Asymptotic Equivalence of Quantum Statistical Models

量子力学的一个基本观点是，随机性是微观层面上物理世界的固有特征。对原子或光脉冲等量子系统进行的任何观测，都会产生不确定的、随机的结果。研究从系统的S态或制备到测量结果的概率分布的直接映射一直是传统量子理论的核心问题之一。在许多量子协议中，实验者对系统及其环境没有完全的了解和控制，或者对影响系统动力学的外场参数的估计感兴趣。在这种情况下，人们处理的是一个统计反问题，即从通过探测大量单个量子系统获得的测量数据推断未知的状态参数。解决这些问题所产生的理论和实践正在形成量子统计领域，该领域位于量子理论和统计推理的交叉点上。目前的项目旨在更好地理解无限维量子系统的统计推断，这一领域可以被视为非参数统计的量子对应。最终目标是发展量子统计模型的比较和收敛理论，从而实现有效的估计技术，并建立可靠的统计方法来计算可靠的误差栏。这个项目与美国国家科学基金会的量子飞跃计划非常契合。该奖项支持的研究生将在PI.的监督下进行量子统计的研究。一个特别感兴趣的领域是希尔伯特空间中带有参数的纯态的不同量子统计模型的局部渐近等价，以及显式构造的量子通道。这一理论的一些元素已经由提出者和合作者最近的结果提出，用相干态(移位真空态)的高斯模型来近似大量独立量子系统的模型。我们的目标将是在纠缠的情况下进一步发展这些概念，首先专注于高斯平稳序列的量子模拟。其中有两种情况：一种是量子光学中常用于模拟压缩真空的零均值纯高斯态的定态模型，另一种是规范不变的定态高斯态，它似乎表现出经典的限制行为。在一个相关的主题中，关于尖锐的非参数风险渐近的结果，如Pinsker界，最初在经典的白噪声模型中被证明，推动了等价理论的发展。为估计纯量子态及其自适应可达性，建立这种类型的极小极大风险界是有意义的。在对称量子假设检验或两个量子态之间的区分问题中，提出者和合作者解决了长期存在的与错误概率的指数衰减率有关的量子Chernoff下界问题。在这方面，出现了几个新的问题，如区分复合假设的最佳误差指数，以及量子光学中可实现接收器的界限可达性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A fundamental insight of quantum mechanics is that randomness is a inherent feature of the physical world at the microscopic level. Any observation made on a quantum system such as an atom or a light pulse, results in a nondeterministic, stochastic outcome. The study of the direct map from the system?s state or preparation to the probability distribution of the measurement outcomes has been one of the core topics in traditional quantum theory. In many quantum protocols, the experimenter has incomplete knowledge and control of the system and its environment, or is interested in estimating an external field parameter which affects the system dynamics. In this case, one deals with a statistical inverse problem of inferring unknown state parameters from the measurement data obtained by probing a large number of individual quantum systems. The theory and practice arising from tackling such questions is shaping up into the field of quantum statistics, which lies at the intersection of quantum theory and statistical inference. The current project aims at a better understanding of statistical inference for infinite dimensional quantum systems, an area which can be seen as a quantum counterpart of nonparametric statistics. The ultimate goal is to develop a theory of comparison and convergence of quantum statistical models, thereby enabling efficient estimation techniques and establishing solid statistical methodology for computing reliable error bars. This project is well aligned with NSF's Quantum Leap Big Idea. The graduate student supported by this award will carry out research on quantum statistics, under the supervision of the PI.An area of particular interest is local asymptotic equivalence, with explicitly constructed quantum channels, of different quantum statistical models of pure states with a parameter in Hilbert space. Some elements of such a theory have already been put forward by a recent result of the proposer and collaborators, approximating a model of a large number of independent quantum systems by a Gaussian model of coherent states (shifted vacuum states). The goal will be to further develop these notions in cases of entanglement, focusing at first on quantum analogs of Gaussian stationary sequences. Among these, two cases are singled out: a stationary model of zero mean pure Gaussian states commonly used to model the squeezed vacuum in quantum optics, and gauge invariant stationary Gaussian states which appear to exhibit classical limiting behavior. In a related topic, results on sharp nonparametric risk asymptotics like the Pinsker bound, proved initially in classical white noise models, have motivated the development of equivalence theory. It is of interest to establish minimax risk bounds of this type for estimating pure quantum states, and their adaptive attainability. In the problem of symmetric quantum hypothesis testing, or discrimination between two quantum states, the proposer and collaborators solved the longstanding problem of the quantum Chernoff lower bound pertaining to the exponential rate of decay of the error probability. In that connection, several new problems appear, such as the optimal error exponent for discriminating composite hypotheses, and attainability of the bound by realizable receivers in quantum optics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.