Some Trace Inequalities Of Quantum Computation And Quantum Information

In this article, we mainly discuss some inequalities about entropy. The full text is divided into three chapters.In Chapter 1, we introduce some linear algebra knowledge under the quantum information background, and led to go into the Dirac sign adopt in quantum mechanics and then show powerful mathematics tool relating with the quantum information and the quantum calculation. For example, the basis and linear independence, density operator and density matrix, inner product, outside product and tensor product, properties of them. These mathematics knowledge is a tool specially used in research of the quantum calculation and the quantum information.In Chapter 2, first we discuss Shannon entropy and its basic importance properties, mainly discuss simple concepts such as 2 dollar entropy, such as opposite entropy, the condition entropy and the mutual information and a little bit related inequalities. Secondly discuss Von Neumann entropy, and then prove its properties. For example, Von Neumann concavity, Klein inequality, the Klein inequality of the broad sense, Araki-Lieb inequality and the Fannes inequality about continuity of entropy.In Chapter 3, we discuss the Yanagi's inequality...