Theroblem of estimating aarameter of a quantum system through a series of measurementserformed sequentially on a quantumrobe is analyzed in the general setting where the underlying statistics is explicitly non-i.i.d. Weresent a generalization of the central limit theorem in theresent context, which under fairly general assumptions shows that as the number N of measurement data increases therobability distribution of functionals of the data (e.g., the average of the data) through which the targetarameter is estimated becomes asymptotically normal and independent of the initial state of therobe. At variance with therevious studies (Guţə M 2011 Phys. Rev. A 83 062324; van Horssen M and Guţə M 2015 J. Math. Phys. 56 022109) we take a diagrammatic approach, which allows one to compute not only the leading orders in N of the moments of the average of the data but also those of the correlations among subsequent measurement outcomes. Inarticular our analysisoints out that the latter, which are not available in usual i.i.d. data, can be exploited in order to improve the accuracy of thearameter estimation. An explicit application of our scheme is discussed by studying how the temperature of a thermal reservoir can be estimated via sequential measurements on a quantumrobe in contact with the reservoir.
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