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Physical Review A

Constructions of multipartite entanglement resistant to particle loss

Wanchen Zhang, Zicheng Han, Fei Shi, Xiande Zhang

Published Jul 17, 2025• 2 citations
Publisher

Abstract

An entangled state is called $m$ resistant if it remains entangled after losing an arbitrary subset of $m$ particles but becomes fully separable after losing any number of particles larger than $m$. Quinta et al. [Phys. Rev. A 100, 062329 (2019)] conjectured that for any $N$-particle systems, there always exists an $m$-resistant pure state of some local dimension $d$. In this paper, we give two general constructions of $m$-resistant pure states. One is from the mixtures of Dicke states, which provides strong $(N\ensuremath{-}k)$-resistant pure $N$-qubit states with $k=4$ or 5. The other is from classical error-correcting codes, which provides new $m$-resistant pure qudit states for certain $m<N/2$.

DOI
10.1103/g4bk-gfhs

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•Oct 28, 2019•4 citations
An entangled state is called mmm resistant if it remains entangled after losing an arbitrary subset of mmm particles but becomes fully separable after losing any number of particles larger than mmm. Quinta et al. [Phys. Rev. A 100, 062329 (2019)] conjectured that for any NNN-particle systems, there always exists an mmm-resistant pure state of some local dimension ddd. In this paper, we give two general constructions of mmm-resistant pure states. One is from the mixtures of Dicke states, which provides strong (N\ensuremath−k)(N\ensuremath{-}k)(N\ensuremath−k)-resistant pure NNN-qubit states with k=4k=4k=4 or 5. The other is from classical error-correcting codes, which provides new mmm-resistant pure qudit states for certain m<N/2m<N/2m<N/2.
An entangled state is called m-resistant if it remains entangled after losing an arbitrary subset of mparticles but becomes fully separable after losing any number of particles larger than m. Quinta et al. [Phys. Rev. A (2019)] conjectured that for any N-particle systems, there always exists an m-resistant pure state. In this paper, we give two general constructions of m-resistant pure states. One is from the mixtures of Dicke states, which provides strong (N - k)-resistant pure N-qubit states with k = 4 or 5. The other is from classical error correcting codes, which provides new m-resistant qudit states for certain m < N/2.
Links of rings offer an intuitive picture of multipartite entanglement. Cutting one ring can free some of the others, just as losing a particle, i.e. tracing it out, can disentangle some of the remaining ones. This analogy classifies multipartite entanglement according to which groups of particles stay entangled when others are lost, a question at the heart of robust quantum networks. However, pure states realizing a given link class were known only in special cases. Here, we close this gap. Recasting link polynomials as hypergraphs, we give an explicit formula for pure qudit states and prove that, for every link class, their entanglement under any pattern of particle loss matches the link exactly. The same construction also controls genuine multipartite entanglement, determining exactly which of the groups that stay entangled are genuinely multipartite entangled, and all of them can be made so. As a consequence, we prove the conjecture that mmm-resistant pure qudit states, which stay entangled after the loss of any mmm particles but become fully separable after the loss of m+1m+1m+1, exist for any number of particles and any mmm.
In this work, we explore the space of quantum states composed of NNN particles. To investigate entanglement resistant to particle loss, we introduce the notion of mmm-resistant states. A quantum state is mmm resistant if it remains entangled after losing an arbitrary subset of mmm particles but becomes separable after losing any number of particles larger than mmm. We establish an analogy to the problem of designing a topological link consisting of NNN rings such that, after cutting any (m+1)(m+1)(m+1) of them, the remaining rings become disconnected. We present a constructive solution to this problem, which allows us to exhibit several NNN-particle states with the desired property of entanglement resistance to particle loss.
A pure quantum state of NNN subsystems with ddd levels each is called kkk-multipartite maximally entangled state, which we call a kkk-uniform state, if all its reductions to kkk qudits are maximally mixed. These states form a natural generalization of NNN-qudit Greenberger-Horne-Zeilinger states which belong to the class 1-uniform states. We establish a link between the combinatorial notion of orthogonal arrays and kkk-uniform states and prove the existence of several classes of such states for NNN-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of N>5N>5N>5 qubits. We show that finding a different class of 2-uniform states would imply the Hadamard conjecture, so the full classification of 2-uniform states seems to be currently out of reach. Furthermore, we establish links between the existence of kkk-uniform states and classical and quantum error correction codes and provide a graph representation for such states.
We study particle-loss resistant entanglement within the framework of stabilizer and graph states. A pure state is mmm-resistant if it remains entangled after the loss of any mmm particles and becomes fully separable after the loss of any m+1m+1m+1 particles. The smallest previously unresolved qubit case was the existence of a five-qubit 111-resistant pure state, which is resolved here by the five-cycle graph state ∣C5⟩\ket{C_5}∣C5​⟩. A stabilizer-subgroup method is also developed for verifying mmm-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 111-resistant stabilizer states are exactly the local Clifford class of C5C_5C5​. Six-qubit 222-resistant stabilizer states exist in three distinct local Clifford classes, whereas no seven-qubit stabilizer state is mmm-resistant for any nonzero admissible mmm. Finally, we prove that the cycle graph states ∣CN⟩\ket{C_N}∣CN​⟩ with N≥7N\ge 7N≥7 are not mmm-resistant for any 0≤m≤N−20\le m\le N-20≤m≤N−2.
Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols and the construction of quantum error correcting codes. To our knowledge, the only known systematic construction of these states is based on a class of classical error correction codes known as maximum distance separable. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from maximum distance separable codes. Furthermore, we used our method to construct several examples of absolutely maximally entangled states whose existence was open so far.