The optimality of Dantzig estimator for estimating low rank density matrices

The famous Dantzig estimator(DE) has been applied in quantum state tomography for estimating an unknown low rank density matrix. The density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. The main contribution of this paper is deriving the low rank oracle inequality of DE and proving its convergence rates in the Schatten p-norms for all p ∈ [1,+∞]. In particular, when considering Pauli measurements, the convergence rate in Schatten p-norm is of the order (up to logarithmic factors) √ m n r ∧(√m n )1− 1 p ∧ 1, for 1 ≤ p ≤ +∞, with m being the dimension of the underlying density matrix, r being its rank and n being the number of Pauli measurements. These rates match the minimax lower bounds established in our previous paper and thus are optimal. When the objective function of DE is replaced by the negative von Neumann entropy, we can also obtain sharp convergence rate in Kullback-Leibler divergence. Our main technical tool is a new upper bound for product empirical processes. ∗Supported in part by NSF Grants DMS-1207808 and CCF-1523768