Quantum statistics: decision problems and entropic functionals on state spaces

量子统计：状态空间上的决策问题和熵函数

基本信息

批准号：
190280289
负责人：
Dr. Arleta Szkola
金额：
--
依托单位：
Abteilung Funktionalanalysis
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2011
资助国家：
德国
起止时间：
2010-12-31 至 2013-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/190280289?language=en
关键词：
Quantum statistics decision problems entropic

项目摘要

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.

一般目标是提供解决量子统计领域内基本状态识别和指定问题的概念和方法。这些自然出现在各种情况下，同时解决了量子信息处理任务。遵循操作观点采用的量子统计数据，我们假设应推断出有关量子状态的丢失信息，请从考虑到考虑系统上执行的测量结果。这是数学统计方法的对应方法，在该方法中，统计决策，推理或估计问题在型上相对于被观察到的数据所示，被认为是随机变量的实际值，与未知分布相关。 C* - 代数的状态空间的概念，该概念模拟了与经典或量子的物理系统相关的可观测值代数。假设有可用于观察系统的副本副本的任意劳通，那么经典统计问题的解决方案通常与某些熵量有关。总的来说，这是真实的。虽然，找到经典结果和证明的量子通常需要新的数学方法是非琐事。我们专注于渐近（多重）状态歧视的问题，测试复合量子量表和量子量子性最大值推断，最大值推断，以及在umegaki相对范围内的量化距离和相对距离的距离。我们的数学方法主要结合了矩阵分析和数学统计的结果以及（非共同）ergodic理论。