Resource theory for work and heat

Several recent results on thermodynamics have been obtained using the tools of quantum information theory and resource theories. So far, the resource theories utilized to describe thermodynamics have assumed the existence of an infinite thermal reservoir, by declaring that thermal states at some background temperature come for free. Here, we propose a resource theory of quantum thermodynamics without a background temperature, so that no states at all come for free. We apply this resource theory to the case of many noninteracting systems and show that all quantum states are classified by their entropy and average energy, even arbitrarily far away from equilibrium. This implies that thermodynamics takes place in a two-dimensional convex set that we call the energy-entropy diagram. The answers to many resource-theoretic questions about thermodynamics can be read off from this diagram, such as the efficiency of a heat engine consisting of finite reservoirs, or the rate of conversion between two states. This allows us to consider a resource theory which puts work and heat on an equal footing, and serves as a model for other resource theories.

Figure 1

The energy-entropy diagram representing the state space of a quantum system with Hamiltonian $H$, degenerate in the ground state (vertical line). The physical points are inside the gray area. Each point $\left(E,S\right)$ represents an equivalence class of microstates, i.e., a single macrostate. Inequality (7), postulating the non-negativity of entropy, is satisfied by the points above the $E$ axis. For a given $\beta$, Ineq. (8) is satisfied by those points below the drawn line which is tangent to the physical region and goes through the point $(E\left({\tau }_{\beta }\right),S\left({\tau }_{\beta }\right))$ with slope $\beta$.

Figure 2

Combining two systems in normalized macrostates $x_1$ and $x_2$ results in a total system in the normalized macrostate $x$, which is a convex combination of $x_1$ and $x_2$, where the coefficient of $x_1$—proportional to the distance between $x$ and $x_2$—is equal to the size of the first system relative to the total system, and similarly for $x_2$.

Figure 3

Decomposing a normalized macrostate into a combination of the thermal macrostate at temperature ${\beta }^{-1}$ and a pure macrostate, together with the set of all states that have such a decomposition at the given $\beta$ (hatched).

Figure 4

The visualization of the effective inverse temperature of the reservoir, ${\beta }_{\text{eff}}$. The thermal reservoir initially consists of $m$ copies of ${\tau }_{{\beta }_1}$, which are each turned into ${\tau }_{{\beta }_2}$ by the state transformation. The value of ${\beta }_{\text{eff}}$ is given by the slope of the line connecting the two corresponding points in the energy-entropy diagram. When ${\beta }_2={\beta }_1+\varepsilon$, for $\left|\varepsilon \right|\to 0$, the two points get closer, and the line approaches the tangent to the curve of thermal states. In this case, ${\beta }_{\text{eff}}={\beta }_1+O\left(\varepsilon \right)$, by Remark t3.

Figure 5

The two effective temperatures that determine engine efficiency interpreted as slopes in the energy-entropy diagram. For ${\beta }_{\text{less-cold}}$ very close to ${\beta }_{\text{cold}}$ and ${\beta }_{\text{less-hot}}$ very close to ${\beta }_{\text{hot}}$, i.e., when the reservoirs are very large compared to the battery size, then the lines approximate tangents and the resulting efficiency approaches the Carnot efficiency.