Resource theory under conditioned thermal operations

The thermal operations framework developed in past works is used to model the evolution of microscopic quantum systems in contact with thermal baths. Here we extend this model to bipartite devices with one part acting as a control external to the system-bath setup. We define the operations of such hybrid devices as conditioned thermal operations. We examine the resource under these operations, which we call conditional athermality. In the quasiclassical limit, we quantify this resource and find the conditions for its conversion between different forms.

Figure 1

In a hybrid classical-quantum thermal device (left), a classical circuit (associated with probability distribution ${\vec{p}}_{\mathrm{X}}$) determines the classical setting $x$ of the device, which in turn prepares the state ${\mathbf{u}}^{|x}$ on the quantum system S. The rest of the figure shows a schematic of a conditioned thermal operation (CTO): A classical measurement-based transformation $R$ takes ${\vec{p}}_{\mathrm{X}}$ to ${\vec{q}}_{\mathrm{Y}}$, simultaneously outputting a classical value $j$. A corresponding set of thermal operations $T^{\left(j\right)}$ determined by $j$ take S from conditional states ${\mathbf{u}}^{|x}$ to ${\mathbf{v}}^{|y}$.

Figure 2

The Lorenz curve construction for a subnormalized vector $\mathbf{w}$ on a system with Gibbs state $\mathbf{g}$: First index the energy eigenstates such that $w_1/g_1\ge w_2/g_2\ge \cdots \ge w_d/g_d$. Then connect the points $(0,0), (g_1,w_1), (g_1+g_2,w_1+w_2),\cdots , (1,1)$ with line segments to obtain the Lorenz curve $\mathcal{L}\left[\mathbf{w}\right]$. The curve defines a function $\mathcal{L}\left[\mathbf{w}\right]\left(s\right)$, which we call the Lorenz function of $\mathbf{w}$.

Figure 3

Thermomajorization: On the left, $\mathbf{u}$ thermomajorizes $\mathbf{v}$; on the right, neither state thermomajorizes the other.

Figure 4

If $H$ is a Hamiltonian whose Gibbs state $\mathbf{g}$ has mutually rational components, then for every $\mathbf{w}$ under $H$ we can find a higher-dimensional state $\tilde{\mathbf{w}}$ under a Hamiltonian $\tilde{H}$ whose Gibbs state is uniform, such that $\mathcal{L}[\tilde{\mathbf{w}},\tilde{H}]=\mathcal{L}[\mathbf{w},H]$.