Steepest entropy ascent solution for a continuous-time quantum walker

We consider the steepest entropy ascent (SEA) ansatz to describe the nonlinear thermodynamic evolution of a quantum system. Recently this principle has been dubbed the fourth law of thermodynamics [Beretta, Phil. Trans. R. Soc. A 378, 20190168 (2020)]. A unique global equilibrium state exists in this context, and any other state is driven by the maximum entropy generation principle towards this equilibrium. We study the SEA evolution of a continuous-time quantum walker (CTQW) on a cycle graph with $N$ nodes. SEA solutions are difficult to find analytically. We provide an approximate scheme to find a general single-particle evolution equation governed by the SEA principle, whose solution produces dissipation dynamics. We call this scheme the fixed Lagrange's multiplier (FLM) method. In the Bloch sphere representation, we find trajectories traced out by the Bloch vector within the sphere itself. We have discussed these trajectories under various initial conditions for the case of a qubit. A similar dissipative motion is also observed in the case of CTQW, where probability amplitudes have been used to characterize decoherence. Our FLM scheme shows good agreement with numerical results. As we report, in CTQW, a strong delocalization exists for low system relaxation time.

Figure 1

A Bloch sphere representation of a qubit. The purple arrow denotes the vector $\vec{r}$, and the green one $\widehat{h}$. The isoentropic concentric surfaces and the constant energy planes are labeled in the diagram.

Figure 2

Plot of the radial distance of the state operator in the Bloch sphere over time ($t=50$) for a qubit under various $\tau$ (first column of the legend), with $\omega =5$, and $\varepsilon =0.999$. As discussed in the paper, the higher $\tau$ states fall slowly compared to lower valued ones. Solid lines represent computation done using FLM, dot-dash lines GPB are plotted using Eq. (23) [43], and the dotted NUM lines are plotted using direct numerical simulation of Eq. (8).

Figure 3

(a) The spiraling trajectory of the state operator in the equatorial plane of the Bloch sphere, as viewed from its north pole after $t=30$. Different $\tau$ values show the difference between each trajectory. (b) Evolution of states after $t=30$, on the surface of revolution generated from the entropy functional expression. The legend identifies the $\tau$ values considered for this plot. The case with $\tau =400$ corresponds to the case with $\varepsilon =0.5$, while the rest of the cases have $\varepsilon =0.999$. Lower $\tau$ states rise faster, and the lowest has the steepest ascent.

Figure 4

A schematic depiction of a qubit state's general energy dissipative evolution (purple spiral in the online colored version). This schematic is for high $\tau$ values; for low values of the same, the trajectory will follow a straighter path, as indicated in Ref. [58], which will also be the steepest ascent.

Figure 5

Probability distribution for a single quantum walker on a ring of 100 nodes. The walker was initiated at node 50, and the probability distribution is after $t=10$. In this diagram, node 50 is shifted to node 0.

Figure 6

Plot of ${\beta }_i$ vs time. In the legend, ${\beta }_I$ is tagged, as it varies for different $N$ and $\varepsilon$ values. ${\beta }_H$ is the constant line $y=0$.

Figure 7

Probability distribution for a single quantum walker on a ring of 100 nodes under SEA conditions, initiated at node 50 (shifted to node 0 for symmetry), after $t=20$. The relaxation time $\tau$ is $0.2,50.11$, and 100.02 (first column in the legend), respectively. $\varepsilon$ is 0.99. The plot contains both results from plotting the analytic solution found using the FLM method (solid lines in the plot) and those (dotted lines in the plot above) from the numerical solution to Eq. (8) using a CTQW Hamiltonian and other relevant substitutions.

Figure 8

${\mathrm{\Pi }}_S$ distribution for a CTQW of $N=50$ over different $\tau$ and $\varepsilon$ values after time $t=1$ (a) and $t=3$ (b). The color bars provide the range and contrast of ${\mathrm{\Pi }}_S$ values. As discussed in the main text, the high ${\mathrm{\Pi }}_S$-valued zones are concentrated around high $\varepsilon$ and low $\tau$ values. These deep purple areas bounded in cyan represent the maximum entropy generation area. Insets of panels (a) and (b): zoomed-in view of the bounded region displaying max ${\mathrm{\Pi }}_S$.

Figure 9

(a) Plot of average energy vs time and (b) entropy vs time for a CTQW on a cycle graph of 100 nodes for various $\tau$ (first column of the legend) values and $\varepsilon =0.99$. The walk was performed up to $t=100$. FLM (solid lines) denotes the analytically computed results, while NUM (dotted lines) denotes numerical results.

Figure 10

Plot of rate of change of entropy vs time for a CTQW on a cycle graph of 100 nodes for (a) $\tau =0.2$ and (b) $\tau =(50.11,100.02)$ with $\varepsilon =0.99$. FLM (solid lines) denotes the analytically computed results, while NUM (dotted lines) denotes numerical results.

Figure 11

Geometric interpretation of the SEA principle. The manifold $\mathcal{L}$ is spanned by the set $\left\{\right|{\mathrm{\Psi }}_i\left)\right\}$, which is shown as encoded on a surface, whose spanning vectors are shown along the edges. $\left|\mathrm{\Phi }\right)$ is the entropy gradient functional whose projection on the plane perpendicular to $\mathcal{L}$ provides the direction of motion under SEA (purple arrow), while the magnitude of ${\mathrm{\Pi }}_{\gamma }$ is constrained by a suitable value set to Eq. (A2), $\frac{dl}{dt}=\dot{\epsilon }$, which defines a circle in the orthogonal plane, shown dotted in the diagram [52].