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

We introduce entanglement-based effective thermodynamics for multipartite entangled pure states and derive an upper bound on locally extractable energy by quantum measurement and feedback control. In our effective thermodynamics, the energy extraction process is considered as a nonequilibrium process under a local Hamiltonian at an effective temperature defined with entanglement. Our bound is represented as the sum of the nonequilibrium free energy reduction and the information gain from the measurement, and can be seen as the second law of information thermodynamics in the effective thermodynamics. We also derive a more general bound that is determined only by an initial state of the whole system and the local Hamiltonian. This bound shows how the entanglement structure of the initial state contributes to the energy extraction. In addition, we investigate the tightness of the bounds and demonstrate that the bounds can be achieved in a simple four-qubit example.