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 (\ket{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 (\ket{C_N}) with (N\ge 7) are not (m)-resistant for any (0\le m\le N-2).

• The paper establishes the unique five-qubit 1-resistant graph state using the five-cycle graph state C5 and precise stabilizer-subgroup certificates.
• It employs explicit computations of NPT conditions to certify entangled four-qubit marginals while proving three-qubit marginals are fully separable.
• It systematically classifies m-resistant stabilizer states for N=5,6,7, revealing key limitations in constructing robust multipartite entangled states.

A Five-Qubit 1-Resistant Graph State and Stabilizer Marginal Certificates

Overview and Motivation

The paper establishes the explicit existence of a five-qubit 1-resistant pure quantum state in the context of multipartite entanglement, utilizing the stabilizer and graph state frameworks. In particular, it identifies the five-cycle graph state $\ket{C_5}$ as a five-qubit 1-resistant state—a case that remained unresolved in prior literature. Alongside, the authors present a stabilizer-subgroup approach for certifying $m$-resistance, which leverages algebraic properties of local stabilizer groups for separability and entanglement via exact NPT (Negative Partial Transpose) conditions. Systematic enumeration of graph states for $N = 5, 6, 7$ is conducted, leading to a complete classification (up to local Clifford equivalence) of $m$-resistance among stabilizer states for these sizes.

Particle-Loss Resistant Entanglement and $m$-Resistance

Within multipartite entanglement theory, the $m$-resistant property is defined as follows: an $N$-party pure state is $m$-resistant if every marginal state after tracing out any $m$ subsystems remains entangled, but becomes fully separable after tracing out any $m+1$ subsystems. This concept formalizes and sharpens the notion of entanglement robustness under particle loss—the key fragility property of multipartite states. For the five-qubit, $m$0 case, the issue had been open since previous constructions did not yield an explicit solution, and all symmetric-state searches had failed.

Stabilizer Formalism and Marginal Entanglement Certification

The authors adopt the stabilizer formalism for quantum states, focusing on graph states whose stabilizer generators are naturally encoded by simple graphs. For an $m$1-qubit graph state, each vertex corresponds to a qubit, and controlled-$m$2 edges encode entanglement. The stabilizer of the full state determines the support and algebraic properties of all possible marginals. Two core technical tools are introduced for certifying the separability and entanglement of reduced density matrices (marginals):

• Separability Certificate: The existence of a local stabilizer subgroup, whose nontrivial single-site factors are of fixed Pauli type across each site, ensures that the corresponding marginal is fully separable. If the dimension of the local stabilizer group is $m$3, full separability also follows.
• NPT (Negative Partial Transpose) Entanglement Certificate: For each marginal, compute the spectrum of its partial transpose with respect to every bipartition by explicit Walsh transform of sign functions determined by the stabilizer. Any negative eigenvalue certifies entanglement via the Peres-Horodecki criterion.

These algebraic certificates are exhaustive for the examined cases and are compatible with exact computation, avoiding ambiguous or numerically unstable cases.

Existence and Uniqueness of the Five-Qubit 1-Resistant Graph State

The five-cycle graph state $m$4 is shown to be 1-resistant using the above certificates. Under the symmetries of $m$5, it is proven that:

• All three-qubit marginals (upon tracing out any two qubits) are fully separable by explicit calculation of their local stabilizers, each having dimension 1 or less.
• All four-qubit marginals (upon tracing out any one qubit) are entangled, certified by finding NPT eigenvalues for the partial transpose of the reduced state.

Exhaustive enumeration over all inequivalent five-vertex graph states confirms the uniqueness of $m$6 up to local Clifford (and thus local complementation) equivalence; no other five-qubit graph states possess the 1-resistance property.

General Classification for Small $m$7 and Resistance Parameters

For $m$8, all non-isomorphic graph states are checked for $m$9-resistance across all possible $N = 5, 6, 7$0. The results, classified by local Clifford orbits, are as follows:

• $N = 5, 6, 7$1:
  • 0-resistant states correspond to the GHZ orbit.
  • 1-resistant state is unique (the $N = 5, 6, 7$2 orbit).
  • No states for $N = 5, 6, 7$3 resistance.
• $N = 5, 6, 7$4:
  • 2-resistant states exist in three distinct local Clifford orbits, including the six-qubit AME (absolutely maximally entangled) state and cycle $N = 5, 6, 7$5.
  • No graph states with $N = 5, 6, 7$6-resistance.
• $N = 5, 6, 7$7:
  • For all $N = 5, 6, 7$8, no $N = 5, 6, 7$9-resistant graph states exist.

All $m$0-resistant properties are proven invariant under local unitary (specifically, local Clifford) transformation; the classification thus extends directly to all stabilizer states via their correspondence with graph states.

Nonexistence of Large Cycle Graph State Resistance

A general structural result is established: for every $m$1, the cycle graph state $m$2 fails to show $m$3-resistance for any $m$4. This is proven by exhibiting, for any $m$5, explicit $m$6-qubit marginals that remain entangled; and for smaller $m$7, explicit marginals that are fully separable, precluding the necessary threshold condition.

Practical and Theoretical Implications

The resolution of the five-qubit case demonstrates that stabilizer/graph states can display robust (yet highly constrained) resistance properties, supporting proposals for error-tolerant entangled resource states in quantum information protocols. The explicit stabilizer-subgroup certification method not only automates and streamlines the search for such states, but delineates the precise structural limitations of the stabilizer formalism under particle loss. The negative results for larger $m$8 and for $m$9 in small systems show that stabilizer/graph state constructions cannot realize the full range of resistance properties theoretically possible in the space of pure states—a constraint with implications for the design of fault-tolerant quantum memories and multipartite communication schemes.

The methodology provides a template for systematic investigations in larger systems or with more general classes of (non-stabilizer) pure states. Additionally, the local-complementation approach assists in rapid recognition of equivalence classes, crucial for scalable enumeration schemes.

Conclusion

This work conclusively identifies the unique five-qubit 1-resistant graph state as the five-cycle $m$0, introduces a robust stabilizer-subgroup-based framework for marginal entanglement certification, and systematically classifies $m$1-resistant stabilizer states for $m$2 up to local Clifford equivalence. The results show strict structural limitations for particle-loss-resistance in the stabilizer formalism, motivating further investigation into more general quantum state constructions and resistance properties beyond the graph/stabilizer setting. Open questions remain on the realization of stronger (i.e., genuinely multipartite) resistance and the existence of non-stabilizer pure states with high resistance not captured by the present classification.

Reference: "A five-qubit 1-resistant graph state and stabilizer marginal certificates" (2606.08561)