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arXiv:1908.10390 · Physical Review A

Cut-resistant links and multipartite entanglement resistant to particle loss

Gonçalo M. Quinta, Rui André, Adam Burchardt, Karol Życzkowski

Published Dec 23, 2019• 11 citations
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Abstract

In this work, we explore the space of quantum states composed of $N$ particles. To investigate entanglement resistant to particle loss, we introduce the notion of $m$-resistant states. A quantum state is $m$ resistant if it remains entangled after losing an arbitrary subset of $m$ particles but becomes separable after losing any number of particles larger than $m$. We establish an analogy to the problem of designing a topological link consisting of $N$ rings such that, after cutting any $(m+1)$ of them, the remaining rings become disconnected. We present a constructive solution to this problem, which allows us to exhibit several $N$-particle states with the desired property of entanglement resistance to particle loss.

DOI
10.1103/physreva.100.062329

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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.
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.
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 $(N\ensuremath{-}k)$-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.
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.
We propose an alternative classification scheme for quantum entanglement based on topological links. This is done by identifying a nonrigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for a given number of rings. To determine all different possibilities, we develop a formalism that associates any link to a polynomial, with each polynomial thereby defining a distinct equivalence class. To demonstrate the use of this classification scheme, we choose qubit quantum states as our example of physical system. A possible procedure to obtain qubit states from the polynomials is also introduced, providing an example state for each link class. We apply the formalism for the quantum systems of three and four qubits and demonstrate the potential of these tools in a context of qubit networks.
One of the key manifestations of quantum mechanics is the phenomenon of quantum entanglement. While the entanglement of bipartite systems is already well understood, our knowledge of entanglement in multipartite systems is still limited. This dissertation covers various aspects of the quantification of entanglement in multipartite states and the role of symmetry in such systems. Firstly, we establish a connection between the classification of multipartite entanglement and knot theory and investigate the family of states that are resistant to particle loss. Furthermore, we construct several examples of such states using the Majorana representation as well as some combinatorial methods. Secondly, we introduce classes of highly-symmetric but not fully-symmetric states and investigate their entanglement properties. Thirdly, we study the well-established class of Absolutely Maximally Entangled (AME) quantum states. On one hand, we provide construction of new states belonging to this family, for instance, an AME state of 4 subsystems with six levels each, on the other, we tackle the problem of equivalence of such states. Finally, we present a novel approach for the general problem of verification of the equivalence between any pair of arbitrary quantum states based on a single polynomial entanglement measure.