Geometric Power Capacity of Coherent Ergotropy in Quantum Batteries

We explore coherent ergotropy extraction in quantum batteries from a resource-geometric point of view. For an initial state $\rho$, we quantify the coherent extraction process by the coherent ergotropy $\mathcal{E}_c(\rho)$ and the coherent extraction distance $D_c^{\rm ext}(\rho)$ between the active state $\sigma_\rho$ and the passive state $P_\rho$. This defines the geometric power capacity $\Pi_c(\rho)=\mathcal{E}_c(\rho)/D_c^{\rm ext}(\rho)$, which measures the coherent ergotropy released unit minimal unitary distance. We prove that, for any driving Hamiltonian satisfying $\|V_t\|\leq\nu$, the actual coherent discharging power is bounded by $P_c^{\rm ext}(\rho;V_t)\leq \nu\Pi_c(\rho)$, showing that $\Pi_c(\rho)$ is a capacity under unit driving norm rather than the power of a particular protocol. General bounds on $\Pi_c(\rho)$ are derived by combining relative entropy bounds on coherent ergotropy with geometric bounds on the coherent extraction distance. We also formulate coherence measure induced bounds and protocol-corrected capacities involving the effective speed of a given Hamiltonian. Qubit and qutrit examples demonstrate that $\Pi_c(\rho)$ captures a resource-geometric feature of coherent discharging beyond coherent ergotropy or coherence measures alone.