A five-qubit 1-resistant graph state and stabilizer marginal certificates

We study particle-loss resistant entanglement within the framework of stabilizer and graph states. A pure state is m-resistant if it remains entangled after the loss of any m particles and becomes fully separable after the loss of any m+1 particles. The smallest previously unresolved qubit case was the existence of a five-qubit 1-resistant pure state, which is resolved here by the five-cycle graph state ket{C₅}. A stabilizer-subgroup method is also developed for verifying m-resistance in graph states, using local stabilizers to certify full separability and exact negative partial transpose (NPT) witnesses to certify entanglement. Applying this to all graph states associated with non-isomorphic graphs on five, six, and seven vertices, we obtain a graph state classification up to local Clifford equivalence, which also classifies stabilizer states up to local Clifford equivalence. Thus, the five-qubit 1-resistant stabilizer states are exactly the local Clifford class of C₅. Six-qubit 2-resistant stabilizer states exist in three distinct local Clifford classes, whereas no seven-qubit stabilizer state is m-resistant for any nonzero admissible m. Finally, we prove that the cycle graph states ket{C_N} with Nge 7 are not m-resistant for any 0le mle N-2.