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

quant-pharXiv:2606.08561Jun 9, 20260 citations

arXiv:1605.07929 · The European Physical Journal D

Identifying non-k-separability of a class of N-qubit complete graph states using correlation tensors

N. Ananth, Murugaian Senthilvelan

Published Jul 1, 2016• 3 citations

arXiv HTML PDF

DOI: 10.1140/epjd/e2016-70056-2

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

quant-pharXiv:2606.08561Jun 9, 20260 citations

q\ge2

, which include the graph-state resources of measurement-based quantum computation. Dividing a single-tensor variation by the nonvanishing amplitude turns it into an arbitrary function on the closed neighborhood of its vertex. Overlapping neighborhoods give the complete kernel. Its non-bond part is generated on periodic regular-polygon tilings by triangles, chordless squares, and edge-sharing diamonds. Under stated period conditions, generic gauge completeness on all thirty-nine families studied identifies this nullity as a WGS rank drop: for thirty-three families at every

q\ge2

, and for six at

q=2

. Supports outside every neighborhood give the missing directions: at qubit graph states on the eleven Archimedean tilings every nearest-neighbor

XX

and

YY

direction is missing, including on honeycomb, where the non-bond kernel is empty. On large honeycomb and square tori, small nonzero coherent

XX

errors leave the bond-two PEPS set on the same graph. Removing redundant parameters and supplying missing directions are therefore distinct operations.

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