Asymptotic Methods in Quantum Statistics

对统计学家来说，量子信息的处理是一个具有挑战性的新领域。虽然内在随机性的概念是量子力学的核心，但它不能仅用传统的概率来描述，即观察到的随机变量、样本空间等概念是不够的。在最简单的层面上，有限概率定律必须被状态所取代，状态被定义为轨迹为1的复杂正定厄米矩阵。算子代数理论提供了一个非常一般的框架。在量子统计决策理论中，状态族概括了概率测度族（统计实验），许多状态的张量积取代了经典的简单随机样本。在经典环境中已知的一个问题是对称假设检验的风险渐近性，或具有相等先验权重的两种状态的贝叶斯判别（错误概率的指数衰减率的Chernoff界）。最近，研究者和合作者在量子水平上解决了这个问题，他们开发了一种新的方法，通过在纯化上测量将一对概率分布与一对状态关联起来，从而降低量子风险到经典风险。本项目旨在进一步探索这种新方法在量子测试、估计和可能的量子统计实验中的更广泛适用性。进一步的研究课题是信息理论概念的量子统计应用，如信道容量、柯尔莫哥洛夫复杂度和速率失真。在非常小的（亚原子）水平上的自然现象受量子理论的支配，其中的物理定律和因果关系与普通人类经验的世界完全不同。物理学家和计算机科学家不久前意识到，这些现象可以用来制造速度超常的计算机，也可以使密码学迅速发展，比如破解目前已知的所有密码，或者构建新的无法破解的密码。虽然量子计算机长期以来一直是一个抽象的概念，到目前为止还只是在萌芽阶段建立起来的，但在量子信息理论的跨学科领域，已经为深远的应用奠定了理论基础。为了寻找最佳性能的基准，该领域的研究人员最近开始利用一些发达的信号处理和统计理论。目前的项目正好位于传统统计学和量子理论之间的这个新前沿。其目的是对量子计算和通信领域有更好的数学和统计理解，这些领域一旦进入应用阶段，将产生巨大的技术影响。

The processing of quantum information is emerging as a challenging new field for statisticians. While the concept of inherent randomness is central to quantum mechanics, it cannot be described in terms of traditional probability alone, i.e. notions such as observed random variables, sample spaces etc. are not sufficient. On the simplest level, finite probability laws have to be replaced by states, which are defined as complex positive definite Hermitian matrices of trace one. A very general framework is provided by the theory of operator algebras. In quantum statistical decision theory, families of states generalize families of probability measures (statistical experiments), and the tensor product of many states replaces the classical simple random sample. One problem which is already known in the classical context is the risk asymptotics for symmetric hypothesis testing, or Bayesian discrimination of two states with equal prior weights (the Chernoff bound on the exponential rate of decay of the error probability). In the recent solution of this problem on the quantum level by the investigator and coauthors, a new method has been developed to reduce the quantum risk to a classical one, via associating a pair of probability distributions to a pair of states by measurement on a purification. The present project aims at exploring further this new method, with regard to wider applicability in quantum testing, estimation and possibly in approximation of quantum statistical experiments. Further subjects of study are quantum statistical applications of information theoretic concepts like channel capacity, Kolmogorov complexity, and rate distortion.Natural phenomena on the very small (subatomic) level are governed by quantum theory, where physical laws and cause-effect relationships are thoroughly different from the world of ordinary human experience. Physicists and computer scientists realized some time ago that these phenomena might be harnessed to build computers of extraordinary speed, and also allow rapid advances in cryptography such as breaking all presently known secret codes or constructing new unbreakable ones. While quantum computers have long remained an abstract idea and have only been built at an embryonic stage so far, the theoretical groundwork for far-reaching applications is already being laid in the interdisciplinary field of Quantum Information Theory. In the need for finding benchmarks for optimal performance, researchers in this field have recently begun to exploit some well developed theories of signal processing and statistics. The present project is situated precisely at this new frontier between traditional statistics and quantum theory. The aim is to achieve a better mathematical and statistical understanding of quantum computing and communication, areas which promise to be of great technological impact once they reach an applied stage.