In classical statistics, a statistical experiment consisting of $n$ i.i.d observations from d-dimensional multinomial distributions can be well approximated by a $d-1$ dimensional Gaussian distribution. In a quantum version of the result it has been shown that a collection of $n$ qudits 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. We also outline a two-stage procedure for the estimation of the low-rank qudit, where we obtain an estimator which is sharp minimax optimal. For the estimation of a linear functional of the quantum state, we construct an estimator, analyze the risk and use quantum LAN to show that our estimator is also optimal in the minimax sense.
翻译:在古典统计中,由d-d-i-i-d观测值组成的统计实验,从d-d-d-d-l/2调热状态中得出的观测值,完全可以被一美元-美元-美元-一元-高斯分布值所近似。在一项结果的量子版中,结果显示,一个包含一个典型部分的量子系统,即一美元-美元-一美元-一美元-维高斯分布值,以及一个包含一元-一元-一元-二美元调热状态组合的量子部分,可以被一元-美元-一元-美元-多元分布值所近似近似。在本文中,当夸特不是完全等级时,我们获得了这一结果的概括性。我们表明,当夸特的等级为一元-一元-一元-一元,那么限制实验就包括一元-一元-一元的量级分布式分布式分布式和一整调热状态的组合。我们还概述了一个用于估计低级调调热状态的两阶段程序,在这个阶段中,我们得到了一个精度-最优化的估测算器。对于量-直径功能的估算,我们在最佳的量度的量度的量度的估测测度中,我们所测测度的量度的量度也是一种测度的量度的量度,我们所测的量度的量度,我们所测度的测的测的测的测的量的测的测的测的量值也是用于。