Spin network entanglement: coarse-graining

In loop quantum gravity, partitioning graph introduces boundaries and entanglement between spin sub-networks, reflecting non-local degrees of freedom and correlation amongst spatial regions. This gives rise to the view of coarse-graining, reducing the degrees of freedom that are unnecessary to be considered. The present work sets coarse-graining in the framework of bulk-boundary relation. We investigates the spin network entanglement, showing that the entanglement is invariant under the coarse-graining at kinematical level. Moreover, we build the transformation between holonomy operators based on finer graph and coarser graph, and reveal the preservation of the coarse-graining method under the evolution generated by the holonomy operator. This leads to a holographical perspective for the entanglement issue in loop quantum gravity.