Fig. 3: Heat distributions for a qubit undergoing the thermalization Step (III).
From: Energetic footprints of irreversibility in the quantum regime
Histograms of classical heat Qcl (red circles) and quantum heat Qqu (blue squares) for a an initial state \({\rho }_{\tilde{\theta }}\) that hosts classical non-thermality: \(nonth({\rho }_{\tilde{\theta }})=\mathrm{log}\,(0.2/0.3)\) and \(coh({\rho }_{\tilde{\theta }})=0\), and for b an initial state \({\rho }_{\tilde{\theta }}\) that hosts quantum coherence: \(coh({\rho }_{\tilde{\theta }})={\sin }^{2}(\pi /6)=1/4\) and \(nonth({\rho }_{\tilde{\theta }})=0\). For comparison, gray circles and gray diamonds in both panels show the classical and quantum heat histograms, respectively, for when Step (III) is fully reversible, i.e., \({\rho }_{\tilde{\theta }}={\tau }_{1}\) and hence \(coh({\rho }_{\tilde{\theta }})=0=nonth({\rho }_{\tilde{\theta }})\). Note that even then the system can exchange heat with the bath leading to a classical heat distribution with non-zero but symmetrical values (dashed line) that give a zero average classical heat. In a the only quantum heat value with non-zero probability is 0 (no quantum heat when thermalizing a classical state), while in b four nontrivial quantum heat values occur since \(coh({\rho }_{\tilde{\theta }}) \, \ne \, 0\).
