Tripartite entanglement in qudit stabilizer states and application in quantum error correction

Consider a stabilizer state on n qudits, each of dimension D with D being a prime or squarefree integer, divided into three mutually disjoint sets or parts. Generalizing a result of Bravyi et al.[J. Math. Phys. 47, 062106 (2006)] for qubits (D=2), we show that up to local unitaries, the three parts of the state can be written as tensor product of unentangled signle-qudit states, maximally entangled Einstein-Podolsky-Rosen (EPR) pairs, and tripartite Greenberger-Horne-Zeilinger (GHZ) states. We employ this result to obtain a complete characterization of the properties of a class of channels associated with stabilizer error-correcting codes, along with their complementary channels.