On The Fixed Points Of Generalized Quantum Operations

In this dissertation, we study the fixed points of generalized quantum operations and related topics in the background of quantum information theory. The theory of quantum information is a combination of quantum mechanics and information theory, in which information is stored in a state of a quantum system and one uses quantum operations to describe the information processing. The dynamic of a quantum system is not close in general due to the inevitable coupling of the system to its environment. A consequence of non-unitary dynamic of an open system is decoherence which makes the quantum system turning into a classical system. However, the advantages of processing information in quantum systems heavily rely on the unique properties of quantum systems. Thus, the structures of quantum systems which can prevent the decoherence are of fundamental importance. Those structures are called the information preserving structure and have been intensively studied recently. Specifically, it was proved by Blume-kohout et al in 2010 that every information preserving structure is isometric to a set of fixed points of some quantum operation. This is a profound result in the study of fixed points of quantum operations and reveals the importance of studying this problem. In fact, since 2002, Arias et al have raised questions on this problem and for some special cases, the answers to these questions were gained satisfactorily. Now, we shall study the problem in the case of generalized quantum operations. The main results of this dissertation are as follows:●With the aid of dilation theory: we get some necessary conditions and a nec-essary and sufficient condition in some settings for the set of fixed points of a generalized quantum operation to be non-trivial.●We characterize the set of fixed points of commutative generalized quantum operations. As an application, we give another proof of the important problem raised by Arias et al. We characterize the set of fixed points of an arbitrary generalized quantum operation.