Continuous thermomajorization and a complete set of laws for Markovian thermal processes

The standard dynamical approach to quantum thermodynamics is based on Markovian master equations describing the thermalization of a system weakly coupled to a large environment, and on tools such as entropy production relations. Here we develop a framework overcoming the limitations that the current dynamical and information theory approaches encounter when applied to this setting. More precisely, we introduce the notion of continuous thermomajorization and employ it to obtain necessary and sufficient conditions for the existence of a Markovian thermal process transforming between given initial and final energy distributions of the system. These lead to a complete set of generalized entropy production inequalities including the standard one as a special case. Importantly, these conditions can be reduced to a finitely verifiable set of constraints governing nonequilibrium transformations under master equations. What is more, the framework is also constructive, i.e., it returns explicit protocols realizing any allowed transformation. These protocols use as building blocks elementary thermalizations, which we prove to be universal controls. Finally, we also present an algorithm constructing the full set of energy distributions achievable from a given initial state via Markovian thermal processes and provide a Mathematica implementation solving $d=6$ on a laptop computer in minutes.

Figure 1

Thermodynamic entropy landscape for a three-level incoherent system. The state of an incoherent system is represented by a vector of populations $\mathit{p}=(p_1,p_2,1-p_1-p_2)$, and the chosen thermal state is $\mathit{\gamma }=(0.7,0.2,0.1)$. (a) Entropy functional $\mathrm{\Sigma }$. The black dashed trajectory is the path of constant $\mathrm{\Sigma }$. The constraint of positive entropy production cannot exclude that the thermodynamic transition from $\mathit{p}\left(0\right)$ to $\mathit{p}\left(t\right)$ is allowed. (b) Entropy functional ${\mathrm{\Sigma }}_2$. The black dashed trajectory is the path of constant $\mathrm{\Sigma }$. The constraint of positive collisional entropy production implies that one cannot transform $\mathit{p}\left(0\right)$ into $\mathit{p}\left(t\right)$.

Figure 2

Thermodynamic $\alpha$ entropy for a two-level system. Value of the relative entropy functional ${\mathrm{\Sigma }}_{\alpha }$ along the trajectory specified in Eq. (23) connecting $\mathit{p}\left(0\right)=(0.1,0.9)$ with $\mathit{p}\left(t_f\right)$ given by Eq. (19). The chosen thermal state is $\mathit{\gamma }=(0.6,0.4)$. Note that, for every $\alpha$, the path connecting $\lambda =0$ to $\lambda =1$ necessarily passes through a region in which entropy ${\mathrm{\Sigma }}_{\alpha }$ decreases. Hence, no Markovian thermal process can transform $\mathit{p}\left(0\right)$ into $\mathit{p}\left(t_f\right)$.

Figure 3

Continuous thermomajorization. The continuous thermomajorization relation $\mathit{p}{⪼}_{\mathit{\gamma }}\mathit{q}$ holds if and only if there is a continuous path of probability distributions $\mathit{r}\left(t\right)$ connecting $\mathit{p}$ and $\mathit{q}$ such that $\mathit{r}\left(t_1\right){\succ }_{\mathit{\gamma }}\mathit{r}\left(t_2\right)$ whenever $t_1\le t_2$.

Figure 4

Verification of continuous thermomajorization for $d=3$. Simplex representing the state space of all three-dimensional probability distributions with regions of fixed $\mathit{\gamma }$ orderings indicated by $\{·,·,·\}$. The optimal paths connecting a state $\mathit{p}\left(0\right)$ with $\mathit{\gamma }$ ordering $\{3,2,1\}$ to states ${\mathit{f}}_1, {\mathit{f}}_2$ of $\mathit{\gamma }$ ordering $\{1,2,3\}$ are realized by elementary thermalizations $T^{i,j}$ (indicated by red and blue arrows). The set of states with $\mathit{\gamma }$ ordering $\{1,2,3\}$ achievable from $\mathit{p}\left(0\right)$ by Markovian thermal processes is finally obtained as the union of the set of $\mathit{q}$ thermomajorized by ${\mathit{f}}_1$ and the set of $\mathit{q}$ thermomajorized by ${\mathit{f}}_2$ (for this last construction see, e.g., Appendix E of Ref. [50]). Note that the set of achievable states is nonconvex. Here the thermal state was chosen to be $\mathit{\gamma }=[1/3,1/3,1/3]$, which corresponds to the infinite temperature limit.

Figure 5

The defining points of the thermomajorization curve are labeled according to Lemma (12), to visualize the action of the two-level partial level thermalization $T^{3,4}\left(\lambda \right)$. This brings down $y_3$ until, at $\lambda =1$, the slopes of the third and fourth segment are equalized (brown segment connecting 2, $3^{\prime }$, and 4).