Entropy of eigenfunctions on quantum graphs

We consider families of finite quantum graphs of increasing size and we are interested in how eigenfunctions are distributed over the graph. As a measure for the distribution of an eigenfunction on a graph we introduce the entropy, it has the property that a large value of the entropy of an eigenfunction implies that it cannot be localised on a small set on the graph. We then derive lower bounds for the entropy of eigenfunctions which depend on the topology of the graph and the boundary conditions at the vertices. The optimal bounds are obtained for expanders with large girth, the bounds are similar to the ones obtained by Anantharaman et.al. for eigenfunctions on manifolds of negative curvature, and are based on the entropic uncertainty principle. For comparison we compute as well the average behaviour of entropies on Neumann star graphs, where the entropies are much smaller. Finally we compare our lower bounds with numerical results for regular graphs and star graphs with different boundary conditions.Comment: 28 pages, 3 figure

Holographic coherent states from random tensor networks

Random tensor networks provide useful models that incorporate various important features of holographic duality. A tensor network is usually defined for a fixed graph geometry specified by the connection of tensors. In this paper, we generalize the random tensor network approach to allow quantum superposition of different spatial geometries. We set up a framework in which all possible bulk spatial geometries, characterized by weighted adjacent matrices of all possible graphs, are mapped to the boundary Hilbert space and form an overcomplete basis of the boundary. We name such an overcomplete basis as holographic coherent states. A generic boundary state can be expanded on this basis, which describes the state as a superposition of different spatial geometries in the bulk. We discuss how to define distinct classical geometries and small fluctuations around them. We show that small fluctuations around classical geometries define "code subspaces" which are mapped to the boundary Hilbert space isometrically with quantum error correction properties. In addition, we also show that the overlap between different geometries is suppressed exponentially as a function of the geometrical difference between the two geometries. The geometrical difference is measured in an area law fashion, which is a manifestation of the holographic nature of the states considered.Comment: 33 pages, 8 figures. An error corrected on page 14. Reference update

Spanning trees short or small

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page