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arXiv:quant-ph/0511263 · Quantum Probability and Infinite Dimensional Analysis

COMPARISON OF SOME METHODS OF QUANTUM STATE ESTIMATION

Thomas Baier, D�nes Petz, Katalin Mária Hangos, Attila Magyar

Published Jul 1, 2007• 5 citations
arXivHTMLPDF

Abstract

In the paper the Bayesian and the least squares methods of quantum state tomography are compared for a single qubit. The quality of the estimates are compared by computer simulation when the true state is either mixed or pure. The fidelity and the Hilbert-Schmidt distance are used to quantify the error. It was found that in the regime of low measurement number the Bayesian method outperforms the least squares estimation. Both methods are quite sensitive to the degree of mixedness of the state to be estimated, that is, their performance can be quite bad near pure states.

DOI
10.1142/9789812770271_0007

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In the paper the Bayesian and the least squares methods of quantum state tomography are compared for a single qubit. The quality of the estimates are compared by computer simulation when the true state is either mixed or pure. The fidelity and the Hilbert-Schmidt distance are used to quantify the error. It was found that in the regime of low measurement number the Bayesian method outperforms the least squares estimation. Both methods are quite sensitive to the degree of mixedness of the state to be estimated, that is, their performance can be quite bad near pure states.
The tomographic reconstruction of the state of a quantum-mechanical system is an essential component in the development of quantum technologies. We present an overview of different tomographic methods for determining the quantum-mechanical density matrix of a single qubit: (scaled) direct inversion, maximum likelihood estimation (MLE), minimum Fisher information distance and Bayesian mean estimation (BME). We discuss the different prior densities in the space of density matrices, on which both MLE and BME depend, as well as ways of including experimental errors and of estimating tomography errors. As a measure of the accuracy of these methods, we average the trace distance between a given density matrix and the tomographic density matrices it can give rise to through experimental measurements. We find that the BME provides the most accurate estimate of the density matrix, and suggest using either the pure-state prior, if the system is known to be in a rather pure state, or the Bures prior if any state is possible. The MLE is found to be slightly less accurate. We comment on the extrapolation of these results to larger systems.
Multi-qubit state tomography is a key problem in quantum information technology, which has been studied extensively. In this work, we focus on multi-qubit state tomography based on neural network estimators and typical conventional estimation approaches. For multi-qubit pure states, fully connected neural networks and restricted Boltzmann machine networks are designed, respectively, to carry out state tomography of 2-qubit (low-dimensional) systems and 5-qubit (high-dimensional) systems. Comparisons are made with maximum likelihood estimation and least squares estimation, where performance indicators are selected as reconstruction accuracy, time cost and the number of parameters. Simulation results indicate that intelligent approaches have significant advantages over conventional approaches for state tomography of low-dimensional systems; for high-dimensional systems, however, the conventional approach is still dominant when the measurement is complete, while the restricted Boltzmann machine network can achieve higher reconstruction accuracy when the measurement is incomplete.
We give bounds on the average fidelity achievable by any quantum state estimator, which is arguably the most prominently used figure of merit in quantum state tomography. Moreover, these bounds can be computed online—that is, while the experiment is running. We show numerically that these bounds are quite tight for relevant distributions of density matrices. We also show that the Bayesian mean estimator is ideal in the sense of performing close to the bound without requiring optimization. Our results hold for all finite dimensional quantum systems.
Quantum state estimation is one of the most extensively studied topics in the field of quantum theory and quantum information. This process is essential in quantum protocols to ensure that the estimated state matches the true state. Typically, the quantum state of identical systems can be estimated from the statistics of measurement outcomes across different measurement bases. In this research, we study adaptive-measurement strategies for quantum state estimation used to identify a pure state of two-level systems and compare the obtained estimated state to the state obtained when the measurement bases are chosen randomly. The adaptive measurements in this study include the maximum-information-gain strategy, and the confirmation strategy. In this work, we simulate the outcome of the measurement on a system and use the Bayesian estimation method to estimate the obtained state. We then adaptively change the basis of the next measurement in accordance with the estimated state and the used adaptive strategy. We find that the maximum-information-gain strategy provides the fastest convergence and the most accurate state estimation compared to the confirmation strategy and the randomly-chosen-measurement-basis strategy, especially in the presence of measurement noise. Keywords: Quantum state estimation, State tomography, Adaptive measurements
An analytical solution for the posterior estimate in Bayesian tomography of the unknown quantum state of an arbitrary quantum system (with a finite-dimensional Hilbert space) is found. First, we derive the Bayesian estimate for a pure quantum state measured by a set of arbitrary rank-1 POVMs under the uninformative (i.e. the unitary invariant or Haar) prior. The expression for the estimate involves the matrix permanents of the Gram matrices with repeated rows and columns, with the matrix elements being the scalar products of vectors giving the measurement outcomes. Second, an unknown mixed state is treated by the Hilbert-Schmidt purification. In this case, under the uninformative prior for the combined pure state, the posterior estimate of the mixed state of the system is expressed through the matrix ααα-permanents of the Gram matrices of scalar products of vectors giving the measurement outcomes. In the mixed case, there is also a free integer parameter -- the Schmidt number -- which can be used to optimise the Bayesian reconstruction (for instance, in case of Schimdt number being equal to 1, the mixed state estimates reduces to the pure state estimate). We also discuss the perspectives of approximate numerical computation and asymptotic analytical evaluation of the Bayesian estimate using the derived formula.
Quantum tomography is the most informative tool for estimating the quality of preparation and transformation of quantum states. Its development is crucially necessary for debugging of developed quantum processors. Many existing methods of quantum tomography differ in types of performed measurements and in procedures of their processing. The practical implementation of quantum tomography requires the comparison of different methods, which is complicated because of the absence of a general methodology of estimation. A universal methodology based on numerical experiments has been proposed in this work to estimate the quality of quantum state tomography methods. The developed methodology has been applied to three quantum tomography methods (root approach, compressed sensing, and adaptive tomography) efficiently operating with almost pure states, which is relevant for the current technological foundation of the experiments.