Quantum State Discrimination And Quantum State Transformation Based On Semidefinite Programming

With the development of quantum information technology, quantum information science exhibits more and more advantages than its classical counterpart. At the same time, it is unavoidable to face many challenging problems.As two fundamental problems of them, quantum state discrimination and quantum state transformation play an important role in quantum communication and quantum computation. Based on the fundamental features of quantum mechanics, the thesis discusses these two problems using the theory of semidefinite programming.The quantum state discrimination problem comprises two problems: optimum unambiguous discrimination of quantum states and minimum-error quantum state discrimination. Now explicit results have been obtained only in some specific cases and we must resort to numerical methods in the most general case. In fact, both of the general problems of quantum state discrimination belong to semidefinite programming problems and can be solved by using the theory of semidefinite programming which is well-developed. Therefore, based on this, we demonstrate that both of the general problems can be formulated as standard semidefinite programming problems. Then, we propose semidefinite programming numerical methods for the two general problems respectively. Furthermore, based on the optimality conditions for the semidefinite program, we derive the optimality conditions for these problems.Quantum state transformation problem is involved in devising physically realizable quantum operation which is optimal. It is hard to attack the general problem by analytical means and we must resort to numerical methods. As well as the general problem of quantum state discrimination, the problem can be solved by using the theory of semidefinite programming. Therefore, we demonstrate that the general problem of quantum state transformation can be formulated as a standard semidefinite programming problem. Then, we propose a semidefinite programming method for the general problem. Furthermore, based on the optimality conditions for the semidefinite program, we derive the optimality conditions for the general problem.Simulation results indicate that these semidefinite programming methods we have proposed are quite efficient. These methods have many advantages. Firstly, they are simple to implement in computer. Secondly, they converge fastly and need few iteration numbers. Finally, the results are very reliable.