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Estimation of low rank density matrices: bounds in Schatten norms and other distances
Let ${\mathcal S}_m$ be the set of all $m\times m$ density matrices (Hermitian positively semi-definite matrices of unit trace).
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.
Nuclear norm penalization and optimal rates for noisy low rank matrix completion
We show that the obtained rates are optimal up to logarithmic factors in a minimax sense and also derive, for any fixed matrix $A_0$, a non-minimax lower bound on the rate of convergence of our estimator, which coincides with the upper bound up to a constant factor.
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