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arXiv:1610.04811 · Electronic Journal of Statistics

Estimation of low rank density matrices by Pauli measurements

Dong Xia

Published Jan 1, 2017• 4 citations
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Abstract

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 $p$-norms with $1\leq p\leq+\infty$ 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 $p$-norms with $1\leq p\leq2$ and, under a stronger condition, the optimal convergence rates for the projection estimator in Schatten $p$-norms with $1\leq p\leq+\infty$. 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.

DOI
10.1214/16-ejs1222

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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.
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters. Such bounds are established for several statistically relevant distances, including quantum versions of Kullback-Leibler divergence (relative entropy distance) and of Hellinger distance (so called Bures distance), and Schatten ppp-norm distances. Sharp upper bounds and oracle inequalities for least squares estimator with von Neumann entropy penalization are obtained showing that minimax lower bounds are attained (up to logarithmic factors) for these distances.
Let Sm\mathcal{S}_{m}Sm​ be the set of all m×mm\times mm×m density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix ρ∈Sm\rho\in\mathcal{S}_{m}ρ∈Sm​ based on outcomes of nnn measurements of observables X1,…,Xn∈HmX_{1},\dots,X_{n}\in\mathbb{H}_{m}X1​,…,Xn​∈Hm​ (Hm\mathbb{H}_{m}Hm​ being the space of m×mm\times mm×m Hermitian matrices) for a quantum system identically prepared nnn times in state ρ.\rho.ρ. Outcomes Y1,…,YnY_{1},\dots,Y_{n}Y1​,…,Yn​ of such measurements could be described by a trace regression model in which Eρ(Yj∣Xj)=tr(ρXj),j=1,…,n.\mathbb{E}_{\rho}(Y_{j}|X_{j})={\rm tr}(\rho X_{j}),j=1,\dots,n.Eρ​(Yj​∣Xj​)=tr(ρXj​),j=1,…,n. The design variables X1,…,XnX_{1},\dots,X_{n}X1​,…,Xn​ are often sampled at random from the uniform distribution in an orthonormal basis {E1,…,Em2}\{E_{1},\dots,E_{m^{2}}\}{E1​,…,Em2​} of Hm\mathbb{H}_{m}Hm​ (such as Pauli basis). The goal is to estimate the unknown density matrix ρ\rhoρ based on the data (X1,Y1),…,(Xn,Yn).(X_{1},Y_{1}),\dots,(X_{n},Y_{n}).(X1​,Y1​),…,(Xn​,Yn​). Let Z^:=m2n∑j=1nYjXj\hat{Z}:=\frac{m^{2}}{n}\sum_{j=1}^{n}Y_{j}X_{j}Z^:=nm2​∑j=1n​Yj​Xj​ and let ρˇ\check{\rho}ρˇ​ be the projection of Z^\hat{Z}Z^ onto the convex set Sm\mathcal{S}_{m}Sm​ of density matrices. It is shown that for estimator ρˇ\check{\rho}ρˇ​ the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten ppp-norm distances, p∈[1,∞]p\in[1,\infty]p∈[1,∞] and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator ρˇ\check{\rho}ρˇ​ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.
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.
Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. Firstly, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e. the sample complexity of tomography decreases with the rank. Secondly, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certified without making any assumptions about the unknown state. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we compare the performance of two compressed sensing estimators—the matrix Dantzig selector and the matrix Lasso—with standard maximum-likelihood estimation (MLE). We find that, given comparable experimental resources, the compressed sensing estimators consistently produce higher fidelity state reconstructions than MLE. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. Finally, we show how to certify the accuracy of a low-rank estimate using direct fidelity estimation, and describe a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements.