Thermodynamic closure of quantum measurements and the limits of the indirect measurement model

We investigate the consequences of requiring that a quantum measurement admit an adiabatic enclosure, so that it can be assigned a genuine thermodynamic description in which energy exchange is meaningfully resolved into work and heat. We identify two inequivalent levels at which such thermodynamic closure can be imposed: at the level of the measurement instrument acting on the system, or at the level of the indirect measurement process realising the instrument, namely the interaction between the system and a measuring apparatus, possibly including arbitrarily large environments. Although every instrument admits a unitary dilation, we show that such a construction does not generally provide an admissible thermodynamic closure. This distinction is especially dramatic for efficient measurements, i.e., measurements that are completely purity-preserving and described by a single Kraus operator, including von Neumann--L\"uders and square-root state-update rules. While efficient measurements are compatible with the laws of thermodynamics when thermodynamic closure is imposed at the level of the instrument, they are categorically forbidden when closure is imposed at the level of the measurement process. Our results therefore reveal a fundamental tension between thermodynamics and the universal applicability of the unitary interaction-based indirect measurement model.