Resource theory of heat and work with non-commuting charges

TL;DR

The paper develops a resource theory of quantum thermodynamics for multiple conserved quantities, including non-commuting charges, in the asymptotic regime of many copies. It maps states to a phase diagram $(\underline{a},s)$ via average charges $\underline{a}$ and entropy $s$, and proves the Asymptotic Equivalence Theorem (AET) that asymptotically equivalent sequences are connected by almost-commuting unitaries and generalized Gibbs states $\tau(\underline{a})$. The second law is formulated in terms of a free-entropy bound $\widetilde{F}$, showing that extractable charges satisfy $\sum_j \beta_j W_j \le -\Delta\widetilde{F}_S$, with the bath size determining achievability; quantum effects such as negative conditional entropy $S(B|S)$ can enable transformations beyond the classical phase diagram. The analysis introduces extended phase diagrams to treat finite baths and derives bath-rate expressions tied to the bath’s heat capacity, revealing a fundamental tradeoff between bath resources and work extraction and providing a mechanism to store multiple charges in physically separated batteries.

Abstract

We consider a theory of quantum thermodynamics with multiple conserved quantities (or charges). To this end, we generalize the seminal results of Sparaciari et al. [PRA 96:052112, 2017] to the case of multiple, in general non-commuting charges, for which we formulate a resource theory of thermodynamics of asymptotically many non-interacting systems. To every state we associate the vector of its expected charge values and its entropy, forming the phase diagram of the system. Our fundamental result is the Asymptotic Equivalence Theorem (AET), which allows us to identify the equivalence classes of states under asymptotic approximately charge-conserving unitaries with the points of the phase diagram. Using the phase diagram of a system and its bath, we analyze the first and the second laws of thermodynamics. In particular, we show that to attain the second law, an asymptotically large bath is necessary. In the case that the bath is composed of several identical copies of the same elementary bath, we quantify exactly how large the bath has to be to permit a specified work transformation of a given system, in terms of the number of copies of the elementary bath systems per work system (bath rate). If the bath is relatively small, we show that the analysis requires an extended phase diagram exhibiting negative entropies. This corresponds to the purely quantum effect that at the end of the process, system and bath are entangled, thus permitting classically impossible transformations. For a large bath, system and bath may be left uncorrelated and we show that the optimal bath rate, as a function of how tightly the second law is attained, can be expressed in terms of the heat capacity of the bath. Our approach solves a problem from earlier investigations about how to store the different charges under optimal work extraction protocols in physically separate batteries.

Figures (5)

• Figure 1: Schematic of the phase diagrams $\mathcal{P}^{(1)}$, $\mathcal{P}^{(2)}$ and $\overline{\mathcal{P}}$. As seen, $\mathcal{P}^{(1)}$ is not convex, having a hollow on the underside.
• Figure 2: State change of the bath for a given work transformation under extraction of $j$-type work $W_j$, viewed in the phase diagram of the bath $\overline{{\cal P}}_B$. The blue line represents the tangent hyperplane at the corresponding point of the generalized thermal state $\tau(\underline{\beta})_B$, $R$ is the number of copies of the elementary baths in the proof of Theorem \ref{['asymptotic second law']}, and $F$ is the point corresponding to the final state of the bath.
• Figure 3: Schematic of the extended phase diagram $\overline{{\cal P}}_{|s_0}$. Depending on the value of $s_0$, whether it is smaller or larger than $\log|B|$, the diagram acquires either the left hand or the right hand one of the above shapes.
• Figure 4: State change of the bath for a given work transformation under the extraction of $j$-type work $W_j$, viewed in the extended phase diagram of the bath, which initially is in the thermal state $\tau(\underline{\beta})_B$, the blue line at the corresponding point in the diagram representing the tangent hyperplane of the diagram. The final states $\{\sigma_{S^nB^n}\}$ give rise to the point $F$ in the extended diagram, whose charge values are those of $\{\sigma_{B^n}\}$, while the entropy is $\frac{1}{n} S(B^n|S^n)_\sigma$.
• Figure 5: Graphical illustration of $R^*$, the minimum bath rate for a work transformation $\{\rho_{S^n}\} \rightarrow \{\sigma_{S^n}\}$ satisfying the second law, according to Theorem \ref{['thm:optimal-rate']}. The initial state is the generalized thermal state $\tau(\underline{\beta})$, its corresponding point marked on the upper boundary of the phase diagram. The final bath states correspond to points on the line denoted $f$, and they are feasible if and only they fall into the phase diagram. Consequently, $F^*$ is the point corresponding to the minimum rate.

Theorems & Definitions (42)

• Lemma 1
• proof
• Definition 2
• Definition 3
• Theorem 4: Asymptotic (approximate) Equivalence Theorem -- AET
• Definition 5
• Theorem 6: First Law
• proof
• Remark 7
• Remark 8