Computation Of Decoherence-free Subspaces And Adjoint Matrices For Quantum System And System Control

This paper is mainly concerned with computational problems related with the analysis and con-trol of multi-qubit quantum systems and control problem of open system. For multi-qubit quan-tum systems coupled with environment undergoing the collective decoherence, there may exist some decoherence-free states. These states span a decoherence-free subspace (DFS). A set of basis states of the DFS consist of the singlets which are states of zero spin, i.e., of zero quantum number. From this, quantum number plays an much important role in calculating the singlet and then influences the computation of DFS. By analysing and researching DFS and constructing Lyapunov function, we ob-tain a group of the real control fields for open quantum system. They can drive an open system to the target state in the DFS. For multi-spin 1/2 quantum systems, in order to transform the Liouville-von Neumann equation into the coordinate differential equation, computation of the adjoint matrices is much important. So we calculate some adjoint and anti-adjoint matrices in three-spin 1/2 systems and analyze the characters of these matrices. The main results obtained in this paper consist of three parts.In the first part, quantum number of the spin angular momentum on representations of Lie al-gebra g3 is researched mainly. Firstly, some basic definitions and conclusions of representations are listed. Secondly,by extending the theorems of representations of lie algebra g3, selecting suitable irreducible representation and according to the known theorem, the values of spin quantum number can be derived. At the same time, the derived conclusions can also be extended to the orbital angular momentum. Finally, we make a summary of the obtained conclusions.In the second part, a generalized method of computing decoherence-free subspaces is provided for K-qubit quantum system undergoing the collective decoherence; Lyapunov control is given in DFS for open quantum system under Lindblad-semigroup formulation. Based on the Young tableaux, the Clebsch-Gordan decomposition for SU(n) is given and some examples are listed. According to the decomposition, we can get the dimensionality of the DFS. The DFS Conditions Under Hamil-tonian Formulation and Lindblad-Semigroup Formulation are listed respectively. According to the condition, we can acquire the singlet space. And the computation of decoherence-free subspaces for the collective decoherence of the 4-qubit quantum system coupled with environment is derived. Fi-nally, we obtain s set of real control fields by constructing Lyapunov fuction for open quantum system such that open system is driven to a given target state in DFS.In the final part, the adjoint matrices play an much important role in the coordinate differential equation transformed from the Liouville-von Neumann equation for a density operator. In order to characterize the dynamics of the coordinates of the density operator in three-spin 1/2 systems,64 adjoint matrices, which are 64 dimensional, need to be computed. Based on the established compu-tational formulas of adjoint and anti-adjoint matrices in multi-spin 1/2 systems, the computational examples of some adjoint and anti-adjoint matrices are given in three-spin 1/2 systems. The results of the examples reveal these matrices are sparse. All the nonzero entries of these sparse matrices are listed in tables and the distribution of the nonzero entries in each 64 x 64 matrix is discussed as well.