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

Abstract

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 $|C_5⟩$. 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 $C_5$. 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 $|C_N⟩$ with $N\ge 7$ are not $m$-resistant for any $0\le m\le N-2$.