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

In classical statistics, a well known paradigm consists in establishing asymptotic equivalence between an experiment of i.i.d. observations and a Gaussian shift experiment, with the aim of obtaining optimal estimators in the former complicated model from the latter simpler model. In particular, a statistical experiment consisting of $n$ i.i.d observations from d-dimensional multinomial distributions can be well approximated by an experiment consisting of $d-1$ dimensional Gaussian distributions. In a quantum version of the result, it has been shown that a collection of $n$ qudits (d-dimensional quantum states) of full rank can be well approximated by a quantum system containing a classical part, which is a $d-1$ dimensional Gaussian distribution, and a quantum part containing an ensemble of $d(d-1)/2$ shifted thermal states. In this paper, we obtain a generalization of this result when the 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 Gaussian distribution and an ensemble of both shifted pure and shifted thermal states. For estimation purposes, we establish an asymptotic minimax result in the limiting Gaussian model. Analogous results are then obtained for estimation of a low-rank qudit from an ensemble of identically prepared, independent quantum systems, using the local asymptotic equivalence result. We also consider the problem of estimation of a linear functional of the quantum state. We construct an estimator for the functional, analyze the risk and use quantum local asymptotic equivalence to show that our estimator is also optimal in the minimax sense.