Volume 39, Issue 2, 2018
Monday, January 29, 2018
We examine the general form of the generators of time evolution on density matrices of damped oscillators and clarify the conditions that must be fulfilled by them. The generators must be adjoint-symmetric to preserve the hermiticity of density matrices and trace-preserving to conserve probability. The conditions lead to a set of basis generators where generic ones are formed by taking linear sums of the basis over real coefficients. The requirement on the positivity of density matrices then limit the range of the coefficients. In ordinary symmetry on state vectors (pure states) in the Hilbert space, symmetry operators are unitary and factorized. The symmetry on density matrices is more general in the sense that it is not limited to unitary and factorized transformation. The time evolution operators of quantum systems are examples of general transformation that form semigroups, but the generic general transformation is not limited to semigroup.