VON NEUMANN ENTROPY PENALIZATION AND LOW-RANK MATRIX ESTIMATION

We study a problem of estimation of a Hermitian nonnegatively definite matrix . of unit trace (e.g., a density matrix of a quantum system) based on n i.i.d. measurements (X., Y.),..., (X n , Y n ), where Y j = tr (.X j ) + . j , j = 1,...,n, {X j } being random i.i.d. Hermitian matrices and {. j } being i.i.d. random veriables with ..(. j |X j ) = 0. The estimator $\hat \rho ^\varepsilon : = \mathop {\arg \min }\limits_{S \in S} \left[ {n^{ - 1} \sum\limits_{j = 1}^n {(y_j - tr(SX_j ))^2 + \varepsilon tr(S\log S)} } \right]$ is considered, where S is the set of all nonnegatively definite Hermitian m . m matrices of trace 1. The goal is to derive oracle inequalities showing how the estimation error depends on the accuracy of approximation of the unknown state . by low-rank matrices.