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Once the fractional quantum Hall (FQH) state for a finite size system is puton the surface of a cylinder, the distance between the two ends with openboundary conditions can be tuned as varying the aspect ratio $\gamma$. Itscales linearly as increasing the system size and therefore has a largeradjustable range than that on disk. The previous study of the quasi-holetunneling amplitude on disk in Ref.~\cite{Zk2011} indicates that the tunnelingamplitudes have a scaling behavior as a function of the tunneling distance andthe scaling exponents are related to the scaling dimension and the charge ofthe transported quasiparticles. However, the scaling behaviors poorly due tothe narrow range of the tunneling distance on disk. Here we systematicallystudy the quasiparticle tunneling amplitudes of the Laughlin state in thecylinder geometry which shows a much better scaling behavior. Especially, thereare some corssover behaviors at two length scales when the two open edges areclose to each other. These lengths are also reflected in the bipartiteentanglement and the electron Green's function as either a singularity or acrossover. These two critical length scales of the edge-edge distance,$L_x^{c_1}$ and $L_x^{c_2}$, are found to be related to the dimension reductionand back scattering point respectively.

The length scale measurements of the Fractional quantum Hall state on cylinder