We propose a new classification scheme for quantum entanglement based on topological links. This is done by identifying a non-rigid 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 which associates any link to a polynomial, with each polynomial thereby defining a distinct equivalence class. In order 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 new tools in a context of qubit networks.

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In this work, we explore the space of quantum states composed of N particles. To investigate the entanglement resistant to particles 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 a 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 distinguished N -particles states with the desired property of entanglement resistance to a particle loss.