  Thermodynamics as a fundamental branch of physics examines the relationships
between heat, work, and energy. The maximum energy extraction can be
characterized by using the passive states that has no extracted energy under
any cyclic unitary process. In this paper, we focus on the concept of marginal
passive state energy and derive polygon inequalities for multi-qubit entangled
pure states. We show that the marginal passive state energies collectively form
a convex polytope for each class of quantum states that are equivalent under
SLOCC. We finally introduce multipartite passive state energy criteria to
classify multipartite entanglement under SLOCC. The present result provides a
thermodynamic method to witness multipartite entanglement.
