  Quantum spin liquids can be faithfully represented and efficiently
characterized within the framework of Projected Entangled Pair States (PEPS).
Guided by extensive exact diagonalization and density matrix renormalization
group calculations, we construct an optimized symmetric PEPS for a SU$(3)_1$
chiral spin liquid on the square lattice. Characteristic features are revealed
by the entanglement spectrum (ES) on an infinitely long cylinder. In all three
$\mathbb{Z}_3$ sectors, the level counting of the linear dispersing modes is in
full agreement with SU$(3)_1$ Wess-Zumino-Witten conformal field theory
prediction. Special features in the ES are shown to be in correspondence with
bulk anyonic correlations, indicating a fine structure in the holographic
bulk-edge correspondence. Possible universal properties of topological
SU$(N)_k$ chiral PEPS are discussed.
