Asymptotic Equivalence of Quantum Statistical Models

量子力学的一个基本观点是，在微观层面上，随机性是物理世界的固有特征。对量子系统（如原子或光脉冲）的任何观察都会导致不确定的随机结果。从系统直接映射的研究？S态或对测量结果概率分布的准备一直是传统量子理论的核心课题之一。在许多量子协议中，实验者对系统及其环境的了解和控制不完全，或者对影响系统动力学的外场参数有兴趣。在这种情况下，我们处理的是从探测大量单个量子系统获得的测量数据推断未知状态参数的统计逆问题。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，它处于量子理论和统计推断的交叉点。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物。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条。这个项目与美国国家科学基金会的“量子飞跃大构想”非常吻合。该奖学金资助的研究生将在PI的指导下进行量子统计研究。一个特别感兴趣的领域是局部渐近等价，具有显式构造的量子通道，在希尔伯特空间中具有参数的纯态的不同量子统计模型。这个理论的一些要素已经在最近的一个结果中被提出，这个结果是通过相干态（移位真空态）的高斯模型来近似大量独立量子系统的模型。我们的目标是在纠缠的情况下进一步发展这些概念，首先关注高斯平稳序列的量子类似物。其中，选择了两种情况：通常用于量子光学中模拟压缩真空的零平均纯高斯态的平稳模型，以及表现出经典极限行为的规范不变平稳高斯态。在一个相关的主题中，尖锐非参数风险渐近的结果，如Pinsker界，最初在经典白噪声模型中得到证明，推动了等价理论的发展。建立这种类型的极大极小风险界来估计纯量子态及其自适应可达性是一个有趣的问题。在对称量子假设检验或两个量子态的区分问题中，作者和合作者解决了长期存在的关于误差概率指数衰减率的量子切尔诺夫下界问题。在这方面，出现了几个新问题，如鉴别复合假设的最优误差指数，以及量子光学中可实现接收器界的可得性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.