Abstract
Quantum information science, a interdisciplinary field, develops many interesting connections between the fields of physics, mathematics, computer science and other related subjects. The intuitive ideas in quantum information also has different influence in these areas. In this thesis, we use no-signalling condition in special relativity theory which is required to be satisfied in quantum information science, even under the condition that a maximally entangled state is given, to test parity-time symmetry quantum theory ($\mathcal{PT}$-symmetric theory). Our result shows that improper interpretation of $\mathcal{PT}$-symmetry quantum mechanics will cause the violation of no-signalling principle and set the constraints which should be obeyed for developing this novel theory. In the second part of thesis, we study the error correction in quantum information science on Kitaev honeycomb model, and establish the connection with the researches done by physicists. Our result formally proves the quasi-degeneracy and the approximate local-indistinguishability of the ground states of Kitaev honeycomb model. We further point out that the simulation of the information lifetime under the thermalization process is possible. By combining the methods from physics society and quantum information society, we shows that there are more possibility in the connection between these two fields.