Optimal Estimation of Low Rank Density Matrices

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

Perturbation of linear forms of singular vectors under Gaussian noise

Let $Ainmathbb{R}^{mtimes n}$ be a matrix of rank $r$ with singular value decomposition (SVD) $A=sum_{k=1}^rsigma_k (u_kotimes v_k),$ where ${sigma_k, k=1,ldots,r}$ are singular values of $A$ (arranged in a non-increasing order) and $u_kin {mathbb R}^m, v_kin {mathbb R}^n, k=1,ldots, r$ are the corresponding left and right orthonormal singular vectors. Let $tilde{A}=A+X$ be a noisy observation of $A,$ where $Xinmathbb{R}^{mtimes n}$ is a random matrix with i.i.d. Gaussian entries, $X_{ij}simmathcal{N}(0,tau^2),$ and consider its SVD $tilde{A}=sum_{k=1}^{mwedge n}tilde{sigma}_k(tilde{u}_kotimestilde{v}_k)$ with singular values $tilde{sigma}_1geqldotsgeqtilde{sigma}_{mwedge n}$ and singular vectors $tilde{u}_k,tilde{v}_k,k=1,ldots, mwedge n.$ The goal of this paper is to develop sharp concentration bounds for linear forms $langle tilde u_k,xrangle, xin {mathbb R}^m$ and $langle tilde v_k,yrangle, yin {mathbb R}^n$ of the perturbed (empirical) singular vectors in the case when the singular values of $A$ are distinct and, more generally, concentration bounds for bilinear forms of projection operators associated with SVD. In particular, the results imply upper bounds of the order $Obiggl(sqrt{frac{log(m+n)}{mvee n}}biggr)$ (holding with a high probability) on $$max_{1leq ileq m}big|big<tilde{u}_k-sqrt{1+b_k}u_k,e_i^mbig>big| {rm and} max_{1leq jleq n}big|big<tilde{v}_k-sqrt{1+b_k}v_k,e_j^nbig>big|,$$ where $b_k$ are properly chosen constants characterizing the bias of empirical singular vectors $tilde u_k, tilde v_k$ and ${e_i^m,i=1,ldots,m}, {e_j^n,j=1,ldots,n}$ are the canonical bases of $mathbb{R}^m, {mathbb R}^n,$ respectively.