Fundamental limits on the trade-off relation between work fluctuation and energy dissipation of heat engines

Reducing work fluctuation and energy dissipation in heat engines is crucially important to increase the efficiency of a given thermodynamic task. Near equilibrium, the fluctuation-dissipation theorem provides a linear relation between work fluctuation and energy dissipation. An information heat engine utilizes information as a resource to extract work from the system. The engine ends up in an out-of-equilibrium feedback-controlled state in general. A heat engine that operates for an arbitrary initial state can extract more work than its counterpart starting at equilibrium. In a state preparation, one has to consider a thermodynamic cost of creating the final state of a target starting from an easily prepared initial state. For such general situations, we derive the fundamental trade-off relation between work fluctuation and energy dissipation. In the vanishing work fluctuation regime, we recover the single-shot results with deterministic work extraction protocols. Near the zero dissipation regime, we recover thermodynamically reversible protocols. In an intermediate regime, the minimum amount of dissipation is characterized by the Renyi divergence of order $\alpha$, where $\alpha$ is given by the degree of work fluctuation. We give an explicit protocol that achieves the lower bound of the trade-off relation.