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Statistica Sinica

Density matrix estimation in quantum homodyne tomography

Yazhen Wang, Chenliang Xu

Published Jun 5, 2014• 9 citations
Publisher

Abstract

Modern scientiflc studies often involve complex quantum systems, and scientists need to learn the systems from experiment data. As density matrices are usually employed to characterize the quantum states of the systems, this paper investigates estimation of density matrices for the quantum systems. We propose statistical methodologies to construct density matrix estimators and establish an asymptotic theory for the estimation methods. We show that the proposed density matrix estimators are consistent and have high convergence rates. A numerical study is conducted to demonstrate the good flnite sample performances of the proposed estimators.

DOI
10.5705/ss.2013.300

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Modern scientiflc studies often involve complex quantum systems, and scientists need to learn the systems from experiment data. As density matrices are usually employed to characterize the quantum states of the systems, this paper investigates estimation of density matrices for the quantum systems. We propose statistical methodologies to construct density matrix estimators and establish an asymptotic theory for the estimation methods. We show that the proposed density matrix estimators are consistent and have high convergence rates. A numerical study is conducted to demonstrate the good flnite sample performances of the proposed estimators.
Quantum state tomography aims to determine the state of a quantum system as represented by a density matrix. It is a fundamental task in modern scientific studies involving quantum systems. In this paper, we study estimation of high-dimensional density matrices based on Pauli measurements. In particular, under appropriate notion of sparsity, we establish the minimax optimal rates of convergence for estimation of the density matrix under both the spectral and Frobenius norm losses; and show how these rates can be achieved by a common thresholding approach. Numerical performance of the proposed estimator is also investigated.
In the framework of noisy quantum homodyne tomography with efficiency parameter 0 < η ⩽ 1, we propose two estimators of a quantum state whose density matrix elements ρ m , n decrease as , for fixed known B > 0 and 0 < r ⩽ 2. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for 0 < η ⩽ 1/2), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in risk.
Summary We describe quantum tomography as an inverse statistical problem in which the quantum state of a light beam is the unknown parameter and the data are given by results of measurements performed on identical quantum systems. The state can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We present consistency results for pattern function projection estimators and for sieve maximum likelihood estimators for both the density matrix of the quantum state and its Wigner function. We illustrate the performance of the estimators on simulated data. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. rates of convergence, adaptation and studying other estimators; a main purpose of the paper is to bring these to the attention of the statistical community.
The quantum state of a light beam can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We describe quantum tomography as an inverse statistical problem in which the state is the unknown parameter and the data is given by results of measurements performed on identical quantum systems. We present consistency results for Pattern Function Projection Estimators as well as for Sieve Maximum Likelihood Estimators for both the density matrix of the quantum state and its Wigner function. Finally we illustrate via simulated data the performance of the estimators. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. rates of convergence, adaptation, studying other estimators, etc., and a main purpose of the paper is to bring these to the attention of the statistical community.
Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in quantum state tomography (QST) is to estimate the unknown low rank density matrix of a quantum system by conducting Pauli measurements. Our main contribution is twofold. First, we establish the minimax lower bounds in Schatten ppp-norms with 1≤p≤+∞1\leq p\leq+\infty1≤p≤+∞ for low rank density matrices estimation by Pauli measurements. In our previous paper [14], these minimax lower bounds are proved under the trace regression model with Gaussian noise and the noise is assumed to have common variance. In this paper, we prove these bounds under the Binomial observation model which meets the actual model in QST. Second, we study the Dantzig estimator (DE) for estimating the unknown low rank density matrix under the Binomial observation model by using Pauli measurements. In our previous papers [14] and [25], we studied the least squares estimator and the projection estimator, where we proved the optimal convergence rates for the least squares estimator in Schatten ppp-norms with 1≤p≤21\leq p\leq21≤p≤2 and, under a stronger condition, the optimal convergence rates for the projection estimator in Schatten ppp-norms with 1≤p≤+∞1\leq p\leq+\infty1≤p≤+∞. In this paper, we show that the results of these two distinct estimators can be simultaneously obtained by the Dantzig estimator. Moreover, better convergence rates in Schatten norm distances can be proved for Dantzig estimator under conditions weaker than those needed in [14] and [25]. When the objective function of DE is replaced by the negative von Neumann entropy, we obtain sharp convergence rate in Kullback-Leibler divergence.
Density estimation is a fundamental task in statistics and machine learning that aims to estimate, from a set of samples, the probability density function of the distribution that generated them. There are different methods for addressing this problem but recently deep-neural density estimation methods have emerged as a powerful alternative. This paper presents a novel method for neural density estimation based on density matrices and adaptive Fourier features. Density matrices are commonly used in quantum mechanics to represent the quantum state of a physical system. In this work, they are used to estimate probability densities using an operation called quantum measurement. The proposed method can be trained without optimization using an averaging operation over the samples of the training dataset. It can also be integrated with deep learning architectures and trained using gradient descent. The performance of the proposed method was evaluated on a range of synthetic and real datasets and compared with fast kernel density estimation and state-of-the-art neural density estimation methods. The results demonstrate that the proposed method achieves competitive performance while being faster and more efficient than existing methods.