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.

翻译：在经典统计学中，一个广为人知的范式在于建立独立同分布观测实验与高斯平移实验之间的渐近等价性，其目的是从后者较简单的模型中获得前者复杂模型中的最优估计量。特别地，由d维多项分布的n个独立同分布观测构成的统计实验，可被近似为由d-1维高斯分布构成的实验。该结果的量子版本表明，满秩的n个qudit（d维量子态）集合可被一个包含经典部分（即d-1维高斯分布）与量子部分（即d(d-1)/2个平移热态系综）的量子系统良好近似。本文在qudit非满秩的情形下推广了这一结果。我们证明，当qudit的秩为r时，极限实验由r-1维高斯分布与同时包含平移纯态和平移热态的系综构成。针对估计问题，我们在极限高斯模型中建立了渐近极小极大结果。随后，利用局部渐近等价性结果，我们从全同制备独立量子系统系综中获取低秩qudit的估计量，并得到了类似结论。我们还研究了量子态线性泛函的估计问题：为泛函构造了一个估计量，分析了其风险，并利用量子局部渐近等价性证明该估计量同样在极小极大意义下最优。