4,810 research outputs found
Characterizing the quantum field theory vacuum using temporal Matrix Product states
In this paper we construct the continuous Matrix Product State (MPS)
representation of the vacuum of the field theory corresponding to the
continuous limit of an Ising model. We do this by exploiting the observation
made by Hastings and Mahajan in [Phys. Rev. A \textbf{91}, 032306 (2015)] that
the Euclidean time evolution generates a continuous MPS along the time
direction. We exploit this fact, together with the emerging Lorentz invariance
at the critical point in order to identify the matrix product representation of
the quantum field theory (QFT) vacuum with the continuous MPS in the time
direction (tMPS). We explicitly construct the tMPS and check these statements
by comparing the physical properties of the tMPS with those of the standard
ground MPS. We furthermore identify the QFT that the tMPS encodes with the
field theory emerging from taking the continuous limit of a weakly perturbed
Ising model by a parallel field first analyzed by Zamolodchikov.Comment: The results presented in this paper are a significant expansion of
arXiv:1608.0654
Tensor Network Models of Unitary Black Hole Evaporation
We introduce a general class of toy models to study the quantum
information-theoretic properties of black hole radiation. The models are
governed by a set of isometries that specify how microstates of the black hole
at a given energy evolve to entangled states of a tensor product
black-hole/radiation Hilbert space. The final state of the black hole radiation
is conveniently summarized by a tensor network built from these isometries. We
introduce a set of quantities generalizing the Renyi entropies that provide a
complete set of bipartite/multipartite entanglement measures, and give a
general formula for the average of these over initial black hole states in
terms of the isometries defining the model. For models where the dimension of
the final tensor product radiation Hilbert space is the same as that of the
space of initial black hole microstates, the entanglement structure is
universal, independent of the choice of isometries. In the more general case,
we find that models which best capture the "information-free" property of black
hole horizons are those whose isometries are tensors corresponding to states of
tripartite systems with maximally mixed subsystems.Comment: 22 pages, 4 figure
Spectral Theory for Networks with Attractive and Repulsive Interactions
There is a wealth of applied problems that can be posed as a dynamical system
defined on a network with both attractive and repulsive interactions. Some
examples include: understanding synchronization properties of nonlinear
oscillator;, the behavior of groups, or cliques, in social networks; the study
of optimal convergence for consensus algorithm; and many other examples.
Frequently the problems involve computing the index of a matrix, i.e. the
number of positive and negative eigenvalues, and the dimension of the kernel.
In this paper we consider one of the most common examples, where the matrix
takes the form of a signed graph Laplacian. We show that the there are
topological constraints on the index of the Laplacian matrix related to the
dimension of a certain homology group. In certain situations, when the homology
group is trivial, the index of the operator is rigid and is determined only by
the topology of the network and is independent of the strengths of the
interactions. In general these constraints give upper and lower bounds on the
number of positive and negative eigenvalues, with the dimension of the homology
group counting the number of eigenvalue crossings. The homology group also
gives a natural decomposition of the dynamics into "fixed" degrees of freedom,
whose index does not depend on the edge-weights, and an orthogonal set of
"free" degrees of freedom, whose index changes as the edge weights change. We
also present some numerical studies of this problem for large random matrices.Comment: 27 pages; 9 Figure
The de Soto Effect
This paper explores the consequences of creating and improving property rights so thatfixed assets can be used as collateral. This has become a cause célèbre of Hernando de Sotowhose views are influential in debates about policy reform concerning property rights.Hence, we refer to the economic impact of such reforms as the de Soto effect. We explore thelogic of the argument for credit contracts, both in isolation, and in market equilibrium. Weshow that the impact will vary with the degree of market competition. Where competition isweak, it is possible that borrowers will be worse off when property rights improve. Wediscuss the implications for optimal policy and the political economy of policy reform.
Calibrating Generative Models: The Probabilistic Chomsky-Schützenberger Hierarchy
A probabilistic Chomsky–Schützenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning
Embedding and approximation theorems for echo state networks
Echo State Networks (ESNs) are a class of single layer recurrent neural
networks that have enjoyed recent attention. In this paper we prove that a
suitable ESN, trained on a series of measurements of an invertible dynamical
system, induces a C1 map from the dynamical system's phase space to the ESN's
reservoir space. We call this the Echo State Map. We then prove that the Echo
State Map is generically an embedding with positive probability. Under
additional mild assumptions, we further conjecture that the Echo State Map is
almost surely an embedding. For sufficiently large, and specially structured,
but still randomly generated ESNs, we prove that there exists a linear readout
layer that allows the ESN to predict the next observation of a dynamical system
arbitrarily well. Consequently, if the dynamical system under observation is
structurally stable then the trained ESN will exhibit dynamics that are
topologically conjugate to the future behaviour of the observed dynamical
system. Our theoretical results connect the theory of ESNs to the
delay-embedding literature for dynamical systems, and are supported by
numerical evidence from simulations of the traditional Lorenz equations. The
simulations confirm that, from a one dimensional observation function, an ESN
can accurately infer a range of geometric and topological features of the
dynamics such as the eigenvalues of equilibrium points, Lyapunov exponents and
homology groups.Comment: 24 pages, 9 figure
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