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 ρ, we quantify the coherent extraction process by the coherent ergotropy E c(ρ) and the coherent extraction distance D c ext (ρ) between the active state σ ρ and the passive state P ρ. This defines the geometric power capacity Π c(ρ)= E c(ρ)/D c ext (ρ), which measures the coherent ergotropy released unit minimal unitary distance. We prove that, for any driving Hamiltonian satisfying |V t| at mostν, the actual coherent discharging power is bounded by P c ext (ρ;V t) at most νΠ c(ρ), showing that Π c(ρ) is a capacity under unit driving norm rather than the power of a particular protocol. General bounds on Π c(ρ) 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 Π c(ρ) captures a resource-geometric feature of coherent discharging beyond coherent ergotropy or coherence measures alone.