Linear and Multilinear Algebra

Schatten q-norms and determinantal inequalities for matrices with numerical ranges in a sector

Junjian Yang, Linzhang Lu, Zhen Chen

Published Dec 22, 2017• 5 citations

Abstract

When A is a density matrix in , Audenaert [J Math Phys. 2007;48] proved an interesting inequality for Schatten q-norms where and stand for the first and second partial trace, respectively, a notion from quantum information theory. As an analogue of this result, Lin [Canad Math Bull. 2016;59] recently presented the following determinantal inequality In this note, we first correct a result of Kuai [Linear Multilinear Algebra. doi:10.1080/03081087.2017.1304521] and then generalize the above two inequalities to a larger class of matrices whose numerical ranges are contained in a sector.

DOI: 10.1080/03081087.2017.1417351

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Linear and Multilinear Algebra

Schatten q-norms and determinantal inequalities for matrices with numerical ranges in a sector

Junjian Yang, Linzhang Lu, Zhen Chen

Published Dec 22, 2017• 5 citations

Publisher

Abstract

DOI: 10.1080/03081087.2017.1417351

Related papers

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arXiv:0711.0721Journal of Mathematical PhysicsJan 1, 20080 citations

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Dong Xia

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.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.

arXiv:1610.04811Electronic Journal of StatisticsJan 1, 20174 citations

Let

\mathcal{S}_{m}

be the set of all

m\times m

density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix

\rho\in\mathcal{S}_{m}

based on outcomes of

n

measurements of observables

X_{1},\dots,X_{n}\in\mathbb{H}_{m}

(

\mathbb{H}_{m}

being the space of

m\times m

Hermitian matrices) for a quantum system identically prepared

n

times in state

\rho.

Outcomes

Y_{1},\dots,Y_{n}

of such measurements could be described by a trace regression model in which

\mathbb{E}_{\rho}(Y_{j}|X_{j})={\rm tr}(\rho X_{j}),j=1,\dots,n.

The design variables

X_{1},\dots,X_{n}

are often sampled at random from the uniform distribution in an orthonormal basis

\{E_{1},\dots,E_{m^{2}}\}

\mathbb{H}_{m}

(such as Pauli basis). The goal is to estimate the unknown density matrix

\rho

based on the data

(X_{1},Y_{1}),\dots,(X_{n},Y_{n}).

Let

\hat{Z}:=\frac{m^{2}}{n}\sum_{j=1}^{n}Y_{j}X_{j}

and let

\check{\rho}

be the projection of

\hat{Z}

onto the convex set

\mathcal{S}_{m}

of density matrices. It is shown that for estimator

\check{\rho}

the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten

p

-norm distances,

p\in[1,\infty]

and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator

\check{\rho}

the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.

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.