Geometric structure of thermal cones

The second law of thermodynamics imposes a fundamental asymmetry in the flow of events. The so-called thermodynamic arrow of time introduces an ordering that divides the system's state space into past, future, and incomparable regions. In this work, we analyze the structure of the resulting thermal cones, i.e., sets of states that a given state can thermodynamically evolve to (the future thermal cone) or evolve from (the past thermal cone). Specifically, for a $d$-dimensional classical state of a system interacting with a heat bath, we find explicit construction of the past thermal cone and the incomparable region. Moreover, we provide a detailed analysis of their behavior based on thermodynamic monotones given by the volumes of thermal cones. Results obtained apply also to other majorization-based resource theories (such as that of entanglement and coherence), since the partial ordering describing allowed state transformations is then the opposite of the thermodynamic order in the infinite temperature limit. Finally, we also generalize the construction of thermal cones to account for probabilistic transformations.

Figure 1

Majorization curve. For three different states $\mathit{p}, \mathit{q}$, and $\mathit{c}$, we plot their majorization curves $f_{\mathit{p}}\left(x\right),f_{\mathit{q}}\left(x\right)$ and $f_{\mathit{c}}\left(x\right)$, respectively. While $\mathit{p}$ majorizes $\mathit{q}$ [since $f_{\mathit{p}}\left(x\right)$ is never below $f_{\mathit{q}}\left(x\right)$], both states are incomparable with $\mathit{c}$, as their majorization curves cross with $f_{\mathit{c}}\left(x\right)$.

Figure 2

Thermomajorization curve. For three different states $\mathit{p}, \mathit{q}$, and $\mathit{c}$, with a thermal Gibbs state $\mathit{\gamma }=(1,e^{-\beta },e^{-2\beta })$, and $\beta >0$, we plot their thermomajorization curves $f_{\mathit{p}}^{\beta }\left(x\right),f_{\mathit{q}}^{\beta }\left(x\right)$ and $f_{\mathit{c}}^{\beta }\left(x\right)$, respectively. While $\mathit{p}$ thermomajorizes $\mathit{q}$ (since $f_{\mathit{p}}^{\beta }\left(x\right)$ is never below $f_{\mathit{q}}^{\beta }\left(x\right)$), both states are incomparable with $\mathit{c}$, as their thermomajorization curves cross with $f_{\mathit{c}}^{\beta }\left(x\right)$.

Figure 3

Majorization cones and Weyl chambers. (a) Probability simplex ${\mathrm{\Delta }}_3$ and a state $\mathit{p}=(0.6,0.3,0.1)$ represented by a black dot $•$ together with its majorization cones. The division of ${\mathrm{\Delta }}_3$ into different Weyl chambers is indicated by dashed lines with the central state $\mathit{\eta }=(1/3,1/3,1/3)$ denoted by a black star $★$. (b) The past cone of a state $\mathit{p}$ restricted to a given Weyl chamber is convex with the extreme points given by ${\mathit{t}}^{\left(n\right)}$ from Eq. (15) and the sharp state. (c) The causal structure induced by bistochastic matrices (i.e., thermal operations in the infinite temperature limit) in a given Weyl chamber.

Figure 4

Partial-order diagrams for majorization. Graphical representation of the partially ordered set formed by the extreme points of the past cone in the probability simplex ${\mathrm{\Delta }}_d$. Each arrow indicates that one of the elements precedes the other in the majorization ordering. (a) Diagram for $d=3$. (b) Diagram for $d=4$. Note that any convex combination between ${\mathit{t}}_{\text{proj}}^{\left(2\right)}$ and ${\mathit{t}}^{\left(3\right)}$, here denoted by ${\mathit{t}}_{\lambda }^{(2,3)}$, results in an incomparable vector with respect to ${\mathit{t}}_{\text{proj}}^{\left(1\right)}$ and ${\mathit{t}}^{\left(4\right)}$.

Figure 5

Entanglement cone in the simplex of the Schmidt coefficients of a $3\times 3$ system. Conversely to thermodynamics, the past of entanglement transformations is the thermodynamic future and viceversa. The black dot $•$ indicates the Schmidt vector of the initial state $\mathit{p}=(0.7,0.2,0.1)$, whereas the black star $★$ represents the maximally entangled state $\mathit{\eta }=(1/3,1/3,1/3)$.

Figure 6

Probabilistic majorization cones for $d=3$. For a three-level system with a state given by $\mathit{p}=(0.7,0.2,0.1)$ represented by a black dot $•$ and a maximally entangled state $\mathit{\eta }=(1/3,1/3,1/3)$ represented by a black star $★$, we plot its probabilistic majorization cone for probabilities of transformation $\mathcal{P}$ decreasing from 1 to 0.5 with 0.125 steps (a–e), respectively. Observe that, for $\mathcal{P}=1$, we recover the structure of the standard majorization cones, while as $\mathcal{P}$ decreases the interconvertible region ${\mathcal{T}}_{\leftrightarrow }(\mathit{p},\mathcal{P})$ expands and the incomparable region ${\mathcal{T}}_{\varnothing }(\mathit{p},\mathcal{P})$ shrinks, disappearing altogether between panels (d) and (e).

Figure 7

Embedded majorization curve. Embedding $\mathfrak{M}:{\mathrm{\Delta }}_d\to {\mathrm{\Delta }}_{2^d-1}$ of $d$-dimensional probability distribution $\mathit{p}$ (here $d=3$) into a $2^d-1=7$-dimensional space is most easily understood by noting that each thermomajorization curve (left) has elbows corresponding to a subset of ${\mathit{\Gamma }}^{\mathfrak{M}}$ on the horizontal axis. The embedding includes all entries of ${\mathit{\Gamma }}^{\mathfrak{M}}$ by subdividing the Lorenz curve into $2^d-1$ fragments. Conversely, the projection ${\mathfrak{P}}_{{\mathit{\pi }}_{\mathit{p}}}$ corresponds to selecting only a subset of of elbows that correspond to a selected order, in this case the original order ${\mathit{\pi }}_{\mathit{p}}$. In the $x$ axes we used the shorthand notation ${\gamma }_{ij}={\gamma }_i+{\gamma }_j$.

Figure 8

Thermal cones for $d=3$. For a three-level system with population given by $\mathit{p}=(0.4,0.36,0.24)$, represented by a black dot $•$, and energy spectrum $E_1=0,E_2=1$ and $E_3=2$, we plot its thermal cone for (a) $\beta =0$, (b) $\beta =0.5$, (c) $\beta =1.0$, and (d) $\beta \to \infty$. By increasing $\beta$, the thermal state (black star $★$) tends toward the ground state $E_1$, and the past thermal cone becomes convex.

Figure 9

Thermal cones for $d=4$. For a four-level system with population given by $\mathit{p}=(25,13,7,3)/48$, represented by a black dot $•$, and energy spectrum $E_1=0,E_2=1,E_3=2$, and $E_4=3$, we plot its thermal cone for (a) $\beta =0$, (b) $\beta =0.2$, and (c) $\beta =1.0$. The thermal state depicted by a black star $★$, and its trajectory (dashed line) is also shown. By increasing the inverse temperature $\beta$, the geometry of each region drastically changes.

Figure 10

Extreme points of ${\mathcal{T}}_{-}^{\beta }\left(\mathit{p}\right)$. The past thermal cone for a three-level system with population give by $\mathit{p}=(0.7,0.2,0.1)$, energy spectrum $E_1=0,E_2=2$, and $E_3=3$, and the inverse temperature $\beta =0.5$. At each chamber $\mathit{\pi }$, the nontrivial extreme points of the past are given by ${\mathit{t}}^{(n,\mathit{\pi })}$ (for $n\in \{1,2,3\}$ and $\mathit{\pi }\in {\mathcal{S}}_d$) and by extreme points of ${\bigcup }_{i=1}^{d-1}conv[({\mathcal{T}}_{+}^{\beta }\left({\mathit{t}}^{(i,\mathit{\pi })}\right)\cup {\mathcal{T}}_{+}^{\beta }\left({\mathit{t}}^{(i+1,\mathit{\pi })}\right)]$ (red dots). The remaining ones are sharp states, points in the boundary between chambers (green dots) and those which are on the edge of the probability simplex, i.e., ${\left({\mathit{\gamma }}_{i,j}\right)}_k=({\gamma }_i{\delta }_{ik}+{\gamma }_j{\delta }_{ik})/({\gamma }_i+{\gamma }_j)$, for $i\ne j$, and $k\in \{1,2,3\}$.

Figure 11

Volume of the thermal cones as a function of $\beta$. For all permutations of the state $\mathit{p}=(0.52,0.12,0.36)$, we plot the volume of the future thermal cone ${\mathcal{V}}_{+}^{\beta }$ (top), the incomparable thermal region ${\mathcal{V}}_{\varnothing }^{\beta }$ (center) and the past thermal cone ${\mathcal{V}}_{-}^{\beta }$ (bottom). Each color corresponds to a permutation associated with a given chamber of the probability simplex. Among all states, two are distinct: the maximally active (red curve) ${\mathit{p}}_{\text{max}}=(0.12,0.36,0.52)$ and the passive (black curve) ${\mathit{p}}_p=(0.52,0.36,0.12)$ states. Any other permutation of the initial state characterizes a different active state. The kinks in ${\mathcal{V}}_{+}^{\beta }$ match with the inverse temperatures at which $\mathit{p}$ changes its $\beta$-ordering (vertical lines of matching colors). The three different simplices at the bottom show how the Weyl chambers change with $\beta$.

Figure 12

Isovolumetric curves. The volume of the future ${\mathcal{V}}_{+}^{\beta }$, incomparable ${\mathcal{V}}_{\varnothing }^{\beta }$, and past thermal regions ${\mathcal{V}}_{-}^{\beta }$, in the space of three-dimensional probability distributions for an equidistant energy spectrum $E_1=0, E_2=1$ and $E_3=2$ and inverse temperature (a) $\beta =0$ and (b) $\beta =0.5$. The thermal state is depicted by a grey star $★$.

Figure 13

Isovolumetric sets for entanglement cones. To calculate the isovolumetric lines for (a) the past cone, (b) the incomparable region, and (c) the future cone for entanglement resource theory of states in ${\mathcal{H}}^{3\times 3}$ one considers the density of the Schmidt coefficients $\mathit{p}$ induced by the Haar measure in the space ${\mathcal{H}}_9$ of pure bipartite states, as depicted in panel (d) by showing a sample of $5\times {10}^4$ points in ${\mathrm{\Delta }}_3$. Observe the scarcity of points in the central region, resulting in characteristic concentration of states with large future in the center of the simplex and large past in the vicinity of the vertices.

Figure 14

Coherent thermal cones. For a two-level system with initial state $\rho$, represented by a black dot $•$, and thermal state represented by a black star $★$ with Bloch vectors ${\mathit{r}}_{\rho }=(0.2,0,0.5)$ and ${\mathit{r}}_{\gamma }=(0,0,1/3)$, respectively, we depict in the real cross-section of the Bloch ball, the coherent thermal cone (a) under Gibbs-preserving operations (b) under thermal operations.

Figure 15

Lorenz curves of the tangent vectors ${\mathit{t}}^{\left(n\right)}$. Majorization curves of the state $\mathit{p}=(0.43,0.37,0.18,0.02)$ (black) and (a) ${\mathit{t}}^{\left(1\right)}$ (b) ${\mathit{t}}^{\left(2\right)}$ (c) ${\mathit{t}}^{\left(3\right)}$ (d) ${\mathit{t}}^{\left(4\right)}$ and (e) all tangent vectors $j\in \{1,2,3,4\}$, respectively.

Figure 16

Isovolumetric sets for entanglement $3\times M$ bipartite systems. The density $P_{3,M}\left(\mathrm{\Lambda }\right)$ of Schmidt coefficients of pure states for qutrit-quMit systems depends heavily on the dimension $M$ of the second system. Panels (a–d) and (e–h) present the isovolumetric lines for past, incomparable, and future regions and the density of the states for $M=6$ and $M=30$, respectively. Note that for larger $M$ the density $P_{3,M}$ is more and more concentrated around the regions close to the center [compare panels (d) and (h)]. This affects the subset of states with large future volume, making it smaller [panels (c) and (g)] as well as the set of states with large past volume, enlarging it [panel (a) with panel (e)].