Quantum statistics: decision problems and entropic functionals on state spaces

总的目标是提供解决量子统计领域内基本状态识别和决策问题的概念和方法。当量子信息处理任务被解决时，这些自然会出现在各种各样的计算机中。根据量子统计中采用的操作观点，我们假设关于量子态的缺失信息应该从对所考虑的系统进行测量的结果中推断出来。这与数理统计的方法相一致，在数理统计中，统计决策、推断或估计问题通常是相对于被认为是具有未知分布的随机变量的实际值的观测数据来制定的。我们使用 *-代数形式主义，它允许处理经典的随机变量和随机过程，以及量子态-与其内在的随机性-在数学上统一的方式。重点是状态空间的概念的C*-代数模型的代数observablesassociated到一个物理系统，无论是经典的还是量子的假设有任意多个感兴趣的系统副本可供观测，经典的热力学问题的解通常与一些熵量有关。这在量子力学中也被证明是正确的。虽然经典结果的量子对应并不简单，而且它们的证明通常需要新的数学方法，但我们主要讨论了渐近（多）态判别、复合量子假设检验和量子最大熵推论，以及相应的态空间熵距离为Umegaki相对熵和量子最大熵距离的推广等问题。我们的数学方法主要是联合收割机结合矩阵分析和数理统计的结果以及（非交换）遍历理论。

The general objective is to provide concepts and methods solving basic state identification anddecision problems within the field of quantum statistics. These naturally appear in various scenarioswhile quantum information processing tasks are addressed. Following an operational viewpoint adoptedin quantum statistics we assume that the missing information about the quantum state should be inferredfrom outcomes of measurements performed on the considered system. This is in correspondence tothe approach of mathematical statistics where statistical decision, inference or estimation problems aretypically formulated with respect to observed data considered as actual values of a random variablewith unknown distribution.We use a *-algebraic formalism which allows to treat classical random variables and stochasticprocesses, as well as quantum states -with their intrinsic randomness- in a mathematically unified way.The emphasis is on the notion of state space of a C*-algebra which models the algebra of observablesassociated to a physical system, either classical or quantum. Assuming that there is an arbitrary largenumber of copies of the system of interest available for observation, solutions of classical statisticalproblems are typically related to some entropic quantities. This in general turns out to be true as wellin quantum mechanics. Although, it is non-trivial to find quantum counterparts of classical results andtheir proofs usually require new mathematical methods.We focus on problems of asymptotic (multiple) state discrimination, testing compound quantumhypotheses and quantum maximum-entropy inference, and corresponding entropic distances on statespaces being Umegaki relative entropy and generalizations of quantum Chernoff distance. Our mathematical methods mainly combine results of matrix analysis and mathematical statistics as well as(non-commutative) ergodic theory.