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.
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.
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.
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 m-resistant pure qudit states, which stay entangled after the loss of any m particles but become fully separable after the loss of m+1, exist for any number of particles and any m.
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.
A pure quantum state is called k-uniform if all its reductions to k-qudit are maximally mixed. We investigate the general constructions of k-uniform pure quantum states of n subsystems with d levels. We provide one construction via symmetric matrices and the second one through the classical error-correcting codes. There are three main results arising from our constructions. First, we show that for any given even n ≥ 2, there always exists an n/2-uniform n-qudit quantum state of level p for sufficiently large prime p. Second, both constructions show that there exist k-uniform n-qudit pure quantum states such that k is proportional to n, i.e., k = Ω(n) although the construction from symmetric matrices in general outperforms the one by error-correcting codes. Third, our symmetric matrix construction provides a positive answer to the open question on whether there exists a 3-uniform n-qudit pure quantum state for all n ≥ 8. In fact, we can further prove that, for every k, there exists a constant Mksuch that there exists a k-uniform n-qudit quantum state for all n ≥ Mk. In addition, by using the concatenation of algebraic geometry codes, we give an explicit construction of k-uniform quantum state when k tends to infinity.
A pure quantum state of N subsystems with d levels each is called k-multipartite maximally entangled state, which we call a k-uniform state, if all its reductions to k qudits are maximally mixed. These states form a natural generalization of N-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 k-uniform states and prove the existence of several classes of such states for N-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of N>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 k-uniform states and classical and quantum error correction codes and provide a graph representation for such states.
We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors.
This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions.
We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.