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In classical statistics, a well known paradigm consists in establishingasymptotic equivalence between an experiment of i.i.d. observations and aGaussian shift experiment, with the aim of obtaining optimal estimators in theformer complicated model from the latter simpler model. In particular, astatistical experiment consisting of $n$ i.i.d observations from d-dimensionalmultinomial distributions can be well approximated by an experiment consistingof $d-1$ dimensional Gaussian distributions. In a quantum version of theresult, it has been shown that a collection of $n$ qudits (d-dimensionalquantum states) of full rank can be well approximated by a quantum systemcontaining a classical part, which is a $d-1$ dimensional Gaussiandistribution, and a quantum part containing an ensemble of $d(d-1)/2$ shiftedthermal states. In this paper, we obtain a generalization of this result whenthe qudits are not of full rank. We show that when the rank of the qudits is$r$, then the limiting experiment consists of an $r-1$ dimensional Gaussiandistribution and an ensemble of both shifted pure and shifted thermal states.For estimation purposes, we establish an asymptotic minimax result in thelimiting Gaussian model. Analogous results are then obtained for estimation ofa low-rank qudit from an ensemble of identically prepared, independent quantumsystems, using the local asymptotic equivalence result. We also consider theproblem of estimation of a linear functional of the quantum state. We constructan estimator for the functional, analyze the risk and use quantum localasymptotic equivalence to show that our estimator is also optimal in theminimax sense.

Minimax estimation of low-rank quantum states and their linear functionals