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In quantum estimation theory and quantum tomography, the quantum stateobtained by sampling converges to the `true' unknown density matrix undertopologies that are different from the natural notion of distance in the spaceof quantum states, i.e. the trace class norm. In this paper, we address suchproblem, finding relations between the rates of convergence in the Schatten$p$-norms and in the trace class norm.

Relations between convergence rates in Schatten p-norms