Super-fidelity, Sub-fidelity And Their Extension

In quantum information theory, an’ essential task is to distinguish two quantum states. If states are pure, then distinguishability of them is not difficult. However, it becomes complicated for two mixed quantum states. This requires us to fide a good distinguish approaches. If it give any two quantum states, the trace distance and fidelity are the two frequently used distances recently. The trace distance ia a good metric, it has been checked in Ref [1], it also has many good qualities, for instance, symmetry, convexity and for trace-preserving quantum operations are contractive. Comparatively trace distance, fidelity is not metric, but it can induce some metrics. We all know, the range of fidelity is0to1, but we hope that the range of fidelity is more accurate. In this paper, We are more concerned about the bound of fidelity. Super-fidelity and sub-fidelity are the important upper and lower bounds of fidelity, respectively. Of course, the study of trace distance and fidelity is not limited to this. In the paper of A.Uhlmann and A.E.Rastegin, they promote the concept of the trace distance and fidelity to partial trace distance and partial fidelity. In this paper, on the basis of their research, we re-define partial fidelity and our goal is to obtain similar results for the lower and upper bounds of partial fidelity. The minimal and maximal of trace distance and Bures distance between two unitary orbits are obtained in [28]. Similar, in the paper, we will discuss the maximal and minimal of sub-fidelity from two unitary orbits.This paper is divided into two chapters, the concrete structure is as follows:Chapter1Firstly, some basic concepts such as quantum bit and quantum state are introduced. Secondly, we give some knowledge about the matrix, the theory of majorization and Schur-convex function theory, which are the main technical support in this paper. Some lemmas to use in the second chapter is proved in the end of this part.Chapter2We first introduced the super-fidelity and sub-fidelity that is founded by J.A.Miszczak et al. Next, the definitions of partial super-fidelity and partial sub-fidelity are provided on the basis of the super-fidelity and sub-fidelity, furthermore, some of their properties is studied. In the last of this paper, we are going to discuss the maximal and minimal of super-and sub-fidelity from two unitary orbits.