Effective Gibbs state for averaged observables

We consider the effective Gibbs state for averaged observables. In particular, we compute the corresponding effective perturbation Hamiltonian. We show that there are many similarities between this effective Hamiltonian and the Hamiltonian of average strength. We also discuss the thermodynamic role of information loss due to the fact that our measurement capabilities are limited only by such averaged observables. The report is based on arXiv: 2110.14407 [quant-ph]. This work was supported by the Ministry of Education and Science of the Russian Federation (agreement No. 075-15-2020-788).