Landauer cost in a continuous vacuum/no-vacuum measurement

We study the thermodynamic cost of maintaining a continuous binary record of a vacuum or no-vacuum measurement. Modeling the monitoring as a time-binned click or no-click process with finite bandwidth, we treat the outcomes as a classical register that is reset after each bin. Landauer's principle then yields an operational lower bound on the dissipated heat rate set by the Shannon entropy rate of the measurement record. We discuss the role of coarse-graining, extend the analysis to many monitored modes, including correlations and compressibility, and provide parameter estimates for circuit-QED photon monitoring, with a speculative horizon-based bookkeeping illustration.