12 research outputs found
Matrix permanent and quantum entanglement of permutation invariant states
We point out that a geometric measure of quantum entanglement is related to
the matrix permanent when restricted to permutation invariant states. This
connection allows us to interpret the permanent as an angle between vectors. By
employing a recently introduced permanent inequality by Carlen, Loss and Lieb,
we can prove explicit formulas of the geometric measure for permutation
invariant basis states in a simple way.Comment: 10 page
Even cycle creating paths
We say that two graphs H1, H2 on the same vertex set are G-creating if the union of the two graphs contains G as a subgraph. Let H (n, k) be the maximum number of pairwise Ck-creating Hamiltonian paths of the complete graph Kn. The behavior of H (n, 2k + 1) is much better understood than the behavior of H (n, 2k), the former is an exponential function of n whereas the latter is larger than exponential, for every fixed k. We study H (n, k) for fixed k and n tending to infinity. The only nontrivial upper bound on H (n, 2k) was proved by Cohen, Fachini, and Körner in the case of k = 2: : (Formula presented.) In this paper, we generalize their method to prove that for every k ≥ 2, (Formula presented.) and a similar, slightly better upper bound holds when k is odd. Our proof uses constructions of bipartite, regular, C2k-free graphs with many edges given in papers by Reiman, Benson, Lazebnik, Ustimenko, and Woldar. © 2019 Wiley Periodicals, Inc