Interaction independent quantum probing

For an open quantum system we assume that we are able to set the system's environment temperature. We fix the time interval and let the system (further referred as the probing system) to evolve during this time in two different temperatures. We make a process tomography of the resulting dynamics (quantum channels "1; "2 related to the temperatures T1 and T2 respectively). We calculate then the values of fidelities for the pair of channels. We derive an inequality between the experimental data and the partition function of environment (hence the spectrum of the environment). If the inequality is not satisfied, it implies that our assumption about the spectrum of the environment is wrong. Notice that there is no dependence on the interaction terms neither on the Hamiltonian of the probing system. We show the power of this method in the following example. Consider a two-level atom passing the one-mode vacuum. We do not know the Hamiltonian of the atom (the probing system) neither the interaction mechanism. We would like to determine the frequency of the vacuum. We will show that wide range of frequencies are forbidden by the inequality.