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

TL;DR

This work investigates how phase, coin, and position disorder affect single-particle entanglement (SPE) in cyclic discrete-time quantum walks (CQWs). By formulating a general CQW on $k$-cycles and analyzing both clean and disordered dynamics, it demonstrates that maximally entangled single-particle states (MESPS) generated at $t=1$ are robust to phase disorder for any initial state and to coin disorder for phase-symmetric initial states, while position disorder breaks parity and drives SPE to a fixed saturation value at large times. Moreover, phase and coin disorder can enhance SPE at specific times and even revive SPE from zero, whereas position disorder tends to reduce recurrence and increase spreading but still can yield revival. These results offer practical insights for lab implementations and potential strategies to harness disorder to control SPE in quantum information processing and cryptography. The study provides analytical proofs, numerical simulations, and an algorithmic framework to explore disorder effects, with implications for error mitigation and entanglement resource engineering in cyclic quantum walks.

Abstract

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.

Figures (14)

• Figure 1: (a) Clean-CQW; (b) CQW with phase or coin disorder, where amplitudes in coin basis change; (c) CQW with position disorder where jump length in shift changes randomly. Sites are shown as green dots, and red (for the same amplitudes) and blue (for different amplitudes) curves denote amplitudes of the particle on 4-cycle.
• Figure 2: (a) SPE $\langle E_{av} \rangle$ vs. time step $t$ for different static-phase-disorder strengths ($\delta$); (b) SPE $\langle E_{av} \rangle$ vs. static-phase-disorder strength $\delta$ for different $t$ for 4-cycle.
• Figure 3: (a) SPE $\langle E_{av} \rangle$ vs. time step $t$ for different dynamic-phase-disorder strength ($\delta$); (b) SPE $\langle E_{av} \rangle$ vs. dynamic-phase-disorder strength $\delta$ for different $t$ for 4-cycle.
• Figure 4: (a) SPE $\langle E_{av} \rangle$ vs. time step $t$ for different position-disorder strength ($\lambda$); (b) SPE $\langle E_{av} \rangle$ vs. position-disorder strength $\lambda$ for different $t$ for 4-cycle.
• Figure 5: (a) Clean-CQW (Hadamard walk) on a 3-cycle ; (b) CQW with phase or coin disorder on a 3-cycle, where amplitudes in both hopping change;(c) CQW with position disorder on a 3-cycle where the hopping length changes randomly. Sites are shown as green dots, and red-blue curves denote the hopping of the particle on a 3-cycle.