Estimation of low rank density matrices: bounds in Schatten norms and other distances
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Let S_m be the set of all m× m density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix ρ∈ S_m based on outcomes of n measurements of observables X_1,..., X_n∈ H_m ( H_m being the space of m× m Hermitian matrices) for a quantum system identically prepared n times in state ρ. Outcomes Y_1,..., Y_n of such measurements could be described by a trace regression model in which E_ρ(Y_j|X_j)= tr(ρ X_j), j=1,..., n. The design variables X_1,..., X_n are often sampled at random from the uniform distribution in an orthonormal basis {E_1,..., E_m^2} of H_m (such as Pauli basis). The goal is to estimate the unknown density matrix ρ based on the data (X_1,Y_1), ..., (X_n,Y_n). Let Ẑ:=m^2/n∑_j=1^n Y_j X_j and let ρ̌ be the projection of Ẑ onto the convex set S_m of density matrices. It is shown that for estimator ρ̌ the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten p-norm distances, p∈ [1,∞] and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator ρ̌ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.
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