title: From entanglement witness to generalized Catalan numbers.
abstract: Being extremely important resources in quantum information and computation, it is vital to efficiently detect and properly characterize entangled states. We analyze in this work the problem of entanglement detection for arbitrary spin systems. It is demonstrated how a single measurement of the squared total spin can probabilistically discern separable from entangled many-particle states. For achieving this goal, we construct a tripartite analogy between the degeneracy of entanglement witness eigenstates, tensor products of SO(3) representations and classical lattice walks with special constraints. Within this framework, degeneracies are naturally given by generalized Catalan numbers and determine the fraction of states that are decidedly entangled and also known to be somewhat protected against decoherence. In addition, we introduce the concept of a "sterile entanglement witness", which for large enough systems detects entanglement without affecting much the system's state. We discuss when our proposed entanglement witness can be regarded as a sterile one.
authors.affiliation: H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, U.K.|School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 6997801, Israel.
authors.forename: E
authors.initials: E
authors.lastname: Cohen
authors.affiliation: Centro de Física do Porto, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal.|II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany.
authors.forename: T
authors.initials: T
authors.lastname: Hansen
authors.affiliation: School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 6997801, Israel.
authors.forename: N
authors.initials: N
authors.lastname: Itzhaki