On The Quantum Entropy

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.