Fig. 5
From: Energetic instability of passive states in thermodynamics
Instability of passive states and their dynamics. a The state space of a qutrit system, where the region of passive states is highlighted in light blue. The black line contained in the passive region is the set of thermal states. We fix an initial state ρ P, represented by the black point in the diagram. Then, we evolve this state by applying the cycle S m,n (where m, n→∞) an infinite number of times. The evolution is then modulated by the parameter \(\alpha = \frac{n}{m}\). For \(\alpha\) equal to \(\frac{{\Delta E_{10}}}{{\Delta E_{21}}}\), the system evolves along the yellow trajectory, and the final state is the thermal state at temperature β min (with same average energy of ρ P). For \(\alpha = \frac{{\beta _{{\mathrm{hot}}}\left( t \right)\Delta E_{10}}}{{\beta _{{\mathrm{cold}}}\left( t \right)\Delta E_{21}}}\), the system evolves along the purple line, and the final state is the thermal state at temperature β max (with same entropy of ρ P). The dark blue region represents the subset of achievable states when the initial state is ρ P. b A partial representation of the state space of a d-level quantum system in the energy–entropy diagram32. In this diagram, quantum states are grouped into equivalence classes defined by their average energy E and their entropy S. Each point between the x-axis (the set of pure states) and the dark blue curve (the set of thermal states) represents one of these equivalence classes. The diagram depends on the Hamiltonian H P of the system. Here, we only represent the states with average energy lower than \(\bar E = {\mathrm{Tr}}\left[ {H_{\rm P}\rho _{{\mathrm{mm}}}} \right]\), where \(\rho _{{\mathrm{mm}}} = \frac{{{{\Bbb I}}}}{{\mathrm{d}}}\) is the maximally-mixed state, since all passive states are contained in this set. For a given initial state ρ P, the light blue region contains all the passive states, which can be achieved with the process
