Tokiro Numasawa

Holographic local quenches and entanglement density

A bstractWe propose a free falling particle in an AdS space as a holographic model of local quench. Local quenches are triggered by local excitations in a given quantum system. We calculate the time-evolution of holographic entanglement entropy. We confirm a logarithmic time-evolution, which is known to be typical in two dimensional local quenches. To study the structure of quantum entanglement in general quantum systems, we introduce a new quantity which we call entanglement density and apply this analysis to quantum quenches. We show that this quantity is directly related to the energy density in a small size limit. Moreover, we find a simple relationship between the amount of quantum information possessed by a massive object and its total energy based on the AdS/CFT.

Dynamics of Entanglement Entropy from Einstein Equation

We study the dynamics of entanglement entropy for weakly excited states in conformal field theories by using anti-de Sitter/conformal field theory (AdS/CFT). This is aimed at a first step to finding a counterpart of the Einstein equation in CFT language. In particular, we point out that the entanglement entropy satisfies differential equations that directly correspond to the Einstein equation in several setups of AdS/CFT. We also define a quantity called entanglement density in higher dimensional field theories and study its dynamical property for weakly excited states in conformal field theories.

Quantum Entanglement of Local Operators in Conformal Field Theories

We introduce a series of quantities which characterize a given local operator in any conformal field theory from the viewpoint of quantum entanglement. It is defined by the increased amount of (Rényi) entanglement entropy at late time for an excited state defined by acting the local operator on the vacuum. We consider a conformal field theory on an infinite space and take the subsystem in the definition of the entanglement entropy to be its half. We calculate these quantities for a free massless scalar field theory in two, four and six dimensions. We find that these results are interpreted in terms of quantum entanglement of a finite number of states, including Einstein-Podolsky-Rosen states. They agree with a heuristic picture of propagations of entangled particles.

Distance between Quantum States and Gauge-Gravity Duality

We study a quantum information metric (or fidelity susceptibility) in conformal field theories with respect to a small perturbation by a primary operator. We argue that its gravity dual is approximately given by a volume of maximal time slice in an anti-de Sitter spacetime when the perturbation is exactly marginal. We confirm our claim in several examples.Introduction The microscopic understanding of black hole entropy in string theory by Strominger and Vafa [1] implies that quantum information plays a crucial role to understand gravitational aspects of string theory. Indeed, quantum information theoretic considerations have provided various useful viewpoints in studies of AdS/CFT [2] or more generally holography [3]. Especially, the idea of quantum entanglement has turned out to be crucially involved in geometries of holographic spacetimes, as typical in the non-trivial topology of eternal black holes [4]. To quantify quantum entanglement we can study the holographic entanglement entropy [5], which is given by the area of codimension two extremal surfaces. In the AdS/CFT, this area is equal to the entanglement entropy in conformal field theories (CFTs). It is natural to wonder if there might be some other information theoretic quantities which are useful to develop studies of holography. As pointed out by Susskind in [6] (see also [7]), it is also intriguing to find a quantity in CFTs which is dual to a volume of a codimension one time slice in AdS. The time slice can connect two boundaries dual to the thermofield doubled CFTs, through the Einstein-Rosen bridge (see Fig.1). In [6], it is conjectured that this quantity is related to a measure of complexity. The main purpose of this letter is to point out a quantum information theoretic quantity which is related to the volume of a time slice. This quantity is called quantum information metric or Bures metric (see e.g.[8]), which we will simply call the information metric. Here we mainly consider the information metric for pure states, though it can be defined for mixed states. Consider one parameter family of quantum states |Ψ(λ)〉 and perturb λ infinitesimally λ → λ+δλ. Then Gλλ is simply defined from the inner product between them as follows:

Quantum dimension as entanglement entropy in two dimensional conformal field theories

Song He, Tokiro Numasawa, Tadashi Takayanagi and Kento Watanabe Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, P. R. China and Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8582, Japan (Dated: March 5, 2014)

Continuous Multiscale Entanglement Renormalization Ansatz as Holographic Surface-State Correspondence

We present how the surface/state correspondence, conjectured in arXiv:1503.03542, works in the setup of AdS3/CFT2 by generalizing the formulation of cMERA. The boundary states in conformal field theories play a crucial role in our formulation and the bulk diffeomorphism is naturally taken into account. We give an identification of bulk local operators which reproduces correct scalar field solutions on AdS3 and bulk scalar propagators. We also calculate the information metric for a locally excited state and show that it is given by that of 2d hyperbolic manifold, which is argued to describe the time slice of AdS3.We present how the surface-state correspondence, conjectured by Miyaji and Takayanagi, works in the setup of AdS(3)/CFT(2) by generalizing the formulation of a continuous multiscale entanglement renormalization group ansatz. The boundary states in conformal field theories play a crucial role in our formulation and the bulk diffeomorphism is naturally taken into account. We give an identification of bulk local operators which reproduces correct scalar field solutions on AdS(3) and bulk scalar propagators. We also calculate the information metric for a locally excited state and show that it reproduces the time slice of AdS(3).

On the definition of entanglement entropy in lattice gauge theories

A bstractWe focus on the issue of proper definition of entanglement entropy in lattice gauge theories, and examine a naive definition where gauge invariant states are viewed as elements of an extended Hilbert space which contains gauge non-invariant states as well. Working in the extended Hilbert space, we can define entanglement entropy associated with an arbitrary subset of links, not only for abelian but also for non-abelian theories. We then derive the associated replica formula. We also discuss the issue of gauge invariance of the entanglement entropy. In the ZN gauge theories in arbitrary space dimensions, we show that all the standard properties of the entanglement entropy, e.g. the strong subadditivity, hold in our definition. We study the entanglement entropy for special states, including the topological states for the ZN gauge theories in arbitrary dimensions. We discuss relations of our definition to other proposals.

Notes on Entanglement Entropy in String Theory

A bstractIn this paper, we study the conical 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 conical entropy in closed superstring is UV finite owing to the string scale cutoff.

Quantum entanglement of fermionic local operators

A bstractIn this paper we study the time evolution of (Rényi) entanglement entropies for locally excited states in four dimensional free massless fermionic field theory. Locally excited states are defined by being acted by various local operators on the ground state. Their excesses are defined by subtracting (Rényi) entanglement entropy for the ground state from those for locally excited states. They finally approach some constant if the subsystem is given by half of the total space. They have spin dependence. They can be interpreted in terms of quasi-particles.

EPR Pairs, Local Projections and Quantum Teleportation in Holography

A bstractIn 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.