arXiv subjects

Bergfinnur Durhuus

Publications and source records attributed to Bergfinnur Durhuus.

At least 19 recordsLinked to original sources

arXiv2025

Fundamental Limit on the Power of Entanglement Assistance in Quantum Communication

The optimal rate of reliable communication over a quantum channel can be enhanced by pre-shared entanglement. Whereas the enhancement may be unbounded in infinite-dimensional settings even when the input power is constrained, a long-standing conjecture asserts that the ratio between the entanglement-assisted and unassisted classical capacities is bounded in finite-dimensional settings [Bennett et al., IEEE Trans. Inf. Theory 48, 2637 (2002)]. In this work, we prove this conjecture by showing that their ratio is upper bounded by $o(d^2)$, where $d$ is the input dimension of the channel. An application to quantum communication with noisy encoders and decoders is given.

quant-ph↗

arXiv2024

The Spectrum of Asymptotic Cayley Trees

We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a Cayley tree and, in general, a non-empty pure point spectrum. We apply our results to studying continuous time quantum walk on these graphs. If the pure point spectrum is nonempty the walk is in general confined with a nonzero probability.

quant-ph↗

arXiv2024

Tip of the Quantum Entropy Cone

Relations among von Neumann entropies of different parts of an $N$-partite quantum system have direct impact on our understanding of diverse situations ranging from spin systems to quantum coding theory and black holes. Best formulated in terms of the set $Σ^*_N$ of possible vectors comprising the entropies of the whole and its parts, the famous strong subaddivity inequality constrains its closure $\overlineΣ^*_N$, which is a convex cone. Further homogeneous constrained inequalities are also known. In this work we provide (non-homogeneous) inequalities that constrain $Σ_N^*$ near the apex (the vector of zero entropies) of $\overlineΣ^*_N$, in particular showing that $Σ_N^*$ is not a cone for $N\geq 3$. Our inequalities apply to vectors with certain entropy constraints saturated and, in particular, they show that while it is always possible to up-scale an entropy vector to arbitrary integer multiples it is not always possible to down-scale it to arbitrarily small size, thus answering a question posed by A. Winter. Relations of our work to topological materials, entanglement theory, and quantum cryptography are discussed.

quant-ph↗

arXiv2022

From Trees to Gravity

In this article we study two related models of quantum geometry: generic random trees and two-dimensional causal triangulations. The Hausdorff and spectral dimensions that arise in these models are calculated and their relationship with the structure of the underlying random geometry is explored. Modifications due to interactions with matter fields are also briefly discussed. The approach to the subject is that of classical statistical mechanics and most of the tools come from probability and graph theory.

hep-th↗

arXiv2022

Local limits of one-sided trees

A finite \emph{one-sided tree} of height $h$ is defined as a rooted planar tree obtained by grafting branches on one side, say the right, of a spine, i.e. a linear path of length $h$ starting at the root, such that the resulting tree has no simple path starting at the root of length greater than $h$. We consider the distribution $τ_N$ on the set of one-sided trees $T$ of fixed size $N$, such that the weight of $T$ is proportional to $e^{-μh(T)}$, where $μ$ is a real constant and $h(T)$ denotes the height of $T$. We show that, for $N$ large, $τ_N$ has a weak limit as a probability measure supported on infinite one-sided trees. The dependence of the limit measure $τ$ on $μ$ shows a transition at $μ_0=-\ln 2$ from a single spine phase for $μ\leq μ_0$ to a multi-spine phase for $μ> μ_0$. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for $μ<μ_0$, to quadratic growth at $μ=μ_0$, and to qubic growth for $μ> μ_0$.

math.PR↗

arXiv2022

Trees with exponential height dependent weight

We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is of exponential form $e^{-μh}$. Defining the total weight for such trees of fixed size to be $Z^{(μ)}_N$, we determine its asymptotic behaviour for large $N$, for arbitrary real values of $μ$. Based on this we evaluate the local limit of the corresponding probability measures and find a transition at $μ=0$ from a single spine phase to a multi-spine phase. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for $μ<0$ to the familiar quadratic growth at $μ=0$ and to cubic growth for $μ> 0$.

math.PR↗

arXiv2022

Trees with power-like height dependent weight

We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is given by a power function $h^α$. Defining the total weight for such trees of fixed size to be $Z_N$, a detailed analysis of the analyticity properties of the corresponding generating function is provided. Based on this, we determine the asymptotic form of $Z_N$ and show that the local limit at large size is identical to the Uniform Infinite Planar Tree, independent of the exponent $α$ of the height distribution function.

math.PR↗

arXiv2021

Critical behaviour of loop models on causal triangulations

We introduce a dense and a dilute loop model on causal dynamical triangulations. Both models are characterised by a geometric coupling constant $g$ and a loop parameter $α$ in such a way that the purely geometric causal triangulation model is recovered for $α=1$. We show that the dense loop model can be mapped to a solvable planar tree model, whose partition function we compute explicitly and use to determine the critical behaviour of the loop model. The dilute loop model can likewise be mapped to a planar tree model; however, a closed-form expression for the corresponding partition function is not obtainable using the standard methods employed in the dense case. Instead, we derive bounds on the critical coupling $g_c$ and apply transfer matrix techniques to examine the critical behaviour for $α$ small.

hep-th↗

arXiv2021

The structure of spatial slices of three-dimensional causal triangulations

We consider causal 3-dimensional triangulations with the topology of $S^2\times [0,1]$ or $D^2\times [0,1]$ where $S^2$ and $D^2$ are the two-dimensional sphere and disc, respectively. These triangulations consist of slices and we show that these slices can be mapped bijectively onto a set of certain coloured two-dimensional cell complexes satisfying simple conditions. The cell complexes arise as the cross section of the individual slices.

math-ph↗

arXiv2020

Recursion relations for chromatic coefficients for graphs and hypergraphs

We establish a set of recursion relations for the coefficients in the chromatic polynomial of a graph or a hypergraph. As an application we provide a generalization of Whitney's broken cycle theorem for hypergraphs, as well as deriving an explicit formula for the linear coefficient of the chromatic polynomial of the $r$-complete hypergraph in terms of roots of the Taylor polynomials for the exponential function.

math.CO↗

arXiv2014

Expoential bounds on the number of causal triangulations

We prove that the number of combinatorially distinct causal 3-dimensional triangulations homeomorphic to the 3-dimensional sphere is bounded by an exponential function of the number of tetrahedra. It is also proven that the number of combinatorially distinct causal 4-dimensional triangulations homeomorphic to the 4-sphere is bounded by an exponential function of the number of 4-simplices provided the number of all combinatorially distinct triangulations of the 3-sphere is bounded by an exponential function of the number of tetrahedra.

math-ph↗

arXiv2012

Generic Ising Trees

The Ising model on an infinite generic tree is defined as a thermodynamic limit of finite systems. A detailed description of the corresponding distribution of infinite spin configurations is given. As an application we study the magnetization properties of such systems and prove that they exhibit no spontaneous magnetization. Furthermore, the values of the Hausdorff and spectral dimensions of the underlying trees are calculated and found to be, respectively, $\bar{d}_h=2$ and $\bar{d}_s=4/3$.

cond-mat.stat-mech↗

arXiv2011

Star product realizations of kappa-Minkowski space

We define a family of star products and involutions associated with $κ$-Minkowski space. Applying corresponding quantization maps we show that these star products restricted to a certain space of Schwartz functions have isomorphic Banach algebra completions. For two particular star products it is demonstrated that they can be extended to a class of polynomially bounded smooth functions allowing a realization of the full Hopf algebra structure on $κ$-Minkowski space. Furthermore, we give an explicit realization of the action of the $κ$-Poincaré algebra as an involutive Hopf algebra on this representation of $κ$-Minkowski space and initiate a study of its properties.

math-ph↗

arXiv2010

The scattering problem for a noncommutative nonlinear Schrödinger equation

We investigate scattering properties of a Moyal deformed version of the nonlinear Schrödinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and also has soliton solutions if the interaction potential is suitably chosen. We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave operators defined for small scattering data in the general case and for arbitrary scattering data in the rotationally symmetric case.

math-ph↗

arXiv2010

On the spectral dimension of causal triangulations

We introduce an ensemble of infinite causal triangulations, called the uniform infinite causal triangulation, and show that it is equivalent to an ensemble of infinite trees, the uniform infinite planar tree. It is proved that in both cases the Hausdorff dimension almost surely equals 2. The infinite causal triangulations are shown to be almost surely recurrent or, equivalently, their spectral dimension is almost surely less than or equal to 2. We also establish that for certain reduced versions of the infinite causal triangulations the spectral dimension equals 2 both for the ensemble average and almost surely. The triangulation ensemble we consider is equivalent to the causal dynamical triangulation model of two-dimensional quantum gravity and therefore our results apply to that model.

math-ph↗

arXiv2009

Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length

We consider pyramids made of one-dimensional pieces of fixed integer length a and which may have pairwise overlaps of integer length from 1 to a. We prove that the number of pyramids of size m, i.e. consisting of m pieces, equals (am-1,m-1) for each a >= 2. This generalises a well known result for a = 2. A bijective correspondence between so-called right (or left) pyramids and a-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids is proportional to the square root of the size.

math.CO↗

arXiv2006

The spectral dimension of generic trees

We define generic ensembles of infinite trees. These are limits as $N\to\infty$ of ensembles of finite trees of fixed size $N$, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension $d_h =2$. Our main result is that their spectral dimension is $d_s=4/3$, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is 1/3.

math-ph↗

arXiv2006

Local limit of labeled trees and expected volume growth in a random quadrangulation

Exploiting a bijective correspondence between planar quadrangulations and well-labeled trees, we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume. The limit random surface can be described in terms of a birth and death process and a sequence of multitype Galton--Watson trees. As a consequence, we find that the expected volume of the ball of radius $r$ around a marked point in the limit random surface is $Θ(r^4)$.

math.PR↗