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.
In the present work, we have investigated the problem of estimating parameters of several exponential distributions with ordered scale parameters under the linex loss function. We have considered estimating ordered scale parameters when the location parameters are known and unknown. For every case, we consider a class of equivariant estimators, and sufficient condition is obtained under which this class of estimators improves upon the usual estimator. Using this result, we have shown that the restricted maximum likelihood estimator is inadmissible. Finally, for every case, we conduct a simulat…
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…
A common problem faced in clinical studies is that of estimating the effect of the most effective (e.g., the one having the largest mean) treatment among $k~(\geq2)$ available treatments. The most effective treatment is adjudged based on numerical values of some statistic corresponding to the $k$ treatments. A proper design for such problems is the so-called "Drop-the-Losers Design (DLD)". We consider two treatments whose effects are described by independent Gaussian distributions having different unknown means and a common known variance. To select the more effective treatment, the two treatm…
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 $O ( ( 1 / k ) \log D )$ rate, where $k$ denotes the number of iterations and $D$ denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is $O ( D ^ 3 + N D ^2 )$, where $N$ denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the $R ρR$ method [A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homod…
National Taiwan University（国立台湾大学）；Hon Hai (Foxconn) Quantum Computing Centre（鸿海（富士康）量子计算中心）
In maximum-likelihood quantum state tomography, both the sample size and dimension grow exponentially with the number of qubits. It is therefore desirable to develop a stochastic first-order method, just like stochastic gradient descent for modern machine learning, to compute the maximum-likelihood estimate. To this end, we propose an algorithm called stochastic mirror descent with the Burg entropy. Its expected optimization error vanishes at a $O ( \sqrt{ ( 1 / t ) d \log t } )$ rate, where $d$ and $t$ denote the dimension and number of iterations, respectively. Its per-iteration time complex…
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 …
Quantum state tomography (QST), the task of estimating an unknown quantum state given measurement outcomes, is essential to building reliable quantum computing devices. Whereas computing the maximum-likelihood (ML) estimate corresponds to solving a finite-sum convex optimization problem, the objective function is not smooth nor Lipschitz, so most existing convex optimization methods lack sample complexity guarantees; moreover, both the sample size and dimension grow exponentially with the number of qubits in a QST experiment, so a desired algorithm should be highly scalable with respect to the…
Quantum state tomography on a d-dimensional system demands resources that grow rapidly with d. They may be reduced by using model selection to tailor the number of parameters in the model (i.e., the size of the density matrix). Most model selection methods typically rely on a test statistic and a null theory that describes its behavior when two models are equally good. Here, we consider the loglikelihood ratio. Because of the positivity constraint $ρ\geq 0$, quantum state space does not generally satisfy local asymptotic normality, meaning the classical null theory for the loglikelihood ratio …
A statistically feasible data post-processing method for the conventional qubit state tomography is studied from an information geometrical point of view. It is shown that the space $(-1,1)^3$ of the Stokes parameters $(ξ_1, ξ_2,ξ_3)$ that specify qubit states should be regarded as a Riemannian manifold endowed with a metric $g_{ij}:=δ_{ij}/(1-(ξ_i)^2)$, and that the data processing based on the maximum likelihood method is realized by the orthogonal projection from the empirical distribution onto the Bloch sphere with respect to the metric $g_{ij}$. An efficient algorithm for computing the ma…
Conventional methods for computing maximum-likelihood estimators (MLE) often converge slowly in practical situations, leading to a search for simplifying methods that rely on additional assumptions for their validity. In this work, we provide a fast and reliable algorithm for maximum likelihood reconstruction that avoids this slow convergence. Our method utilizes the state-of-the-art convex optimization scheme---an accelerated projected-gradient method---that allows one to accommodate the quantum nature of the problem in a different way than in the standard methods. We demonstrate the power of…
Quantum state tomography (QST) is typically performed from a frequentist viewpoint using maximum likelihood estimation (MLE) which seeks to find the best plausible state consistent with the data by maximizing a likelihood function / distribution. The likelihood function holds an implicit assumption that there is suitable data to infer frequency. In data-starved experiments, this may or may not be a feasible assumption. Moreover, MLE returns no error estimates on the final solution and users are forced to rely on alternative approaches involving either additional measurements or simulated data.…
Quantum state tomography (QST) aims at estimating a quantum state from averaged quantum measurements made on copies of the state. Most quantum algorithms rely on QST at some point and it is a well explored topic in the literature, mostly for mixed states. In this paper we focus on the QST of a pure quantum state using parallel unentangled measurements. Pure states are a small but useful subset of all quantum states, their tomography requires fewer measurements and is essentially a phase recovery problem. Parallel unentangled measurements are easy to implement in practice because they allow the…
Bayesian estimation of a mixed quantum state can be approximated via maximum likelihood (MaxLike) estimation when the likelihood function is sharp around its maximum. Such approximations rely on asymptotic expansions of multi-dimensional Laplace integrals. When this maximum is on the boundary of the integration domain, as it is the case when the MaxLike quantum state is not full rank, such expansions are not standard. We provide here such expansions, even when this maximum does not belong to the smooth part of the boundary, as it is the case when the rank deficiency exceeds two. These expansio…
Using a Bayesian methodology, we introduce the maximum a posteriori~(MAP) estimator for quantum state and process tomography. The maximum likelihood, hedged maximum likelihood, 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, however for cases of interest to tomography can take advantage of convex optimization tools making it numerically easier to compute. We show how the MAP and other Bayesian quantum state estimators can be corrected for nois…
Precise reconstruction of unknown quantum states from measurement data, a process commonly called quantum state tomography, is a crucial component in the development of quantum information processing technologies. Many different tomography methods have been proposed over the years. Maximum likelihood estimation is a prominent example, being the most popular method for a long period of time. Recently, more advanced neural network methods have started to emerge. Here, we go back to basics and present a method for continuous variable state reconstruction that is both conceptually and practically …
