The Causal Impedance Inequality: Operationalizing the Heisenberg Principle in the Presence of Causal Latency

We formulate an operator inequality that recasts the energy-time uncertainty principle as a bound on the causal generation of quantum distinguishability. A processing time is defined operationally as the minimal interval required for a quantum state to evolve into a configuration that is distinguishable from its initial state with respect to a fixed geometric criterion. Using the Bures metric and quantum Fisher information, we show that this processing time multiplied by the effective energy dispersion is bounded from below by a quantity of order hbar, independently of the underlying physical realization. The bound holds for unitary dynamics as well as for Markovian open-system evolutions described by completely positive semigroups. In this framework, the Heisenberg uncertainty principle acquires a dynamical meaning: it expresses a universal latency in the actualization of physical distinctions. The inequality thus identifies a fundamental limit on the speed of causal state evolution inherent to quantum theory.