Identifying quantum correlations using explicit SO(3) to SU(2) maps

Quantum state manipulation of two-qubits on the local systems by special unitaries induces orthogonal rotations Bloch spheres. An exact formula is given for determining some rotation sphere. The solution allows easy two-qubit quantum states with a single definition that programmable. With this explicit formula, modifications to correlation matrix are made simple. Using our solution, it possible diagonalize without solving parameters in SU(2) define unitary SO(3). Since diagonalization equivalent interaction Hamiltonian, manipulating important time-optimal control state. relationship between orthogonality conditions and SO(3) when only one qubit can be accessed discussed.