Partial order on passive states and Hoffman majorization in quantum thermodynamics

Passive states, i.e., those states from which no work can be extracted via unitary operations, play an important role in the foundations and applications of quantum thermodynamics. They generalize the familiar Gibbs thermal states, which are the sole passive states being stable under tensor product. Here, we introduce a partial order on the set of passive states that captures the idea of a passive state being virtually cooler than another one. This partial order, which we build by defining the notion of relative passivity, offers a fine-grained comparison between passive states based on virtual temperatures (just like thermal states are compared based on their temperatures). We then characterize the quantum operations that are closed on the set of virtually cooler states with respect to some fixed input and output passive states. Viewing the activity, i.e., nonpassivity, of a state as a resource, our main result is then a necessary and sufficient condition on the transformation of a class of pure active states under these relative passivity-preserving operations. This condition gives a quantum thermodynamical meaning to the majorization relation on the set of nonincreasing vectors due to Hoffman. The maximum extractable work under relative passivity-preserving operations is then shown to be equal to the ergotropy of these pure active states. Finally, we are able to fully characterize passivity-preserving operations in the simpler case of qubit systems, and hence to derive a state interconversion condition under passivity-preserving qubit operations. The prospect of this work is a general resource-theoretical framework for the extractable work via quantum operations going beyond thermal operations.

Figure 1

Schematic of the notion of virtually cooler passive states. In the schematic, ${\left\{E_i\right\}}_{i=0}^{d-1}$ is the set of energy eigenvalues of the Hamiltonian of the system. $N_i$ and ${\tilde{N}}_i$ are populations in energy eigenstates $E_i$ corresponding to the passive states $\rho \left(\vec{r}\right)$ and $\sigma \left(\vec{p}\right)$, respectively. Now, if $\frac{r_i}{r_{i+1}}:=\frac{N_i}{N_{i+1}}\ge \frac{{\tilde{N}}_i}{{\tilde{N}}_{i+1}}:=\frac{p_i}{p_{i+1}}$ for all $i=0,\cdots ,d-2$, we say that $\rho \left(\vec{r}\right)$ is a virtually cooler state than $\sigma \left(\vec{p}\right)$, i.e., $\rho \left(\vec{r}\right){\succ }_{vc}\sigma \left(\vec{p}\right)$.

Figure 2

A schematic for RPPOs. In the schematic, $\mathcal{P}\left(S\right)$ is the set of all passive states. $\sigma \left(\vec{p}\right)$ and $\sigma \left(\vec{q}\right)$ are two fixed passive states. $T_p$ is the set of all passive states that are virtually cooler than $\sigma \left(\vec{p}\right)$ and $T_q$ is the set of all passive states that are virtually cooler than $\sigma \left(\vec{q}\right)$. That is, $T_p:=\{\rho \in \mathcal{P}\left(S\right):\rho {\succ }_{vc}\sigma \left(\vec{p}\right)\}$ and $T_q:=\{\rho \in \mathcal{P}\left(S\right):\rho {\succ }_{vc}\sigma \left(\vec{q}\right)\}$. Then ${\mathfrak{L}}_{p,q}$ denotes RPPOs and is the set of all quantum channels from the set $T_p$ into the set $T_q$.

Figure 3

In above schematic diagram, RPPO is the set of relative passivity preserving operations for two fixed passive state $\sigma \left(\vec{p}\right)$ and $\sigma \left(\vec{q}\right)$, PPO is the set of passivity preserving operations, and IO is the set of incoherent operations. If we choose $\sigma \left(\vec{p}\right)$ and $\sigma \left(\vec{q}\right)$ both to be equal to the maximally mixed state, then RPPO is equal to PPO.