Structure of the thermodynamic arrow of time in classical and quantum theories

In this work we analyze the structure of the thermodynamic arrow of time, defined by transformations that leave the thermal equilibrium state unchanged, in classical (incoherent) and quantum (coherent) regimes. We note that in the infinite-temperature limit, the thermodynamic ordering of states in both regimes exhibits a lattice structure. This means that when energy does not matter and the only thermodynamic resource is given by information, the thermodynamic arrow of time has a very specific structure. Namely, for any two states at present there exists a unique state in the past consistent with them and with all possible joint pasts. Similarly, there also exists a unique state in the future consistent with those states and with all possible joint futures. We also show that the lattice structure in the classical regime is broken at finite temperatures, i.e., when energy is a relevant thermodynamic resource. Surprisingly, however, we prove that in the simplest quantum scenario of a two-dimensional system, this structure is preserved at finite temperatures. We provide the physical interpretation of these results by introducing and analyzing the history erasure process, and point out that quantum coherence may be a necessary resource for the existence of an optimal erasure process.

Figure 1

Thermal cones. The reachability of one state from another via a GP map introduces the ordering of states along the thermodynamic arrow of time. States that can be reached from a given state $\rho$ form its future thermal cone ${\mathcal{T}}_{+}\left(\rho \right)$, whereas states that can be transformed into $\rho$ form its past thermal cone ${\mathcal{T}}_{-}\left(\rho \right)$.

Figure 2

Decomposition of preorder into partial order between equivalence classes. (a) States on the unitary orbit generated by $U\left(t\right)=e^{-iHt}$ are all reversibly interconvertible under GP transformations, as $U\left(t\right)$ is GP. Hence, if a state $\rho$ can thermodynamically evolve to $\sigma$, then any state $U\left(t\right)\rho U^{†}\left(t\right)$ can evolve to any state $U\left(t^{\prime }\right)\sigma U^{†}\left(t^{\prime }\right)$. Thus, without loss of generality one can only study the ordering between single representatives of each orbit. (b) States that are reversibly interconvertible by GP operations can be thought of as living in a plane “perpendicular” to the thermodynamic arrow of time, i.e., members of a thermodynamic equivalence class are at the same stage of the evolution towards equilibrium. The thermodynamic ordering of states is then given by the partial order between planes.

Figure 3

Visualizing the join and meet. (a) The intersection of past thermal cones of $\rho$ and $\sigma$, denoted by ${\mathcal{T}}_{-}(\rho ,\sigma )$, is a set of states that can thermodynamically evolve to both $\rho$ and $\sigma$. The join $\rho \vee \sigma$ is the unique state belonging to ${\mathcal{T}}_{-}(\rho ,\sigma )$ that can be thermodynamically reached from all states in ${\mathcal{T}}_{-}(\rho ,\sigma )$. (b) The intersection of future thermal cones of $\rho$ and $\sigma$, denoted by ${\mathcal{T}}_{+}(\rho ,\sigma )$, is a set of states that can be thermodynamically reached from both $\rho$ and $\sigma$. The meet $\rho \wedge \sigma$ is the unique state belonging to ${\mathcal{T}}_{+}(\rho ,\sigma )$ that can thermodynamically evolve to all states in ${\mathcal{T}}_{+}(\rho ,\sigma )$.

Figure 4

Examples of lattice and nonlattice partial orders. Partially ordered sets can be represented by their Hasse diagrams: an arrow from $a$ to $b$ denotes $a\succ b$. (a) The power set of any set $A$ forms a lattice under the partial order induced by subset inclusion. The join and meet are given by set union and intersection, respectively. Here, we choose a three-element set $A=\{x,y,z\}$. (b) Natural numbers partially ordered by divisibility, i.e., $a\succ b$ if and only if $b$ is a divisor of $a$, form a lattice. The join and meet are given by the operations of taking the least common multiple and greatest common divisor, respectively. Here, we present a sublattice of divisors of 30. (c) A set $\{a,b,c,d,e,f\}$ with partial order defined by the presented Hasse diagram does not form a lattice. Although $b$ and $c$ have common upper bounds $d,e$, and $f$, neither of them is a join, i.e., the least upper bound.

Figure 5

Meet of a majorization lattice. Majorization curves $f_{\mathit{p}}$ and $f_{\mathit{q}}$ (solid lines) for $\mathit{p}=(0.6,0.15,0.15,0.1)$ and $\mathit{q}=(0.5,0.25,0.2,0.05)$, together with their meet $\mathit{p}\wedge \mathit{q}=(0.5,0.25,0.15,0.1)$ (given by the line consisting of triple-crossed segments).

Figure 6

Join of a majorization lattice. Majorization curves $f_{\mathit{p}}$ and $f_{\mathit{q}}$ (solid lines) for $\mathit{p}=(0.6,0.15,0.15,0.1)$ and $\mathit{q}=(0.5,0.25,0.2,0.05)$, together with the curve corresponding to ${\mathit{g}}^{\left(0\right)}=(0.6,0.15,0.2,0.05)$ from Eq. (5) (given by the line consisting of triple-crossed segments). This curve is not concave since ${\mathit{g}}^{\left(0\right)}$ is not arranged in a nonincreasing order. Hence, in order to obtain the join $\mathit{p}\vee \mathit{q}$, one needs to “smooth” the curve between the points $i=1$ and 3 (dashed black line). This leads to the join given by $\mathit{p}\vee \mathit{q}=(0.6,0.175,0.175,0.05)$.

Figure 7

Thermomajorization order is not a lattice. Thermomajorization curves $f_{\mathit{p}}$ and $f_{\mathit{q}}$ (solid lines), together with the candidates for the join $\mathit{p}\vee \mathit{q}$, i.e., optimal curves (plotted with dashed lines) thermomajorizing both $\mathit{p}$ and $\mathit{q}$. Satisfying the inequalities given by Eq. (13) guarantees $f_{\mathit{q}}>f_{\mathit{p}}$ at ${\gamma }_d$; $f_{\mathit{q}}<f_{\mathit{p}}$ at ${\gamma }_{d-1}$; point $A$ lies below 1, which results in the existence of two incomparable candidates for $\mathit{p}\vee \mathit{q}$.

Figure 8

Future thermal cones for qubits (GP operations). A general qubit state $\rho$ and a thermal state $\gamma$ with ${\mathit{r}}_{\gamma }=(0,0,0.5)$ presented in the Bloch sphere. The disk $D_1\left(\rho \right)$ corresponds to a set of states $\left\{\sigma \right\}$ with $R_{-}\left(\sigma \right)\le R_{-}\left(\rho \right)$, whereas the disk $D_2\left(\rho \right)$ corresponds to a set $\left\{\sigma \right\}$ with $R_{+}\left(\sigma \right)\le R_{+}\left(\rho \right)$. The equalities are obtained at the edges of the disks, i.e., on circles $C_1\left(\rho \right)$ and $C_2\left(\rho \right)$. The set of states $\rho$ can be mapped to via GP quantum channels is given by the intersection $D_1\left(\rho \right)\cap D_2\left(\rho \right)$ (which can also be freely revolved around the $z$ axis). (a) A mixed state with ${\mathit{r}}_{\rho }=(0.4,0,0.6)$. (b) A pure state with ${\mathit{r}}_{\rho }=(0.6,0,0.8)$.

Figure 9

Thermodynamic lattice for qubits. A thermal state $\gamma$ and two states $\rho$ and ${\rho }^{\prime }$ presented in the Bloch sphere. (a) States described by ${\mathit{r}}_{\rho }=(0.4,0,0.6),{\mathit{r}}_{{\rho }^{\prime }}=(0.3,0,0.2),{\mathit{r}}_{\gamma }=(0,0,0.5)$. The join $\mathrm{\Omega }$ of $\rho$ and ${\rho }^{\prime }$ lies at the intersection of $C_1\left({\rho }^{\prime }\right)$ and $C_2\left(\rho \right)$, whereas their meet $\omega$ lies at the intersection of $C_1\left(\rho \right)$ and $C_2\left({\rho }^{\prime }\right)$. (b) States described by ${\mathit{r}}_{\rho }=(0,0,-0.8),{\mathit{r}}_{{\rho }^{\prime }}=(0.4,0,0.4),{\mathit{r}}_{\gamma }=(0,0,0.2)$. The join $\mathrm{\Omega }$ of $\rho$ and ${\rho }^{\prime }$ lies at the intersection of $C_1\left(\rho \right)$ and $C_2\left(\rho \right)$ (which coincides with $\rho )$, whereas their meet $\omega$ lies at the intersection of $C_1\left({\rho }^{\prime }\right)$ and $C_2\left({\rho }^{\prime }\right)$ (which coincides with ${\rho }^{\prime })$.

Figure 10

Possible orderings of eigenvalues. The orderings 1–4 of eigenvalues $\{{\lambda }_1,{\lambda }_1^{\prime },{\lambda }_2,{\lambda }_2^{\prime }\}$, together with labels I–V corresponding to different regions of the $[0,1]$ interval.