Estimation of low rank density matrices: bounds in Schatten norms and other distances

Let mathcal Sm 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∈ mathcal Sm based on outcomes of n measurements of observables X1,dots, Xn∈ mathbb Hm (mathbb Hm being the space of m× m Hermitian matrices) for a quantum system identically prepared n times in state ρ. Outcomes Y1,dots, Yn of such measurements could be described by a trace regression model in which mathbb Eρ(Yj|Xj)=rm tr(ρ Xj), j=1,dots, n. The design variables X1,dots, Xn are often sampled at random from the uniform distribution in an orthonormal basis E1,dots, Em2 of mathbb Hm (such as Pauli basis). The goal is to estimate the unknown density matrix ρ based on the data (X1,Y1), dots, (Xn,Yn). Let hat Z:=fracm2nsumj=1n Yj Xj and let check ρ be the projection of hat Z onto the convex set mathcal Sm of density matrices. It is shown that for estimator check ρ 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 check ρ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.