Thermal processes and state achievability

The resource theory of thermal operations aims at describing possible transitions of microscale systems interacting with a macroscale environment under the fundamental assumption of energy conservation. For initial quantum states diagonal in the basis of the local Hamiltonian, these transitions are completely described by thermal processes (TPs), which form a convex set. In this paper, we give a complete characterization of the set of states that can be achieved through TPs, by describing the boundary of the allowed set of states using the so-called thermomajorization curves as a tool. We address the problem of achieving a certain transition through a convex combination of products of extremal TPs. We characterize all extremal TPs by associating them with transportation matrices. It becomes evident that there are extremal TPs that are not required in the implementation of any transition allowed by TPs. The statement holds for every dimension $d\ge 4$ of the state space. A property of the associated graphs, biplanarity, is identified as the distinguishing feature of these extremal TPs that are required for the arbitrary transition allowed by TPs.

Figure 1

Sets of diagonal states (blue polygons) and thermal processes (upper green polygon) for a system in a space of dimension $d$ (not directly manifested in the picture). For different initial states ${\rho }_1$ and ${\rho }_2$, different sets of states achievable by TPs can be obtained: ${\rho }_1^{\mathrm{TP}}$ and ${\rho }_2^{\mathrm{TP}}$, respectively (red polygons). Some extremal points of TPs, $T_2$ and $T_3$, map onto an extremal or nonextremal point of ${\rho }^{\mathrm{TP}}$, depending on the initial state $\rho$. Non-biplanar extremal TPs, like $T_4$, never map onto an extremal point of ${\rho }^{\mathrm{TP}}$, no matter which initial state $\rho$ from the blue polygon is selected. When convex combinations of products of TPs are allowed, these extremal TPs can be discarded with no harm to the attainable set of states (lower green polygon). Non-biplanar extremal TPs exist for all $d\ge 4$.

Figure 2

Thermomajorization diagram for a $d=3$ system, and certain $H_S$ and $\beta$ defining $q_{00}=1, q_{10}$, and $q_{20}$. According to Lemma 1, $\rho ,\sigma \in {\rho }_{\mathrm{init}}^{\mathrm{TP}}$, but $\zeta \notin {\rho }_{\mathrm{init}}^{\mathrm{TP}}$ and ${\rho }_{\mathrm{init}}\notin {\zeta }^{\mathrm{TP}}$. Elbows of curves are indicated by circles. Curves $\beta \left(\sigma \right)$ and $\beta \left(\rho \right)$ are thermomajorized by curve $\beta \left({\rho }_{\mathrm{init}}\right)$. Curve $\beta \left(\sigma \right)$ is tightly thermomajorized by curve $\beta \left({\rho }_{\mathrm{init}}\right)$. States ${\rho }_{\mathrm{init}}$ and $\zeta$ have $\beta$ order (2,3,1), while states $\sigma , \rho$ have $\beta$ order (1,2,3).

Figure 3

(a) First three steps of procedure in Theorem 1 determining three nonzero matrix elements of an extremal point of a transportation polytope set by vectors $\mathit{c}$ and $\mathit{r}$ such that $c_k>r_j, r_i>c_l, r_i-c_l>c_k-r_j$. Entries of the vectors $\mathit{r}$ and $\mathit{c}$ are listed along the corresponding rows and columns, and are updated with each step. (b) Arrows joining nonzero matrix entries fixed in the consecutive steps of procedure in Theorem 1. An arbitrary element with nondetermined value can be fixed in the next step. (c) Arrows joining nonzero matrix entries fixed in the consecutive steps of procedure in Definition 6. Only elements with nondetermined values that lie on the same row or column as the previously picked element can be fixed in the next step.

Figure 4

Forests with no isolated vertices on bipartite graphs for chosen extremal transportation matrices: (a) $\left(\begin{array}{ccc}\hfill 1& \hfill 3& \hfill 4\\ \hfill 2& \hfill 0& \hfill 0\\ \hfill 5& \hfill 0& \hfill 0\end{array}\right)$, with $\mathit{r}=(8,2,5), \mathit{c}=(8,3,4)$; (b) the same as in (a); (c) $\left(\begin{array}{ccc}\hfill 0& \hfill 0& \hfill 2\\ \hfill 0& \hfill 7& \hfill 0\\ \hfill 5& \hfill 0& \hfill 1\end{array}\right)$, with $\mathit{r}=(2,7,6), \mathit{c}=(5,7,3)$; (d) $\left(\begin{array}{cccc}\hfill 4& \hfill 0& \hfill 0& 0\\ \hfill 5& \hfill 3& \hfill 6& 7\\ \hfill 0& \hfill 0& \hfill 2& 0\\ \hfill 0& \hfill 0& \hfill 0& 1\end{array}\right)$, with $\mathit{r}=(4,21,2,1), \mathit{c}=(9,3,8,8)$; (e) $\left(\begin{array}{cccc}\hfill 1& \hfill 1& \hfill 0& 0\\ \hfill 1& \hfill 0& \hfill 1& 0\\ \hfill 1& \hfill 0& \hfill 0& 1\\ \hfill 1& \hfill 0& \hfill 0& 0\end{array}\right)$, with $\mathit{r}=(2,2,2,1), \mathit{c}=(4,1,1,1)$. Notice that the graph in (b) is the plane version of the biplanar graph in (a). Graphs in (d) and (e) cannot be driven into plane forms by isomorphisms that do not switch vertices between sides of the bipartite structure. The graph in (c) can be driven into such form, and, as opposed to the rest of the graphs, is a forest composed of two trees instead of one tree. Labels of the vertices on the left (right) side of the graphs correspond to columns (rows) of the respective extremal transportation matrix.

Figure 5

Geometry of the set of achievable states for $d=3$ and $\beta$ and $H_S$ such that $q_{10}+q_{20}\le 1$. (a) Set ${\rho }_{\mathrm{init}}^{\mathrm{TP}}$ (solid-lined red polygon) for some state ${\rho }_{\mathrm{init}}$ with thermomajorization curve with $\beta$ order (231). Regions of states with different $\beta$ orders are separated by dotted black lines. Vertices 1,2,3 correspond to states $\mathrm{diag}[1,0,0], \mathrm{diag}[0,1,0]$, and $\mathrm{diag}[0,0,1]$, respectively. (b) Thermomajorization curves of extremal points of ${\rho }_{\mathrm{init}}^{\mathrm{TP}}$. Shape and color of elbows within a curve correspond to shape of points in (a). Note that $\beta$ order of the state represented by a yellow square is not determined uniquely, so its rightmost elbow can be placed both on pentagon and diamond elbows. In particular, the yellow rectangle state is achieved by an extremal thermal process $\left(\begin{array}{ccc}\hfill 1-q_{10}-q_{20}& \hfill 1& \hfill 1\\ \hfill q_{10}& \hfill 0& \hfill 0\\ \hfill q_{20}& \hfill 0& \hfill 0\end{array}\right)$ on the diagonal elements of ${\rho }_{\mathrm{init}}$. One can check that this thermal process has a graph representation as in Fig. 4 and Fig. 4, and therefore has associated orders ${\pi }_{\mathrm{init}}=\left(231\right)$ and ${\pi }_{\mathrm{out}}=\left(123\right)$ or ${\pi }_{\mathrm{out}}=\left(132\right)$. Theorem 4 implies then that it is the only process mapping ${\rho }_{\mathrm{init}}$ into the yellow rectangle state. Green triangle, blue diamond, and purple pentagon extremal states can be obtained uniquely from ${\rho }_{\mathrm{init}}$ by the extremal TPs $\left(\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1-q_{21}& \hfill 1\\ \hfill 0& \hfill q_{21}& \hfill 0\end{array}\right)$, $\left(\begin{array}{ccc}\hfill 1-q_{10}& \hfill 1-q_{21}& \hfill 1\\ \hfill q_{10}& \hfill 0& \hfill 0\\ \hfill 0& \hfill q_{21}& \hfill 0\end{array}\right)$, $\left(\begin{array}{ccc}\hfill 1-q_{20}& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill q_{20}& \hfill 0& \hfill 0\end{array}\right)$, respectively.

Figure 6

Construction of an extremal thermal process from thermomajorization diagram.

Figure 7

All states $\rho \in {\rho }_{\mathrm{init}}^{\mathrm{TP}{\left(2\right)}_{\beta ,{\mathrm{H}}_{\mathrm{S}}}}$ can be represented as $\rho \left(\alpha \right)=[(1-\alpha )\mathcal{I}+\alpha \mathcal{E}]{\rho }_{\mathrm{init}}$, with value $\alpha \in [0,1]$ determining the position of the only elbow of thermomajorization curve $\beta (\rho \left(\alpha \right))$. The movement of the elbow associated with continuous increase of $\alpha$ is marked by arrows. For $\alpha =1/(1+q_{10})$ marking a transition to a Gibbs state ${\rho }_{\beta }$, the elbow disappears, to reemerge for higher $\alpha$ on a different vertical line.

Figure 8

Representing a state as a convex composition of extremal points (step 1). Decomposition of a state $\rho =c_{1,1}\rho \left({\gamma }_{1,1}\right)+c_{1,2}\rho \left({\gamma }_{1,2}\right)$, with last elbows of ${\gamma }_{1,1}$ and ${\gamma }_{1,2}$ lying on $\beta \left({\rho }_{\mathrm{init}}\right)$. (a) Validity of a construction comes from a decomposition of states of two-level systems (Fig. 7), trivially extended to states of higher dimension and equal occupations on the added levels. (b) Permuting the segments turns ${\gamma }_{1,1}^{\prime }$ and ${\gamma }_{1,2}^{\prime }$ into ${\gamma }_{1,1}$ and ${\gamma }_{1,2}$, respectively, and asserts that the curves are concave.

Figure 9

Representing a state as a convex decomposition of extremal points (step 2). Decomposition of (a) $\rho \left({\gamma }_{1,1}\right)$ and (b) $\rho \left({\gamma }_{1,2}\right)$ into states represented by thermomajorization curves ${\gamma }_{2,i}, i=1,2,3,4$, with last 2 elbows lying on $\beta \left({\rho }_{\mathrm{init}}\right)$.

Figure 10

A state $\rho$ belongs to a facet $H_S$ of ${\rho }_{\mathrm{init}}^{\mathrm{TP}}$ given by a set of points $S$. A point from this set defines a set of segments $\mathcal{X}=\{x_1,\cdots ,x_n\}$ that lie to the left of it on $\beta \left(\rho \right)$. If ${\rho }_1\notin H_S$, then $\beta \left({\rho }_1\right)$ has an elbow below $S$ (case [A]) or a segment below $S$ (case [B]). Then, the state ${\rho }_2$ from the decomposition $\rho =\lambda {\rho }_1+(1-\lambda ){\rho }_2, \lambda \in [0,1]$, has a curve that has to lie over $S$, both in the case [C] with all segments of $\beta \left({\rho }_2\right)$ to the left from $S$ being taken from the set $\mathcal{X}$, as well as for different arrangements (case [D]). Therefore, ${\rho }_2\notin {\rho }_{\mathrm{init}}^{\mathrm{TP}}$.