© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1405.5350

Quantum state tomography: Mean squared error matters, bias does not

Jiangwei Shang, Hui Khoon Ng, Berthold‐Georg Englert

Published May 21, 2014• 7 citations

Abstract

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.

DOI: 10.48550/arxiv.1405.5350

Related papers

Systematic Errors in Current Quantum State Tomography Tools

Christian Schwemmer, Lukas Knips, Daniel Richart +4 more

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.

arXiv:1310.8465•Physical Review Letters•Feb 26, 2015•114 citations

Investigating bias in maximum-likelihood quantum-state tomography

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

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.

arXiv:1604.00321•Physical Review A•Feb 8, 2017•22 citations

Maximum likelihood quantum state tomography is inadmissible

Christopher Ferrie, Robin Blume-Kohout

Maximum likelihood estimation (MLE) is the most common approach to quantum state tomography. In this letter, we investigate whether it is also optimal in any sense. We show that MLE is an inadmissible estimator for most of the commonly used metrics of accuracy, i.e., some other estimator is more accurate for every true state. MLE is inadmissible for fidelity, mean squared error (squared Hilbert-Schmidt distance), and relative entropy. We prove that almost any estimator that can report both pure states and mixed states is inadmissible. This includes MLE, compressed sensing (nuclear-norm regularized) estimators, and constrained least squares. We provide simple examples to illustrate why reporting pure states is suboptimal even when the true state is itself pure, and why "hedging" away from pure states generically improves performance.

arXiv:1808.01072•Aug 3, 2018•9 citations

Quantum State Tomography via Linear Regression Estimation

Bo Qi, Zhibo Hou, Li Li +3 more

A simple yet efficient state reconstruction algorithm of linear regression estimation (LRE) is presented for quantum state tomography. In this method, quantum state reconstruction is converted into a parameter estimation problem of a linear regression model and the least-squares method is employed to estimate the unknown parameters. An asymptotic mean squared error (MSE) upper bound for all possible states to be estimated is given analytically, which depends explicitly upon the involved measurement bases. This analytical MSE upper bound can guide one to choose optimal measurement sets. The computational complexity of LRE is O(d(4)) where d is the dimension of the quantum state. Numerical examples show that LRE is much faster than maximum-likelihood estimation for quantum state tomography.

arXiv:1304.6827•Scientific Reports•Dec 13, 2013•156 citations

Biased Tomography Schemes: An Objective Approach

Z. Hradil, Dmitri S. Mogilevtsev, Jaroslav Řeháček

We report on an intrinsic relationship between the maximum-likelihood quantum-state estimation and the representation of the signal. A quantum analogy of the transfer function determines the space where the reconstruction should be done without the need for any ad hoc truncations of the Hilbert space. An illustration of this method is provided by a simple yet practically important tomography of an optical signal registered by realistic binary detectors.

arXiv:quant-ph/0606042•Physical Review Letters•Jun 12, 2006•89 citations

Optimal experiment design for quantum state tomography: Fair, precise, and minimal tomography

J. Nunn, Brian J. Smith, Graciana Puentes +2 more

Given an experimental setup and a fixed number of measurements, how should one take data to optimally reconstruct the state of a quantum system? The problem of optimal experiment design (OED) for quantum state tomography was first broached by Kosut et al. [R. Kosut, I. Walmsley, and H. Rabitz, e-print arXiv:quant-ph/0411093 (2004)]. Here we provide efficient numerical algorithms for finding the optimal design, and analytic results for the case of `minimal tomography'. We also introduce the average OED, which is independent of the state to be reconstructed, and the optimal design for tomography (ODT), which minimizes tomographic bias. Monte Carlo simulations confirm the utility of our results for qubits. Finally, we adapt our approach to deal with constrained techniques such as maximum-likelihood estimation. We find that these are less amenable to optimization than cruder reconstruction methods, such as linear inversion.

arXiv:0911.4310•Physical Review A•Apr 22, 2010•39 citations

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.

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.

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.

Maximum likelihood estimation (MLE) is the most common approach to quantum state tomography. In this letter, we investigate whether it is also optimal in any sense. We show that MLE is an inadmissible estimator for most of the commonly used metrics of accuracy, i.e., some other estimator is more accurate for every true state. MLE is inadmissible for fidelity, mean squared error (squared Hilbert-Schmidt distance), and relative entropy. We prove that almost any estimator that can report both pure states and mixed states is inadmissible. This includes MLE, compressed sensing (nuclear-norm regularized) estimators, and constrained least squares. We provide simple examples to illustrate why reporting pure states is suboptimal even when the true state is itself pure, and why "hedging" away from pure states generically improves performance.

A simple yet efficient state reconstruction algorithm of linear regression estimation (LRE) is presented for quantum state tomography. In this method, quantum state reconstruction is converted into a parameter estimation problem of a linear regression model and the least-squares method is employed to estimate the unknown parameters. An asymptotic mean squared error (MSE) upper bound for all possible states to be estimated is given analytically, which depends explicitly upon the involved measurement bases. This analytical MSE upper bound can guide one to choose optimal measurement sets. The computational complexity of LRE is O(d(4)) where d is the dimension of the quantum state. Numerical examples show that LRE is much faster than maximum-likelihood estimation for quantum state tomography.

Given an experimental setup and a fixed number of measurements, how should one take data to optimally reconstruct the state of a quantum system? The problem of optimal experiment design (OED) for quantum state tomography was first broached by Kosut et al. [R. Kosut, I. Walmsley, and H. Rabitz, e-print arXiv:quant-ph/0411093 (2004)]. Here we provide efficient numerical algorithms for finding the optimal design, and analytic results for the case of `minimal tomography'. We also introduce the average OED, which is independent of the state to be reconstructed, and the optimal design for tomography (ODT), which minimizes tomographic bias. Monte Carlo simulations confirm the utility of our results for qubits. Finally, we adapt our approach to deal with constrained techniques such as maximum-likelihood estimation. We find that these are less amenable to optimization than cruder reconstruction methods, such as linear inversion.