【作者】 汪加梅；

【导师】 鲁世杰；

【作者基本信息】 浙江大学 ， 基础数学， 2010， 博士

【摘要】 熵理论在科学史上是一个极其重要的里程碑.从它产生至今已经一百五十余年了,但我们仍然难以对其进行一个科学的描述.伴随着热力学和统计力学的发展,熵的研究显得越发重要,且它的物理意义变得越发隐晦.本文首先研究了序列效应代数的熵理论,主要结果如下：●利用序列效应代数理论,本文提出了量子逻辑的分割以及分割的加细的概念,并给出了各种熵的定义.●本文研究了序列效应代数的各种熵的关系及性质.●本文定义和研究了态关于分割的相对熵.●通过熵的韦恩图,我们说明了序列熵是经典熵的推广.特别地,通过量子测量的实例,本文说明了序列效应代数的熵及态关于分割的相对熵极其依赖于它的序列积的顺序.其次,我们引进并研究了经典和量子的统一(r,s)-相对熵,这种相对熵是若干重要的广义相对熵的推广.特别地,本文的性质是建立在可分的量子系统下的,而不仅仅适用于有限维.其中,尤其重要的性质是,可分量子系统的量子统一(r,s)-相对熵在保迹完全正定映射下是不增的.更多还原

【Abstract】 The entropy theory in the history of science is an extremely important mile-stone. More than one hundred and fifty years have passed since it generates, but we are still difficult to give them a scientific description. Accompanied by the develop-ment of thermodynamics and statistical mechanics, the study of the entropy theory becomes increasingly important, and its physical meaning has become increasingly obscure.First, in this paper, we studies the entropy theory of the sequential effect algebra. Main results are as follows:●Making use of the sequential effect algebra theory, we propose the definitions of the partition and the refinement of partitions, and present different kinds of entropies.●We investigate the relations and properties of different entropies of the se-quential effect algebra.●We define and study the relative entropy of states about partition,●Using the venn diagram, we show that the entropies of sequential effect al-gebra are the generalization of the classical entropies. In particular, by some properties and examples, we display that the entropies of the partitions of sequential effect algebra and relative entropies of states about partition ex-tremely dependent on the order of its sequence product.Second, we also introduce and study the classical and quantum unified (r, s)-relative entropies, which are the generalization of some important generalized rel-ative entropies. Particularly, our properties are based on the separable quantum system, not only suit for the finite dimensional quantum system. The most im-portant one in those properties is that quantum unified (r, s)-relative entropy for a separable quantum system is decreasing under a trace-preserving completely posi-tive mapping.更多还原