A role of constraint in self-organization

In this paper we introduce a neural network model of self-organization. This model uses a variation of Hebb rule for updating its synaptic weights, and surely converges to the equilibrium status. The key point of the convergence is the update rule that constrains the total synaptic weight and this seems to make the model stable. We investigate the role of the constraint and show that it is the constraint that makes the model stable. For analyzing this setting, we propose a simple probabilistic game that models the neural network and the self-organization process. Then, we investigate the characteristics of this game, namely, the probability that the game becomes stable and the number of the steps it takes.Comment: To appear in the Proc. RANDOM'98, Oct. 199

Notes on Entanglement Entropy in String Theory

In this paper, we study the entanglement entropy in string theory in the simplest setup of dividing the nine dimensional space into two halves. This corresponds to the leading quantum correction to the horizon entropy in string theory on the Rindler space. This entropy is also called the conical entropy and includes surface term contributions. We first derive a new simple formula of the conical entropy for any free higher spin fields. Then we apply this formula to computations of conical entropy in open and closed superstring. In our analysis of closed string, we study the twisted conical entropy defined by making use of string theory on Melvin backgrounds. This quantity is easier to calculate owing to the folding trick. Our analysis shows that the entanglement entropy in closed superstring is UV finite owing to the string scale cutoff.Comment: 27 pages, no figures, latex, v2: typos corrected, references adde

EPR Pairs, Local Projections and Quantum Teleportation in Holography

In this paper we analyze three quantum operations in two dimensional conformal field theories (CFTs): local projection measurements, creations of partial entanglement between two CFTs, and swapping of subsystems between two CFTs. We also give their holographic duals and study time evolutions of entanglement entropy. By combining these operations, we present an analogue of quantum teleportation between two CFTs and give its holographic realization. We introduce a new quantity to probe tripartite entanglement by using local projection measurement.Comment: 61 pages, 24 figures. v2: comments and refs added. v3: minor correction

Quantum Dimension as Entanglement Entropy in 2D CFTs

We study entanglement entropy of excited states in two dimensional conformal field theories (CFTs). Especially we consider excited states obtained by acting primary operators on a vacuum. We show that under its time evolution, entanglement entropy increases by a finite constant when the causality condition is satisfied. Moreover, in rational CFTs, we prove that this increased amount of (both Renyi and von-Neumann) entanglement entropy always coincides with the log of quantum dimension of the primary operator.Comment: 5 pages, 3 eps figures, Revte

Out-of-Time-Ordered Correlators in (T2)n/Zn(T^2)^n/\mathbb{Z}_n

In this note we continue analysing the non-equilibrium dynamics in the $(T^2)^n/\mathbb{Z}_n$ orbifold conformal field theory. We compute the out-of-time-ordered four-point correlators with twist operators. For rational $\eta \ (=p/q)$ which is the square of the compactification radius, we find that the correlators approach non-trivial constants at late time. For $n=2$ they are expressed in terms of the modular matrices and for higher $n$ orbifolds are functions of $pq$ and $n$. For irrational $\eta$, we find a new polynomial decay of the correlators that is a signature of an intermediate regime between rational and chaotic models.Comment: 20 pages, 3 figure

Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories

We introduce a new optimization procedure for Euclidean path integrals which compute wave functionals in conformal field theories (CFTs). We optimize the background metric in the space on which the path integration is performed. Equivalently this is interpreted as a position-dependent UV cutoff. For two-dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space and we interpret this as a continuous limit of the conjectured relation between tensor networks and Anti--de Sitter (AdS)/conformal field theory (CFT) correspondence. We confirm our procedure for excited states, the thermofield double state, the Sachdev-Ye-Kitaev model and discuss its extension to higher-dimensional CFTs. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.Comment: 7 pages, Revtex, 2 figures, Version 2 : The version published in PRL, title expanded and typos correcte