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Optics and Spectroscopy

Quantum theory of incompatible observations

Z. Hradil

Published Sep 1, 2001• 1 citation
Publisher

Abstract

Quantum theory allows the sequential measurement of incompatible observables in the course of repeated measurements. To get information about the observed system, all observations must be synthesized. This is the main idea of quantum tomographical methods. As shown in this paper, the maximum likelihood principle provides the best measure for relating the experimental data with predictions of quantum theory. Synthesis of incompatible observations appears to be a new quantum measurement described by a positive operator-valued measure. Besides this, the procedure finds the optimal state of the system, which best fits such a measurement.

DOI
10.1134/1.1405215

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Quantum theory allows the sequential measurement of incompatible observables in the course of repeated measurements. To get information about the observed system, all observations must be synthesized. This is the main idea of quantum tomographical methods. As shown in this paper, the maximum likelihood principle provides the best measure for relating the experimental data with predictions of quantum theory. Synthesis of incompatible observations appears to be a new quantum measurement described by a positive operator-valued measure. Besides this, the procedure finds the optimal state of the system, which best fits such a measurement.
The question of what information about a quantum state may be inferred from a sequence of measurements made on it, is addressed. The main result is that maximum-likelihood estimation gives an arguably natural optimum approach in quantum theory. It singles out a particular choice of basis, in which the state is expressed. Various incompatible and incomplete observations made on the ensemble are treated as a complete generalized measurement. Consequently, the maximum likelihood (relative entropy) is the best measure for relating experimental data with theoretical predictions of quantum theory.
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.
Maximum-likelihood estimation is applied to the determination of an unknown quantum measurement. The calibrated measuring apparatus carries out measurements on many different quantum states and the positive operator-valued measure governing the measurement statistics is then inferred from the collected data via the maximum-likelihood principle. In contrast to the procedures based on linear inversion, our approach always provides a physically sensible result. We illustrate the method on the case of the Stern-Gerlach apparatus.
This thesis studies the field of quantum measurements and more precisely sequential measurements. The aim is to provide answers to questions like when is performing a sequential or a repeated measurement meaningful, or even possible. This is important in order to understand what type of information can be acquired when performing measurements on quantum states. In basic quantum theory observables are identified with self-adjoint operators. We introduce the concept of a positive operator valued measure (POVM) to gain better understanding on observables. The downside to POVMs is that they don't produce a new state after the measurement, only measurement statistics. In order to perform multiple measurements on a quantum state, the concepts of measurement model and instrument are defined in this thesis. Using instruments to perform measurements is essential in order to describe a state after the initial measurement is performed. This way it's possible to make measurements sequentially. After defining the necessary tools for sequential measurements and a few different properties related to them, some applicable situations for these kind of measurements are shown. This way sequential measurements end up being somewhat meaningful in order to gain information from quantum systems.
ABSTRACT The simultaneous inseparable reconstruction of quantum process and states is presented. The method is basedon constrained extremization of the likelihood functional. This statistical approach is ingenuous in quantummechanical frame and ensures physically correct results. Adopted scheme predicts the input density operatorsand the operator of quantum process all at once with help of noisy measured data only. Attention is paid to thecomparison of posed method with sequential measurement of the states and the process. Method is illustratedby numerical simulations.Keywords: quantum measurement, quantum state reconstruction, quantum process reconstruction, maximumlikelihood, completely positive maps 1. INTRODUCTION Measuring of quantum state of physical systems has attracted increased attention recently.15 Topical exper- iments often include extracting of quantum state from measured data as it's native part and many methods were developed for this purpose during recent years. Nevertheless most of them yield reconstructed objectswhich are not bona fide physical states. Only constrained statistical methods can simply integrate various ex-perimental data and guarantee the non-negativity of estimated density matrix. Physically the most importantis maximum likelihood principle61' representing genuine quantum measurement. Maximum likelihood methodensures existence of path in state space along which the likelihood of state monotonically increases.12Discussions of experimental results and understanding quantum nature behind them require however recog-nition of quantum processes, not only input or output quantum states. Examination of quantum communicationchannels and evaluation of performance of quantum gates are examples of practical applicability of quantumprocess reconstruction.1320 And again, maximum likelihood appears as the efficient and exceptional
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.