Masamichi Miyaji

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:

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).

Surface/State Correspondence as a Generalized Holography

We propose a new duality relation between codimension two space-like surfaces in gravitational theories and quantum states in dual Hilbert spaces. This surface/state correspondence largely generalizes the idea of holography such that we do not need to rely on any existence of boundaries in gravitational spacetimes. The present idea is motivated by the recent interpretation of AdS/CFT in terms of the tensor networks so called MERA. Moreover, we study this correspondence from the viewpoint of entanglement entropy and information metric. The Cramer-Rao bound in quantum estimation theory implies that the quantum fluctuations of radial coordinate of the AdS is highly suppressed in the large N limit.

Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT

A bstractWe propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.

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.

Boundary states as holographic duals of trivial spacetimes

A bstractWe study real-space quantum entanglement included in conformally invariant boundary states in conformal field theories (CFTs). We argue that boundary states essentially have no real-space entanglement, except for constant contributions from long range topological entanglement, by computing the entanglement entropy when the system is bipartition into two spatial regions. From the viewpoint of classical gravity duals in holography, this shows that boundary states are dual to trivial spacetimes of zero space-time volume. We also point out that a continuous multiscale entanglement renormalization ansatz (cMERA) for any CFTs can be formulated by employing a boundary state as its infrared unentangled state with an appropriate regularization. Exploiting this idea, we propose an approximation scheme of cMERA construction for general CFTs.

From path integrals to tensor networks for the AdS/CFT correspondence

In this paper, we discuss tensor network descriptions of

Path-integral complexity for perturbed CFTs

\mathrm{AdS}/\mathrm{CFT}

Causal evolutions of bulk local excitations from CFT

from two different viewpoints. First, we start with a Euclidean path-integral computation of ground state wave functions with a UV cutoff. We consider its efficient optimization by making its UV cutoff position dependent and define a quantum state at each length scale. We conjecture that this path integral corresponds to a time slice of anti--de Sitter (AdS) spacetime. Next, we derive a flow of quantum states by rewriting the action of Killing vectors of

Butterflies from information metric

{\mathrm{AdS}}_{3}