Fisher-information retention under local driving in non-Hermitian feed-forward chains

RESUMO

When local drive hardware is installed before a sensing task is revealed, directional non-Hermitian buildup creates a tradeoff between targeted preparation and spatial coverage. We quantify this tradeoff for stable linear scatterers with proper complex Gaussian output statistics. For parameters encoded in the internal generator, and at fixed coherent internal resource, the drive hardware enters the optimized mean-response Fisher information only through the projector onto the internally reachable subspace, while the preparation Gramian sets the source cost. In an on-resonance feed-forward chain, nested Green-function columns decompose into disjoint port intervals, and the site-local retention factors in each interval sum exactly to unity. Directionality concentrates this unit downstream, so the worst-site retention decays exponentially when one port serves many sites; for $m$ equal intervals, a single probe shared by all tasks has a worst-site retention lower by a further factor $m$ than task-specific reoptimization. Joint source-power and internal-resource constraints connect this fixed-resource law to source-limited preparation. A noisy RLC/VCCS model illustrates the retention law with stored electromagnetic energy, covariance Fisher information, and component disorder. The resulting bounds turn directional buildup into a port-layout criterion: source savings for a selected task are accompanied by a spatial coverage cost for tasks unknown at the hardware-design stage.