纯量子态系综与量子高斯白噪声的局部渐近等价性
Local asymptotic equivalence of pure quantum states ensembles and quantum Gaussian white noise
AI总结：
本文研究无穷维纯态系综的量子统计模型，建立其与量子高斯白噪声模型的局部渐近等价性，推导Hermite-Sobolev类纯态估计极小极大速率，发现二次泛函估计的肘效应及纯态检验的精确参数速率。
摘要：
量子技术越来越依赖专门的统计推断方法来分析量子测量数据，这推动了“量子统计学”的发展，该领域正形成于量子物理学与“经典”统计学的交叉领域。目前研究较少的课题之一是无穷维量子系统的统计推断，它可被视为非参数统计的量子对应物。本文分析了由以纯态同制备的量子系统系综构成的量子统计模型的渐近理论。在大系综极限下，我们建立了该独立同分布（i.i.d.）模型与量子高斯白噪声模型的局部渐近等价性（LAE）。我们利用LAE结果，建立了属于波函数埃尔米特-索伯列夫（Hermite-Sobolev）类的纯态估计的极小极大速率。此外，对于相同态的二次泛函估计，我们注意到速率中存在肘效应；而对于纯态检验，在非参数Hermite-Sobolev类上达到了精确的参数速率。
英文摘要：
Quantum technology is increasingly relying on specialised statistical inference methods for analysing quantum measurement data. This motivates the development of "quantum statistics", a field that is shaping up at the overlap of quantum physics and "classical" statistics. One of the less investigated topics to date is that of statistical inference for infinite dimensional quantum systems, which can be seen as quantum counterpart of non-parametric statistics. In this paper we analyse the asymptotic theory of quantum statistical models consisting of ensembles of quantum systems which are identically prepared in a pure state. In the limit of large ensembles we establish the local asymptotic equivalence (LAE) of this i.i.d. model to a quantum Gaussian white noise model. We use the LAE result in order to establish minimax rates for the estimation of pure states belonging to Hermite-Sobolev classes of wave functions. Moreover, for quadratic functional estimation of the same states we note an elbow effect in the rates, whereas for testing a pure state a sharp parametric rate is attained over the nonparametric Hermite-Sobolev class.
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