Some Problems In Quantum Information Theory

In the last two decades, quantum mechanics has not only been deeply applied to many other branches of physics, but also been widely used in some other fields like information science. As a new interdiscipline, quantum information theory is making rapid progress recently, which shows vast potential in both theoretical re-search and technological applications in future development. Quantum information theory not only extend classical information to quantum information, but also uses quantum state in microsystem to represent it. Thus it has entered into the new stage in which quantum state can be artificially manipulated, stored and transmitted.Quantum measurement theory is associated with the foundation of quantum mechanics, and is the starting point of many other problems in quantum informa-tion theory. Non-disturbance between quantum measurements is closely related to other concepts, such as compatibility, joint measurability and coexistence, and it is important to see how to mathematically describe the non-disturbance and discuss its relationship with compatibility of quantum measurements. Quantum gates is the fundamental elements in quantum computation and quantum algorithm design. And great interest has been taken in generalized quantum gates since the duality quantum computer model was put forward.The key issue of quantum detection and quantum estimation theory is to distinguish two alternative density operators by quantum measurements. A convenient approach to this problem is to introduce various distance measures on the space of quantum states.In this thesis, we study several important problems such as the disturbance between quantum measurements, the characterizations of generalized quantum gates and distinguishability measure between quantum ensembles. The main content of this thesis contains the following ones:Firstly, we study the non-disturbance between quantum measurements. Based on the general sequential product on standard effect algebraε(H), we give new interpretations of measurements. and obtained some new theorems concerning non-disturbance criteria between quantum measurements in the sense of Kirkpatrick, which was previously studied by Gudder using the special sequential product. More-over, we show that the general sequential products are unitarily equivalent to Gud-der's sequential product.The second part devotes to some new characterizations of generalized quantum gates (GQGs) by using the techniques of operator matrices. Specifically, we show that A is a generalized quantum gate iff An is a generalized quantum gate for any positive integer n. Furthermore, using the characterizations of some important operator classes, we show that these operators are all GQGs, and some operators can be represented as the convex combination of two unitary operators.Thirdly, we introduce and study the Jensen-Shannon divergence between quan-tum ensembles. Using projection approximation method, we first study some impor-tant properties of Jensen-Shannon divergence in complex separable Hilbert space. Furthermore, by probabilistic coupling technique, we define the QJSD between quantum ensembles and study its properties.