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arXiv:1604.00321 · Physical Review A

Investigating bias in maximum-likelihood quantum-state tomography

G. B. Silva, Scott Glancy, H. M. Vasconcelos

Published Feb 8, 2017• 22 citations
arXivHTMLPDF

Abstract

Maximum-likelihood quantum-state tomography yields estimators that are consistent, provided that the likelihood model is correct, but the maximum-likelihood estimators may have bias for any finite data set. The bias of an estimator is the difference between the expected value of the estimate and the true value of the parameter being estimated. This paper investigates bias in the widely used maximum-likelihood quantum-state tomography. Our goal is to understand how the amount of bias depends on factors such as the purity of the true state, the number of measurements performed, and the number of different bases in which the system is measured. For this, we perform numerical experiments that simulate optical homodyne tomography of squeezed thermal states under various conditions, perform tomography, and estimate bias in the purity of the estimated state. We find that estimates of higher purity states exhibit considerable bias, such that the estimates have lower purities than the true states.

DOI
10.1103/physreva.95.022107

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Maximum-likelihood quantum-state tomography yields estimators that are consistent, provided that the likelihood model is correct, but the maximum-likelihood estimators may have bias for any finite data set. The bias of an estimator is the difference between the expected value of the estimate and the true value of the parameter being estimated. This paper investigates bias in the widely used maximum-likelihood quantum-state tomography. Our goal is to understand how the amount of bias depends on factors such as the purity of the true state, the number of measurements performed, and the number of different bases in which the system is measured. For this, we perform numerical experiments that simulate optical homodyne tomography of squeezed thermal states under various conditions, perform tomography, and estimate bias in the purity of the estimated state. We find that estimates of higher purity states exhibit considerable bias, such that the estimates have lower purities than the true states.
Because of the constraint that the estimators be bona fide physical states, any quantum state tomography scheme - including the widely used maximum likelihood estimation - yields estimators that may have a bias, although they are consistent estimators. Schwemmer et al. (arXiv:1310.8465 [quant-ph]) illustrate this by observing a systematic underestimation of the fidelity and an overestimation of entanglement in estimators obtained from simulated data. Further, these authors argue that the simple method of linear inversion overcomes this (perceived) problem of bias, and there is the suggestion to abandon time-tested estimation procedures in favor of linear inversion. Here, we discuss the pros and cons of using biased and unbiased estimators for quantum state tomography. We conclude that the little occasional benefit from the unbiased linear-inversion estimation does not justify the high price of using unphysical estimators, which are typically the case in that scheme.
Quantum state tomography aims to determine the quantum state of a system from measured data and is an essential tool for quantum information science. When dealing with continuous variable quantum states of light, tomography is often done by measuring the field amplitudes at different optical phases using homodyne detection. The quadrature-phase homodyne measurement outputs a continuous variable, so to reduce the computational cost of tomography, researchers often discretize the measurements. We show that this can be done without significantly degrading the fidelity between the estimated state and the true state. This paper studies different strategies for determining the histogram bin widths. We show that computation time can be significantly reduced with little loss in the fidelity of the estimated state when the measurement operators corresponding to each histogram bin are integrated over the bin width.
We propose an iterative algorithm that computes the maximum-likelihood estimate in quantum state tomography. The optimization error of the algorithm converges to zero at an O((1/k)log⁡D)O ( ( 1 / k ) \log D )O((1/k)logD) rate, where kkk denotes the number of iterations and DDD denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is O(D3+ND2)O ( D ^ 3 + N D ^2 )O(D3+ND2), where NNN denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the RρRR ρRRρR method [A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homodyne tomography. \textit{J. Opt. B: Quantum Semiclass. Opt.} 2004] [G. Molina-Terriza et al. Triggered qutrits for quantum communication protocols. \textit{Phys. Rev. Lett.} 2004.].
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.
Common tools for obtaining physical density matrices in experimental quantum state tomography are shown here to cause systematic errors. For example, using maximum likelihood or least squares optimization to obtain physical estimates for the quantum state, we observe a systematic underestimation of the fidelity and an overestimation of entanglement. Such strongly biased estimates can be avoided using linear evaluation of the data or by linearizing measurement operators yielding reliable and computational simple error bounds.
Using a Bayesian methodology, we introduce the maximum a posteriori probability (MAP) estimator for quantum state and process tomography. We show that the maximum likelihood, the hedged maximum likelihood, and the maximum likelihood-maximum entropy estimator, and estimators of this general type, can be viewed as special cases of the MAP estimator. The MAP, like the Bayes mean estimator, includes prior knowledge. For cases of interest to tomography MAP can take advantage of convex optimization tools, making it numerically tractable. We show how the MAP and other Bayesian quantum state estimators can be corrected for noise produced if the quantum state passes through a noisy quantum channel prior to measurement. Numerical simulations on a single qubit indicate that, on average, including these corrections significantly improves the estimate even when the measurement data are modestly large.