arXiv:2606.08561

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

Zicheng Han, Wanchen Zhang, Xiande Zhang

quant-phSubmitted Jun 7, 2026• Announced Jun 9, 2026• 0 citations

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 $0\le m\le N-2$.

Comments: 10 pages, 1 figure, 1 table

Related papers

Local unitary versus local Clifford equivalence of stabilizer and graph states

Bei Zeng, Hyeyoun Chung, Andrew W. Cross +1 more

The equivalence of stabilizer states under local transformations is of fundamental interest in understanding properties and uses of entanglement. Two stabilizer states are equivalent under the usual stochastic local operations and classical communication criterion if and only if they are equivalent under local unitary (LU) operations. More surprisingly, under certain conditions, two LU-equivalent stabilizer states are also equivalent under local Clifford (LC) operations, as was shown by Van den Nest et al. [Phys. Rev. A 71, 062323 (2005)]. Here, we broaden the class of stabilizer states for which LU equivalence implies LC equivalence $(\mathrm{LU}\ensuremath{\Leftrightarrow}\mathrm{LC})$ to include all stabilizer states represented by graphs with cycles of length neither 3 nor 4. To compare our result with Van den Nest et al.'s, we show that any stabilizer state of distance $\ensuremath{\delta}=2$ is beyond their criterion. We then further prove that $\mathrm{LU}\ensuremath{\Leftrightarrow}\mathrm{LC}$ holds for a more general class of stabilizer states of $\ensuremath{\delta}=2$. We also explicitly construct graphs representing $\ensuremath{\delta}>2$ stabilizer states which are beyond their criterion: we identify all 58 graphs with up to 11 vertices and construct graphs with ${2}^{m}\ensuremath{-}1$ $(m\ensuremath{\geqslant}4)$ vertices using quantum error-correcting codes which have non-Clifford transversal gates.

arXiv:quant-ph/0611214Physical Review AMar 19, 200739 citations

Transformations of Stabilizer States in Quantum Networks

Matthias Englbrecht, Tristan Kraft, Barbara Kraus

Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describes local Clifford transformations among graph states. Then, we introduce a mathematical framework to study PLC equivalence of stabilizer states, relating it to the classification of tuples of bilinear forms. This framework allows us to study decompositions of stabilizer states into tensor products of indecomposable ones, that is, decompositions into states from the entanglement generating set (EGS). While the EGS is finite up to 3 parties [Bravyi et al., J. Math. Phys. 47 , 062106 (2006)], we show that for 4 and more parties it is an infinite set, even when considering party-local unitary transformations. Moreover, we explicitly compute the EGS for 4 parties up to 10 qubits. Finally, we generalize the framework to qudit stabilizer states in prime dimensions not equal to 2 , which allows us to show that the decomposition of qudit stabilizer states into states from the EGS is unique.

arXiv:2203.04202QuantumOct 25, 202211 citations

Distinguishing graph states by the properties of their marginals

Lina Vandré, Jarn de Jong, Frederik Hahn +3 more

Graph states are a class of multipartite entangled quantum states that are ubiquitous in quantum information. We study equivalence relations between graph states under local unitaries (LU) to obtain distinguishing methods both in local and in networked settings. Based on the marginal structure of graph states, we introduce a family of easy-to-compute LU invariants. We show that these invariants uniquely identify the entanglement classes of every graph state up to eight qubits and discuss their reliability for larger numbers of qubits. To handle larger graphs, we generalize tools to test for local Clifford (LC) equivalence of graph states that work by condensing large graphs into smaller graphs. In turn, we show that statements on the equivalence of these smaller graphs (which are easier to compute) can be used to infer statements on the equivalence of the original, larger graphs. We analyze LU equivalence in two key settings, with and without allowing for the permutation of qubits. We identify entanglement classes, whose marginal structure does not allow us to distinguish them. As a result, we increase the bound on the number of qubits where the LU-LC conjecture holds from 8 to 10 qubits in the setting where qubit permutations are allowed.

May 28, 20250 citations

Local Equivalences of Graph States

Nathan Claudet

Graph states form a large family of quantum states that are in one-to-one correspondence with mathematical graphs. Graph states are used in many applications, such as measurement-based quantum computation, as multipartite entangled resources. It is thus crucial to understand when two such states have the same entanglement, i.e. when they can be transformed into each other using only local operations. In this case, we say that the graph states are LU-equivalent (local unitary). If the local operations are restricted to the so-called Clifford group, we say that the graph states are LC-equivalent (local Clifford). Interestingly, a simple graph rule called local complementation fully captures LC-equivalence, in the sense that two graph states are LC-equivalent if and only if the underlying graphs are related by a sequence of local complementations. While it was once conjectured that two LU-equivalent graph states are always LC-equivalent, counterexamples do exist and local complementation fails to fully capture the entanglement of graph states. We introduce in this thesis a generalization of local complementation that does fully capture LU-equivalence. Using this characterization, we prove the existence of an infinite strict hierarchy of local equivalences between LC- and LU-equivalence. This also leads to the design of a quasi-polynomial algorithm for deciding whether two graph states are LU-equivalent, and to a proof that two LU-equivalent graph states are LC-equivalent if they are defined on at most 19 qubits. Furthermore, we study graph states that are universal in the sense that any smaller graph state, defined on any small enough set of qubits, can be induced using only local operations. We provide bounds and an optimal, probabilistic construction.

quant-pharXiv:2511.22271Nov 28, 20250 citations

Distinguishing graph states by the properties of their marginals

Lina Vandré, Jarn de Jong, Frederik Hahn +3 more

arXiv:2406.09956Physical Review AMay 28, 20255 citations

Graph States Under the Action of Local Clifford Group in Non-Binary Case

Mohsen Bahramgiri, Salman Beigi

Graph states are well-entangled quantum states that are defined based on a graph. Of course, if two graphs are isomorphic their associated states are the same. Also, we know local operations do not change the entanglement of quantum states. Therefore, graph states that are either isomorphic or equivalent under the local Clifford group have the same properties. In this paper, we first establish a bound on the number of graph states which are neither isomorphic nor equivalent under the action of local Clifford group. Also, we study graph states in non-binary case. We translate the action of local Clifford group, as well as measurement of Pauli operators, into transformations on their associated graphs. Finally, we present an efficient algorithm to verify whether two graph states, in non-binary case, are locally equivalent or not.

arXiv:quant-ph/0610267Oct 31, 200627 citations