Some Properties Of Kth-partial Renyi Entropy And T~+Majorization

In1948, Shannon put forward the concept of "information entropy". The re-search of entropy has been rapidly developed It measures how much un-certainty there is in the state of a physical system. From then on, many scholars proposed various general entropy on the basic of Shannon entropy, such as Tsallis entropy, Renyi entropy, quasi entropy and so on. They are all have good functional proper-ties, including Non-negativity, Monotonicity, Continuity, Concavity and Convexity and Stability. Quantum entropy are nontrivial mathematical subjects, and active study of them has led to some deep insight.However, quantum entanglement catalysis is also very important in quantum information processing and quantum computation. For qualitive and quantitative research of quantum entanglement is very important. But it is still not perfect to establish a general theory of entanglement of quantum state and quantitative research, we need further research. The main task of this paper is to investigate some properties of the entropy and the quantum entanglement catalysis.This paper discusses some properties of kth-partial Renyi entropy and T+ma-jorization. The main conclusions are as follows:1We first introduce the k-th Renyi entropy for both classical and quantum cases. Then we get some primary properties of them, such as Non-negativity, Mono-tonicity, Stability, Symmetry, Concavity and Convexity, Schur-Conc-avity.2We discusse the connection between quantum measurement, quantum chan-nel and the majorization.3The two main properties associated of T+majorization with entanglement catalysis are studied and the anti-symmetry of T+majorization is proved. They are based on the optimization theory.