  We investigate how the initial and final exchange symmetries between the
two-coin states influence the spatial entanglement dynamics between the two
corresponding quantum walkers. Notably, when the initial state is
anti-symmetric and the final measurement on the coins yields symmetric
outcomes, all the initial entanglement will be transferred to the spatial
degrees of freedom, regardless of when the coins are measured. Conversely, if
the final outcomes are anti-symmetric, the spatial entanglement exhibits damped
oscillation with a period ($T$) being inversely proportional to the coin
operator parameter ($\theta$). These behaviours are reversed for symmetric
initial states. Moreover, we also observe the same spatial entanglement damping
regardless of the initial state when the post-selected results lack symmetry.
Our findings reveal how symmetries affect the entanglement dynamics in quantum
walks, offering potential insights for applications in quantum technology.
