Myška, Johann Summhammer, H. Rauch

Published Dec 19, 2005• 0 citations
Publisher

Abstract

Standard deterministic techniques for quantum state reconstruction, as for example optical homodyne tomography, photon chopping, unbalanced homodyning etc., are based on the deterministic inversion of measured data. Since the frequencies obtained in realistic experiments always differ from probabilities predicted by quantum theory due to fluctuations, imperfections and realistic restrictions, the algorithm of inversion cannot guarantee the positive definiteness of the reconstructed density matrix. Hence the estimation of the noises may appear as doubtful. Quantum states may be successfully reconstructed within quantum and information theories using the maximum likelihood estimation. The question of deterministic schemes: “What quantum state is determined by that measurement?” is replaced by the formulation consistent with quantum theory: “What quantum state(s) seems to be most likely for that measurement?” Nonlinear equation for reconstructed state is formulated. An exact solution may be approached by subsequent iterations. Reconstruction is formulated as a problem of proper normalization of incompatible (nonorthogonal) measurements. The results obtained by this novel method may differ significantly from the standard predictions. Data are fitted better keeping the constraint of positive definiteness of reconstructed density matrix. However, this interpretation may enlarge uncertainty in prediction of quantum state in comparison with deterministic schemes, since, in general, there is a whole family of states which fit the measured data equally well. The novel technique is nonlinear and the reinterpretation of existing reconstruction schemes represents an advanced program. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/0-306-47097-7_23

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Standard deterministic techniques for quantum state reconstruction, as for example optical homodyne tomography, photon chopping, unbalanced homodyning etc., are based on the deterministic inversion of measured data. Since the frequencies obtained in realistic experiments always differ from probabilities predicted by quantum theory due to fluctuations, imperfections and realistic restrictions, the algorithm of inversion cannot guarantee the positive definiteness of the reconstructed density matrix. Hence the estimation of the noises may appear as doubtful. Quantum states may be successfully reconstructed within quantum and information theories using the maximum likelihood estimation. The question of deterministic schemes: “What quantum state is determined by that measurement?” is replaced by the formulation consistent with quantum theory: “What quantum state(s) seems to be most likely for that measurement?” Nonlinear equation for reconstructed state is formulated. An exact solution may be approached by subsequent iterations. Reconstruction is formulated as a problem of proper normalization of incompatible (nonorthogonal) measurements. The results obtained by this novel method may differ significantly from the standard predictions. Data are fitted better keeping the constraint of positive definiteness of reconstructed density matrix. However, this interpretation may enlarge uncertainty in prediction of quantum state in comparison with deterministic schemes, since, in general, there is a whole family of states which fit the measured data equally well. The novel technique is nonlinear and the reinterpretation of existing reconstruction schemes represents an advanced program. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
Self-calibrating states are introduced as quantum resources for the simultaneous reconstruction of a quantum state and measurement device. Simple criteria for a state being self-calibrating are formulated. Theory applies to realistic state estimation scenarios, where the detection model is nonlinear in unknown parameters of the measuring device. Examples discussed include time multiplexed detection with on/off detectors and quantum homodyne tomography. The thermally noised single-photon states and Gaussian states are self-calibrating for these schemes. For homodyne tomography, statistical tests applied to raw measured data are sufficient for judging whether or not the signal state is self-calibrating without any a priori information.
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.
The intrinsic relationship between the maximum-likelihood quantum-state estimation and the representation of the signal is elaborated. A quantum analogy of the transfer function determines the space where a successful reconstruction can be achieved. This provides a tool for reducing the number of dimensions of the observed system based on physical characteristics of the reconstruction scheme rather than some ad hoc truncations. The method is illustrated with two examples of practical importance: an optical quantum homodyne tomography and a simple and robust tomography of an optical signal recorded by realistic binary detectors.
The problem of optical homodyne tomography is considered in the context of a Bayesian model-fitting or inverse problem approach. An algorithm is formulated, based on matrix computation rather than the numerical approximation of an analytic inverse transform. This automatically takes into account the effects of noise, detector inefficiencies and incomplete sampling of the data. The relationships with conventional reconstruction schemes, methods for including various forms of prior information and for calculating error estimates are discussed. The process of reconstructing the photon number distribution and the density matrix are illustrated using both simulated and experimental data.