Quantum states exhibiting single-particle entanglement (SPE) can encode and process quantum information more robustly than their multi-particle analogs. Understanding the vulnerability and resilience of SPE to disorder is therefore crucial. This letter investigates phase, coin, and position disorder via discrete-time quantum walks on odd and even cyclic graphs to study their effect on SPE. The reduction in SPE is insignificant for low levels of phase or coin disorder, showing the resilience of SPE to minor perturbations. However, SPE is seen to be more vulnerable to position disorder. We analytically prove that maximally entangled single-particle states (MESPS) at time step $t=1$ are impervious to phase disorder regardless of the choice of the initial state. Further, MESPS at timestep $t=1$ is also wholly immune to coin disorder for phase-symmetric initial states. Position disorder breaks odd-even parity and distorts the physical time cone of the quantum walker, unlike phase or coin disorder. SPE saturates towards a fixed value for position disorder, irrespective of the disorder strength at large timestep $t$. Furthermore, SPE can be enhanced with moderate to significant phase or coin disorder strengths at specific time steps. Interestingly, disorder can revive single-particle entanglement from absolute zero in some instances, too. These results are crucial in understanding single-particle entanglement evolution and dynamics in a lab setting.

翻译：相较于多粒子纠缠态，展现单粒子纠缠（SPE）的量子态能够以更强的鲁棒性编码和处理量子信息。因此，理解SPE对无序性的脆弱性与鲁棒性至关重要。本文通过在奇偶循环图上进行离散时间量子行走，研究相位、硬币和位置无序性对SPE的影响。对于低强度的相位或硬币无序性，SPE的衰减并不显著，这表明SPE对微小扰动具有鲁棒性。然而，SPE对位置无序性表现出更高的脆弱性。我们通过解析证明，在时间步$t=1$处的最大纠缠单粒子态（MESPS）不受相位无序性的影响，且与初始态的选择无关。此外，对于相位对称的初始态，时间步$t=1$处的MESPS也完全不受硬币无序性的影响。与相位或硬币无序性不同，位置无序性会破坏奇偶宇称性并扭曲量子行走者的物理时间锥。在大的时间步$t$下，无论无序性强弱如何，SPE都会趋于一个固定值。此外，在特定的时间步，中度至显著的相位或硬币无序性强度甚至可以增强SPE。有趣的是，在某些情况下，无序性甚至能够使单粒子纠缠从绝对零值恢复。这些结果对于理解实验室环境中单粒子纠缠的演化与动力学至关重要。