Temporal-Access Resource Theory for Continuous Process Tensors

The uncontrolled endpoint channel can conceal how well a quantum process can protect a logical task under time-resolved control. We formulate a fixed-task resource theory of continuous process tensors with physical access interfaces, entrance-erasing free objects, and budgeted free conversions. Complete-record total-variation error measures performance; the calibrated capability is monotone under conversions that preserve a common free baseline. For unknown-state transmission the baseline is exactly $1-d^{-2}$. Two qubit processes with the same endpoint channel have optimal errors $0$ and $p={\sin }^2(2gh)$ at the same finite pulse budget in an ideal-transparent system-only window architecture, forcing any endpoint-only predictor to err by at least $p\mathrm{/}2$ on one of them. An exactly solvable family shows how the allocation of a fixed interaction phase around a control window sets the attainable error, and a nontrivial perfect echo requires and attains total control action $\pi$ in that architecture. The separation extends directly to every qubit unitary target at a budget that includes its boundary gate, while unitary tasks in any dimension have baseline $1-d^{-2}$ whenever a budget-feasible free realization gives the depolarizing output. Complete-record implementation bounds quantify coherent and dissipative control imperfections. Continuous-process randomized compilation supplies physical instrument pullbacks; compatible completion conditions extend their action, and certified dynamical bounds yield achievable task-error and capability guarantees. The resulting theory identifies temporal structure and control investment that endpoint characterization alone cannot resolve.