Georgia Institute of Technology College of Sciences

We study a problem of estimation of a large Hermitian nonnegatively definite matrix S of unit trace based on n independent measurements Y_j = tr(SX_j ) + Z_j , j = 1, . . . , n, where {X_j} are i.i.d. Hermitian matrices and {Z_j } are i.i.d. mean zero random variables independent of {X_j}. Problems of this nature are of interest in quantum state tomography, where S is an unknown density matrix of a quantum system. The estimator is based on penalized least squares method with complexity penalty defined in terms of von Neumann entropy. We derive oracle inequalities showing how the estimation error depends on the accuracy of approximation of the unknown state S by low-rank matrices. We will discuss these results as well as some of the tools used in their proofs (such as generic chaining bounds for empirical processes and noncommutative Bernstein type inequalities).