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We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. First, we show that the signed join of a negative $2$-clique with any positive $(n,3)$-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as $n$ increases. Next, we prove that a signed complete graph has perfect state transfer if its positive subgraph is a regular graph with perfect state transfer and its negative subgraph is periodic. This shows that signing is useful for creating perfect state transfer since no complete graph (except for the $2$-clique) has perfect state transfer. Also, we show that the double-cover of a signed graph has perfect state transfer if the positive subgraph has perfect state transfer and the negative subgraph is periodic.Here, signing is useful for constructing unsigned graphs with perfect state transfer. Finally, we study perfect state transfer on a family of signed graphs called the exterior powers which is derived from a many-fermion quantum walk on graphs.

A signed graph is a graph whose edges are given ±1 weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal ±1 matrix. For a signed graph Σ on n vertices, its exterior kth power, where k = 1, . . . , n − 1, is a graph k Σ whose adjacency matrix is given bywhere P ∧ is the projector onto the anti-symmetric subspace of the k-fold tensor product space (C n ) ⊗k and Σ k is the k-fold Cartesian product of Σ with itself. The exterior power creates a signed graph from any graph, even unsigned. We prove sufficient and necessary conditions so that k Σ is balanced. For k = 1, . . . , n − 2, the condition is that either Σ is a signed path or Σ is a signed cycle that is balanced for odd k or is unbalanced for even k; for k = n − 1, the condition is that each even cycle in Σ is positive and each odd cycle in Σ is negative.We are interested in graph operators which arise from taking the quotient of a Cartesian product of an underlying graph with itself. More specifically, such operators are defined on a graph G = (V, E) after applying the following three steps. First, we take the k-fold Cartesian product of G with itself, namely G k . Note that the vertex set of G k is the set of k-tuples V k . For the second (possibly optional) step, we remove from V k (via vertex deletions) the set D consisting of all k-tuples of vertices which contain a repeated vertex. We denote the

We give an answer in the "geometric" setting to a question of [dFEI18] asking when local isomorphisms of k-schemes can be detected on the associated maps of local arc or jet schemes. In particular, we show that their ideal-closure operation a → a ac (the arc-closure) on a local k-algebra (R, m, L) is trivial when R is Noetherian and k ֒→ L is separable, and thus that such a germ Spec R has the (embedded) local isomorphism property.

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