Quantum Channels, Fidelity In Infinite Dimensional Systems And Relative Topics

Quantum information and quantum computation is one of the most active subjectsin recent decade years, and can be applied into many areas. Many physicists,computerscientists and Mathematicians have paid their attentions to the theory. In the mathe-matical framework of quantum theory, quantum states are positive operators with unittrace on Hilbert spaces. Distinguishing quantum states is one of important problemsin quantum information science, and fidelity plays a important role. However, it is dif-ficult to calculate the fidelity. Some properties and estimations of fidelity for the finitedimensional quantum systems are known. But there are few discussion for the infinitedimensional case. In the thesis, we study the channels, fidelity and relative topics forinfinite dimensional quantum systems.1. We generalize some properties of quantum fidelity to infinite dimensional case,including the study of the lower and upper bound of fidelity, thai is, super-and sub-fidelities. For any two n-mode Gaussian states, we calculate the super-and sub-fidelities.Furthermore, we give lower and upper bounds of the trace distance.2. The unitary freedom in operator sum representation for quantum channels infinite dimensional systems is a very important basic property for quantum informationtheory. However, analogy for infinite dimensional case is not valid. Instead, we estab-lish the bi-contractive freedom in operator-sum representations for quantum channels.Applying the result, we give a calculation formula of entanglement fidelity. The relationbetween the fidelity and entanglement fidelity in infinite dimensional quantum systemsis also discussed. 3. Fidelity, super-fidelity and sub-fidelity are invariant under unitary and anti-unitary evolutions. We discuss the inverse problems and give a characterization of sur-jective maps on quantum states which preserve any one of the fidelity, the super-fidelityor the sub-fidelity for arbitrary dimensional systems. Our results show that any one ofthese three physical quantities determines the unitary and anti-unitary evolutions.4. We establish a quantum error-correcting theorem for infinite-dimensional quan-tum system.