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The Lindblad equation is widely employed in studies of Markovian quantum opensystems. Here, firstly, a simple result is presented on the time evolution ofthe non Neumann entropy under the Lindblad equation, which enables one toexamine if the entropy increases/decreases. Then, secondly, the followingquestion is posed: In a quantum open system with a time-dependent Hamiltonian,what is the corresponding Lindblad equation for the quantum state that keepsthe internal energy of the system constant in time? This issue is of importancein realizing quasi-stationary states of open systems such as quantum circuitsand batteries. As an illustrative example, the time-dependent harmonicoscillator is analyzed. The Lindbladian operator is uniquely determined withthe help of a Lie-algebraic structure, and the time derivative of the vonNeumann entropy is shown to be nonnegative depending on the time-dependence ofthe Hamiltonian.

Lindbladian operators, von Neumann entropy and energy conservation in time-dependent quantum open systems