Quantum Operation And Related Problems

The changes, for instance, decoherence, occurred over a quantum system, which is generally induced by quantum operations on the quantum system. It distinguishes between classical information theory and quantum information theory. Hence quan-tifying the changes induced over a quantum system by quantum operations is an important problem in quantum information theory.In this thesis, we study several important problems such as the dynamical addi-tivity and strong dynamical additivity of quantum operations, the characterizations of von Neumann entropy-preserving quantum operations, classical and quantum cor-relations in bipartite states and the upper bound of the Holevo quantity. The main content is the following:Firstly, we study the dynamical additivity and strong dynamical additivity of quantum operations. Extension of quantum states without changing its entropy to a larger quantum system is employed to characterize the additivity of map entropy for the composite quantum operations. Then we introduce notion of bi-orthogonality within quantum operations and based on it, we investigate the strong dynamical additivity of quantum operations.The second part devotes to characterizations of von Neumann entropy-preserving quantum operations. We represent the von Neumann entropy of a quantum state as the relative entropy between the target state and the full mixed state. By the satu-ration of Petz’s monotonicity condition of relative entropy and the property of fixed points of quantum operations, we give a complete characterization of von Neumann entropy-preserving unital quantum operations. As a consequence, Poritz’s classical version of corresponding Shannon entropy preserving bi-stochastic matrices can be easily derived from our results.Thirdly, we give an upper bound on the classical correlations in bipartite states in terms of quantum operation language. We also study the quantum correlations in bipartite states by making essential use of the Lindblad inequality and the structure of strong additivity states.Finally, we investigate the upper bound of the Holevo quantity. In the last part of the thesis, we generalize an important inequality which is appeared in the previous chapter. Then we utilize it to construct a bipartite state to prove a conjecture, proposed by W. Roga, on upper bound of the Holevo quantity.