No-improvement conjecture for annihilation-only coherent controllers

Let a coherent-classical estimation scheme be defined by the plant, coherent controller, coherent-classical homodyne detector, and coherent-classical estimator, with estimation cost $\tilde{J}_c$. Assume that the plant is physically realizable and that the coherent controller is a physically realizable annihilation-operator-only system. Let the corresponding purely-classical estimation scheme have cost $\bar{J}_c$.

No-improvement conjecture. The coherent-classical estimation cost satisfies

$$\tilde{J}_c \geq \bar{J}_c.$$

This conjecture asserts that an annihilation-operator-only coherent controller cannot improve estimation accuracy over the corresponding purely-classical estimator. The paper motivates it using the quantum data-processing inequality, but does not establish it in general.