Phase vs coin vs position disorder as a probe for the resilience and revival of single particle entanglement in cyclic quantum walks

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.