Reliability Dynamics in a Two-Site Dissipative Quantum Spin Chain

Abstract

As a key index for applications of a device, the device's reliability is its ability to survive (function normally over time) under the influence of some environment. In this paper we present a quantum energy-storing device model with a quantum spin chain, whose environment influence is described by the Lindblad master equation. Here the device survives if the spin system stays in the state with nonzero excitations; otherwise, it fails. Because the Lindblad dynamics enforces one-way energy decay and strict irreversibility of the failure state, we can investigate the reliability of the quantum device directly using classical reliability theory. Focusing on the minimal nontrivial case -- a two-site spin-1/2 chain -- we derive closed-form expressions for the reliability and the hazard rate. The dynamics exhibit an overdamped-underdamped crossover controlled by the competition between coherent exchange and dissipation inhomogeneity. The exact analytical formulas are in excellent agreement with numerical simulations. More importantly, we establish an experimentally accessible protocol for assessing reliability based on first-passage time statistics.

Figures (6)

• Figure 1: Extremum-count maps of the overdamped hazard in the $(\gamma_1,\gamma_2)$ plane for (a) $J=0.1$, (b) $J=0.3$, and (c) $J=0.5$. Only the overdamped region $|\Delta\gamma|>4J$ is classified; a narrow band around the crossover line $|\Delta\gamma|=4J$ is left blank. Two sectors appear: a $0$-extrema sector, where $h(t)$ increases monotonically toward the plateau $h_\infty=\bar{\gamma}-\Lambda/2$, and a $2$-extrema sector, where $h(t)$ shows a local maximum and a local minimum before relaxing to the same plateau.
• Figure 2: Numerical verification of the closed-form reliability and hazard for the two-site model initialized in $\ket{11}$. Panels (a,b): underdamped regime, showing oscillatory relaxation. Panels (c,d): overdamped regime with monotone hazard. Panels (e,f): overdamped regime with nonmonotone hazard. Parameters are $J=0.5$, $\gamma_1=0.2$, $\gamma_2=0.5$ for (a,b); $J=0.5$, $\gamma_1=3.0$, $\gamma_2=0.5$ for (c,d); and $J=0.1$, $\gamma_1=2.5$, $\gamma_2=1.0$ for (e,f). Dashed lines denote the analytical results and Solid lines denote Lindblad numerics.
• Figure 3: Experimental assessment of the reliability $\widehat{R}(t_k)$ in the underdamped and overdamped regimes, obtained from $N_s=10^6$ independent experimental shots.
• Figure 4: Experimental assessment of the hazard $\widehat{h}(t_k)$ in the underdamped and overdamped regimes, obtained from $N_s=10^6$ independent experimental shots.
• Figure 5: Variance of the discrete-time hazard estimator $\widehat{h}(t_k)$ versus the number of samples $N_s$ at fixed sampling interval $\Delta t=0.1$. Panels (a)--(c) compare the empirical variance ${\rm Var}_{\rm emp}[\widehat{h}(t_k)]$, obtained from $50$ independent Monte Carlo repetitions for each $N_s$, with the theoretical prediction ${\rm Var}_{\rm th}[\widehat{h}(t_k)]$ from Eq. \ref{['eq:var_hhat_asymp']}. Panel (d) shows the theory-only scaling at three representative times, illustrating both the $1/N_s$ decay and the increase of fluctuations at later times due to the shrinking risk set.