The Thermodynamic Geometry of Conditional Control

Information is widely regarded as the resource underlying the thermodynamic advantages enabled by conditional control. We show, however, that informational quantities such as Holevo information and accessible distinguishability, although constraining the achievable advantage, do not uniquely determine its thermodynamic value. The missing ingredient is a passive spectral rearrangement vector that characterizes the effect of conditioning on the ensemble. Specifically, the conditional-control advantage is exactly determined by the geometric pairing between this vector and the Hamiltonian energy-gap structure. This result reveals a thermodynamic geometry of conditional control, explains how informationally equivalent ensembles can possess different thermodynamic values, and identifies the passive spectral rearrangement vector as the minimal operational descriptor required to determine the thermodynamic value of conditional control for a fixed Hamiltonian.