Upper Bound on Locally Extractable Energy from Entangled Pure State under Feedback Control

We introduce an effective thermodynamics for multipartite entangled pure states and derive an upper bound on extractable energy with feedback control from a subsystem under a local Hamiltonian. The inequality that gives the upper bound corresponds to the second law of information thermodynamics in our effective thermodynamics. In addition, we derive a more general bound that is determined only by an initial state and the local Hamiltonian. This bound gives an explicit relationship between the extractable energy and the entanglement structure of the initial state. We also investigate the tightness of the upper bounds and show that the bounds can be achieved in a simple example.