Analytic description of statistics of spectra of quantum graphs

We discuss how to obtain exact and approximate distributions for various statistical characteristics of the spectra of quantum graphs using previously found exact solutions of the spectral problem. We indicate the relation between the appearing spectral decompositions and the theory of weakly dependent random variables and indicate the relation between the known limit theorems for trigonometric sums and the universal statistical properties of the spectra of quantum chaotic systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 38–66, July, 2008.     

Perturbation theory in large order

For many quantum mechanical models, the behavior of perturbation theory in large order is strikingly simple. For example, in the quantum anharmonic oscillator, which is defined by−″ + (x²/4+εx⁴/4−E)y=0, y(±∞)=0,the perturbation coefficients A_n in the expansion for the ground-state energysimplify dramatically as n → ∞:.We use the methods of applied mathematics to investigate the nature of perturbation theory in quantum mechanics and we show that its large-order behavior is determined by the semiclassical content of the theory. In quantum field theory the perturbation coefficients are computed by summing Feynman graphs. We present a statistical procedure in a simple λ⁴ model for summing the set of all graphs as the number of vertices → ∞. Finally, we discuss the connection between the large-order behavior of perturbation theory in quantum electrodynamics and the value of α, the charge on the electron.     

Fractional revival and association schemes

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose–Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.     

Perfect state transfer on bi-Cayley graphs over abelian groups

The study of perfect state transfer on graphs has attracted a great deal of attention during the past ten years because of its applications to quantum information processing and quantum computation. Perfect state transfer is understood to be a rare phenomenon. This paper establishes necessary and sufficient conditions for a bi-Cayley graph having a perfect state transfer over any given finite abelian group. As corollaries, many known and new results are obtained on Cayley graphs having perfect state transfer over abelian groups, (generalized) dihedral groups, semi-dihedral groups and generalized quaternion groups. Especially, we give an example of a connected non-normal Cayley graph over a dihedral group having perfect state transfer between two distinct vertices, which was thought impossible.     

On graphs whose Laplacian matrix’s multipartite separability is invariant under graph isomorphism

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are responsible for many nonintuitive physical phenomena. The entanglement property of normalized Laplacian matrices is in general not invariant under graph isomorphism. In recent papers, graphs were identified whose entanglement and separability properties are invariant under isomorphism. The purpose of this note is to completely characterize the set of graphs whose separability is invariant under graph isomorphism. In particular, we show that this set consists of K_2,2 and its complement, all complete graphs and no other graphs.     

Infinite quantum graphs

Infinite quantum graphs with δ-interactions at vertices are studied without any assumptions on the lengths of edges of the underlying metric graphs. A connection between spectral properties of a quantum graph and a certain discrete Laplacian given on a graph with infinitely many vertices and edges is established. In particular, it is shown that these operators are self-adjoint, lower semibounded, nonnegative, discrete, etc. only simultaneously.     

Algebraic topology and the quantization of fluctuating currents

We give a new approach to the study of statistical mechanical systems: algebraic topology is used to investigate the statistical distributions of stochastic currents generated in graphs. In the adiabatic and low temperature limits we will demonstrate that quantization of current generation occurs.     

基于交叉点排列的几类距离正则图的谱分布

基于距离正则图的交叉点排列,给出了三类距离正则图的谱分布,讨论了一维整数格Z的距离矩阵与第一类切比雪夫多项式之间的关系,体现出研究图的谱分布的量子概率技巧.     

Perfect State Transfer on Weighted Abelian Cayley Graphs*

Recently, there are extensive studies on perfect state transfer (PST for short) on graphs due to their significant applications in quantum information processing and quantum computations. However, there is not any general characterization of graphs that have PST in literature. In this paper, the authors present a depiction on weighted abelian Cayley graphs having PST. They give a unified approach to describe the periodicity and the existence of PST on some specific graphs.     

Quantum Ergodicity for Quantum Graphs without Back-Scattering

We give an estimate of the quantum variance for d-regular graphs quantised with boundary scattering matrices that prohibit back-scattering. For families of graphs that are expanders, with few short cycles, our estimate leads to quantum ergodicity for these families of graphs. Our proof is based on a uniform control of an associated random walk on the bonds of the graph. We show that recent constructions of Ramanujan graphs, and asymptotically almost surely, random d-regular graphs, satisfy the necessary conditions to conclude that quantum ergodicity holds.     

Enumeration of labeled,three-connected graphs

The technique of functional Legendre transformations developed in statistical physics and quantum field theory is used to enumerate labeled graphs and multi-graphs.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 21–24, 1982.     

Perfect state transfer in integral circulant graphs

The existence of perfect state transfer in quantum spin networks based on integral circulant graphs has been considered recently by Saxena, Severini and Shparlinski. We give the simple condition for characterizing integral circulant graphs allowing the perfect state transfer in terms of its eigenvalues. Using that, we complete the proof of results stated by Saxena, Severini and Shparlinski. Moreover, it is shown that in the class of unitary Cayley graphs there are only two of them allowing perfect state transfer.     

Graph complexity and the laplacian matrix in blocked experiments

connections between the efficiency of statistical designs, the Laplacian matrix studied in algebra, and graphs of maximal complexity are being described. Up to a positive scalar multiple the Laplacian (or Kirchhoff) matrix concides with the information matrix that arises out of a statistical additive model with treatment and block effects. By way of the matrix-tree theorem design efficiency is shown to be closely related to graph complexity. Several open questions regarding graphs of maximal complexity are being brought to attention.     

Multifractals,Mumford curves and eternal inflation

We relate the Eternal Symmetree model of Harlow, Shenker, Stanford, and Susskind to constructions of stochastic processes related to quantum statistical mechanical systems on Cuntz-Krieger algebras. We extend the eternal inflation model from the Bruhat-Tits tree to quotients by p-adic Schottky groups, again using quantum statistical mechanics on graph algebras.     

Left and right convergence of graphs with bounded degree

The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence (left‐convergence), or counting homomorphisms into fixed graphs (right‐convergence). Under appropriate conditions, these two ways of defining convergence was proved to be equivalent in the dense case by Borgs, Chayes, Lovász, Sós and Vesztergombi. In this paper a similar equivalence is established in the bounded degree case, if the set of graphs in the definition of right‐convergence is appropriately restricted. In terms of statistical physics, the implication that left convergence implies right convergence means that for a left‐convergent sequence, partition functions of a large class of statistical physics models converge. The proof relies on techniques from statistical physics, like cluster expansion and Dobrushin Uniqueness. © 2012 Wiley Periodicals, Inc. Random Struct. 2012     

Wigner-Yanase skew information vs. quantum Fisher information

Among concepts describing the information contents of quantum mechanical density operators, both the Wigner-Yanase skew information and the quantum Fisher information defined via symmetric logarithmic derivatives are natural generalizations of the classical Fisher information. We will establish a relationship between these two fundamental quantities and show that they are comparable.

     

Graph structural properties of non-Yutsis graphs allowing fast recognition

Yutsis graphs are connected simple graphs which can be partitioned into two vertex-induced trees. Cubic Yutsis graphs were introduced by Jaeger as cubic dual Hamiltonian graphs, and these are our main focus.Cubic Yutsis graphs also appear in the context of the quantum theory of angular momenta, where they are used to generate summation formulae for general recoupling coefficients. Large Yutsis graphs are of interest for benchmarking algorithms which generate these formulae.In an earlier paper we showed that the decision problem of whether a given cubic graph is Yutsis is NP-complete. We also described a heuristic that was tested on graphs with up to 300,000 vertices and found Yutsis decompositions for all large Yutsis graphs very quickly.In contrast, no fast technique was known by which a significant fraction of bridgeless non-Yutsis cubic graphs could be shown to be non-Yutsis. One of the contributions of this article is to describe some structural impediments to Yutsisness. We also provide experimental evidence that almost all non-Yutsis cubic graphs can be rapidly shown to be non-Yutsis by applying a heuristic based on some of these criteria. Combined with the algorithm described in the earlier paper this gives an algorithm that, according to experimental evidence, runs efficiently on practically every large random cubic graph and can decide on whether the graph is Yutsis or not.The second contribution of this article is a set of construction techniques for non-Yutsis graphs implying, for example, the existence of 3-connected non-Yutsis cubic graphs of arbitrary girth and with few non-trivial 3-cuts.