Upper bound on locally extractable energy from entangled pure state under feedback control

In order to investigate locally extractable energy from multipartite entangled pure states, we introduce effective thermodynamics by defining an entanglement-based effective temperature. Based on the effective thermodynamics, we 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. Using the entanglement-based effective temperature, we discuss how our energy extraction protocol on pure states is related to ordinary information thermodynamic processes and show the consistency of our results with previous information thermodynamics. We also investigate the tightness of the upper bounds and show that the bounds can be achieved in a simple example.