Estimation of Low Rank Density Matrices: Bounds in Schatten Norms and Other Distances
نویسندگان
چکیده
Let 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 ∈ Sm based on outcomes of n measurements of observables X1, . . . ,Xn ∈ Hm (Hm being the space of m × m Hermitian matrices) for a quantum system identically prepared n times in state ρ. Outcomes Y1, . . . , Yn of such measurements could be described by a trace regression model in which Eρ(Yj |Xj) = tr(ρXj), j = 1, . . . , n. The design variables X1, . . . ,Xn are often sampled at random from the uniform distribution in an orthonormal basis {E1, . . . , Em2} of Hm (such as Pauli basis). The goal is to estimate the unknown density matrix ρ based on the data (X1, Y1), . . . , (Xn, Yn). Let
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