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Register a new userEstimation of low rank density matrices: bounds in Schatten norms and other distances
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Added by
Dong Xia
Publication date
2016
fields
Mathematical Statistics
and research's language is
English
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Let ${mathcal S}_m$ be the set of all $mtimes m$ density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix $rhoin {mathcal S}_m$ based on outcomes of $n$ measurements of observables $X_1,dots, X_nin {mathbb H}_m$ (${mathbb H}_m$ being the space of $mtimes m$ 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 ${E_1,dots, E_{m^2}}$ of ${mathbb H}_m$ (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, $pin [1,infty]$ 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.
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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 quant
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