LaTeX2e 16 pages, 2 figures, to be published in Frontiers in Physics Research Topic, Qubits and Spacetime. Revised version has new figures, more explanation of HST formalism and Jacobson's derivation of Einstein equations, and added references
The formalism of Holographic Space-time (HST) is a translation of the principles of Lorentzian geometry into the language of quantum information. Intervals along time-like trajectories, and their associated causal diamonds, completely characterize a Lorentzian geometry. The Bekenstein-Hawking-Gibbons-'t Hooft-Jacobson-Fischler-Susskind-Bousso Covariant Entropy Principle, equates the logarithm of the dimension of the Hilbert space associated with a diamond to one quarter of the area of the diamond's holographic screen, measured in Planck units. The most convincing argument for this principle is Jacobson's derivation of Einstein's equations as the hydrodynamic expression of this entropy law. In that context, the null energy condition (NEC) is seen to be the analog of the local law of entropy increase. The quantum version of Einstein's relativity principle is a set of constraints on the mutual quantum information shared by causal diamonds along different time-like trajectories. The implementation of this constraint for trajectories in relative motion is the greatest unsolved problem in HST. The other key feature of HST is its claim that, for non-negative cosmological constant or causal diamonds much smaller than the asymptotic radius of curvature for negative c.c., the degrees of freedom localized in the bulk of a diamond are constrained states of variables defined on the holographic screen. This principle gives a simple explanation of otherwise puzzling features of BH entropy formulae, and resolves the firewall problem for black holes in Minkowski space. It motivates a covariant version of the CKN\cite{ckn} bound on the regime of validity of quantum field theory (QFT) and a detailed picture of the way in which QFT emerges as an approximation to the exact theory.
In this paper, we will analyze the connection between the fidelity susceptibility, the holographic complexity and the thermodynamic volume. We will regularize the fidelity susceptibility and the holographic complexity by subtracting the contribution of the background AdS spacetime from the deformation of the AdS spacetime. It will be demonstrated that this regularized fidelity susceptibility has the same behavior as the thermodynamic volume and that the regularized complexity has a very different behavior. As the information dual to different volumes in the bulk would be measured by the fideli…
The maximum entropy that can be stored in a bounded region of space is in dispute: it goes as volume, implies (non-gravitational) microphysics; it goes as the surface area, asserts the "holographic principle." Here I show how the holographic bound can be derived from elementary flat-spacetime quantum field theory when the total energy of Fock states is constrained gravitationally. This energy constraint makes the Fock space dimension (whose logarithm is the maximum entropy) finite for both Bosons and Fermions. Despite the elementary nature of my analysis, it results in an upper limit on entrop…
We define bulk/boundary maps corresponding to quantum gravity states in the tensorial group field theory formalism, for quantum geometric models sharing the same type of quantum states of loop quantum gravity. The maps are defined in terms of a partition of the quantum geometric data associated to an open graph into bulk and boundary ones, in the spin representation. We determine the general condition on the entanglement structure of the state that makes the bulk/boundary map isometric (a necessary condition for holographic behaviour), and we analyse different types of quantum states, identify…
Certain solutions of Einstein's equations in anti-de Sitter spacetime can be engineered, using extended gravitational thermodynamics, to yield `holographic heat engines', devices that turn heat into useful mechanical work. On the other hand, there are constructions (both experimental and theoretical) where a series of operations is performed on a small quantum system, defining what are known as `quantum heat engines'. We propose that certain holographic heat engines can be considered models of quantum heat engines, and the possible fruitfulness of this connection is discussed. Motivated by fea…
We argue that the flat spacetime with inexact quantum mechanics in it is dual to the de Sitter spacetime with exact quantum mechanics in it, and the positive cosmological constant of this de Sitter spacetime is in the second order of the degree of the violation of the bulk quantum mechanics in the flat spacetime. The flat spacetime is holographic and has a dual time-contracted boundary conformal field theory with two redefined central charges at null infinity. The vanishing smallness of the observed positive cosmological constant suggests the extraordinary exactness of the bulk quantum mechani…
The relation between Loop Quantum Gravity (LQG) and tensor network is explored from the perspectives of bulk-boundary duality and holographic entanglement entropy. We find that the LQG spin-network states in a space $Σ$ with boundary $\partialΣ$ is an exact holographic mapping similar to the proposal in arXiv:1309.6282. The tensor network, understood as the boundary quantum state, is the output of the exact holographic mapping emerging from a coarse graining procedure of spin-networks. Furthermore, when a region $A$ and its complement $\bar{A}$ are specified on the boundary $\partialΣ$, we sho…
Spacetime foam, also known as quantum foam, has its origin in quantum fluctuations of spacetime. Arguably it is the source of the holographic principle, which severely limits how densely information can be packed in space. Its physics is also intimately linked to that of black holes and computation. In particular, the same underlying physics is shown to govern the computational power of black hole quantum computers.
Kerr-Schild (KS) geometry is based on a congruence of twistor null lines which forms a holographic space-time determined by the Kerr theorem. We describe in details integration of the non-stationary Debney-Kerr-Schild equations for electromagnetic excitations of black-holes taking into account the consistent back-reaction to metric. The exact KS solutions have the form of singular beam-like pulses supported on twistor null lines of the Kerr congruence. These twistor-beam pulses have very strong back reaction to metric and BH horizon and produce a fluctuating holographic KS geometry which takes…
The holographic state of matter exists in the quantum gravitational regime, with black holes as the example. In this essay, we provide a microscopic derivation to the holographic thermodynamics and further discuss its implications for quantum gravity. Especially, we establish the link between holographic entropy and dimensional reduction. It seems that the fundamental physics behind black holes and the very early universe is $1+1$ dimensional.
We study the Bell inequality in a holographic model of the casually disconnected Einstein-Podolsky-Rosen (EPR) pair. The Clauser-Horne-Shimony-Holt(CHSH) form of Bell inequality is constructed using holographic Schwinger-Keldysh (SK) correlators. We show that the manifestation of quantum correlation in Bell inequality can be holographically reproduced from the classical fluctuations of dual accelerating string in the bulk gravity. The violation of this holographic Bell inequality supports the essential quantum property of this holographic model of an EPR pair.
It was recently suggested that quantum mechanics and gravity are not fundamental but emerge from information loss at causal horizons. On the basis of the formalism the holographic principle is also shown to arise naturally from the loss of information about bulk fields observed by an outside observer. As an application, Witten's prescription is derived.
We study Renyi entropies $S_n$ in quantum error correcting codes and compare the answer to the cosmic brane prescription for computing $\widetilde{S}_n \equiv n^2 \partial_n (\frac{n-1}{n} S_n)$. We find that general operator algebra codes have a similar, more general prescription. Notably, for the AdS/CFT code to match the specific cosmic brane prescription, the code must have maximal entanglement within eigenspaces of the area operator. This gives us an improved definition of the area operator, and establishes a stronger connection between the Ryu-Takayanagi area term and the edge modes in l…
We compute the time evolution of the mutual information in out of equilibrium quantum systems whose gravity duals are Vaidya spacetimes in three and four dimensions, which describe the formation of a black hole through the collapse of null dust. We find the holographic mutual information to be non monotonic in time and always monogamous in the ranges explored. We also find that there is a region in the configuration space where it vanishes at all times. We show that the null energy condition is a necessary condition for both the strong subadditivity of the holographic entanglement entropy and …
The holographic complexity and fidelity susceptibility have been defined as new quantities dual to different volumes in AdS. In this paper, we will use these new proposals to calculate both of these quantities for a variety of interesting deformations of AdS. We obtain the holographic complexity and fidelity susceptibility for an AdS black hole, Janus solution and a solution with cylindrically symmetry, an inhomogeneous background and a hyperscaling violating background. It is observed that the holographic complexity depends on the size of the subsystem for all these solutions and the fidelity…
From the viewpoint of local quantum field theory, this letter investigates the high-order corrections to the holographic entropy bound. As a result, the logarithmic correction term appears naturally with the definite coefficient $-{1/2}$, which thus provides another semi-classical platform to examine such candidates for quantum gravity as Loop Quantum Gravity and String Theory.
We study a conjectured connection between the AdS/CFT and a real-space quantum renormalization group scheme, the multi-scale entanglement renormalization ansatz (MERA). By making a close contact with the holographic formula of the entanglement entropy, we propose a general definition of the metric in the MERA in the extra holographic direction, which is formulated purely in terms of quantum field theoretical data. Using the continuum version of the MERA (cMERA), we calculate this emergent holographic metric explicitly for free scalar boson and free fermions theories, and check that the metric …
We propose a measure of holographic information based on a causal wedge construction. The motivation behind this comes from an attempt to understand how boundary field theories can holographically reconstruct spacetime. We argue that given the knowledge of the reduced density matrix in a spatial region of the boundary, one should be able to reconstruct at least the corresponding bulk causal wedge. In attempt to quantify the `amount of information' contained in a given spatial region in field theory, we consider a particular bulk surface (specifically a co-dimension two surface in the bulk spac…
We study the holographic mutual information in AdS4 of disjoint spatial domains in the boundary which are delimited by smooth closed curves. A numerical method which approximates a local minimum of the area functional through triangulated surfaces is employed. After some checks of the method against existing analytic results for the holographic entanglement entropy, we compute the holographic mutual information of equal domains delimited by ellipses, superellipses or the boundaries of two dimensional spherocylinders, finding also the corresponding transition curves along which the holographic …
In the article we present explicit expressions for quantum fluctuations of spacetime in the case of $(4+n)$-dimensional spacetimes, and consider their holographic properties and some implications for clocks, black holes and computation. We also consider quantum fluctuations and their holographic properties in ADD model and estimate the typical size and mass of the clock to be used in precise measurements of spacetime fluctuations. Numerical estimations of phase incoherence of light from extra-galactic sources in ADD model are also presented.
It is suggested that quantum entanglement emerges from the holographic principle stating that all of the information of a region (bulk bits) can be described by the bits on its boundary surface. There are redundancy and information loss in the bulk bits that lead to the nonlocal correlation among the bulk bits. Quantum field theory overestimates the independent degrees of freedom in the bulk. The maximum entanglement in the universe increases as the size of the cosmic horizon and this could be related with the arrow of time and dark energy.
We analyze different holographic complexity proposals for black holes that include corrections from bulk quantum fields. The specific setup is the quantum BTZ black hole, which encompasses in an exact manner the effects of conformal fields with large central charge in the presence of the black hole, including the backreaction corrections to the BTZ metric. Our results show that Volume Complexity admits a consistent quantum expansion and correctly reproduces known limits. On the other hand, the generalized Action Complexity picks up large contributions from the singularity, which is modified du…
We study the effect of anisotropy on holographic entanglement entropy (HEE) and holographic mutual information (MI) in the Q-lattice model, by exploring the HEE and MI for infinite strips along arbitrary directions. We find that the lattice always enhances the HEE. The MI, however, is enhanced by lattice for large sub-regions; while for small sub-regions, the MI is suppressed by the lattice. We also discuss how these phenomena result from the deformation of geometry caused by Q-lattices.
In this paper, by using Verlinde's formalism and a modified Padmanabhan's prescription, we have obtained the lowest order quantum correction to the gravitational acceleration and MOND-type theory by considering a nonzero difference between the number of bits of the holographic screen and number of bits of the holographic screen that satisfy the equipartition theorem. We will also carry out a phase transition and critical phenomena analysis in MOND-type theory where critical exponents are obtained.
We argue that classical spacetimes represent amplified information in the holographic theory of quantum gravity. In general, classicalization of a quantum system involves amplification of information at the cost of exponentially reducing the number of observables. In quantum gravity, the geometry of spacetime must be the analogously amplified information. Bulk local semiclassical operators probe this information without disturbing it; these correspond to logical operators acting on code subspaces of the holographic theory. From this viewpoint, we study how bulk local operators may be realized …
We present a fully quantum version of the holographic principle in terms of quantum systems, subsystems, and their interactions. We use the concept of environment induced decoherence to prove this principle. We discuss the conditions under which the standard (semi-classical) holographic principle is obtained from this quantum mechanical version.
We study the conjectured holographic duality between entanglement of purification and the entanglement wedge cross-section. We generalize both quantities and prove several information theoretic inequalities involving them. These include upper bounds on conditional mutual information and tripartite information, as well as a lower bound for tripartite information. These inequalities are proven both holographically and for general quantum states. In addition, we use the cyclic entropy inequalities to derive a new holographic inequality for the entanglement wedge cross-section, and provide numeric…
A new entropy bound, tighter than the standard holographic bound due to Bekenstein, is derived for spacetimes with non-rotating isolated horizons, from the quantum geometry approach in which the horizon is described by the boundary degrees of freedom of a three dimensional Chern Simons theory.
In this note, we establish the 2-D Quantum Inequalities - first proved by Flanagan - for all CFTs with a causal holographic dual. Following the treatment of Kelly \& Wall, we establish that the Boundary Causality Condition in an asymptotic AdS spacetime implies the Quantum Inequalities on the boundary. Our results extend easily to curved spacetime and are stable under deformations of the CFT by relevant operators. We discuss higher dimensional generalizations and possible connections to recent bounds on $a/c$ in 4-D CFTs.
We study the holographic quantum error correcting code properties of a Sierpinski Triangle-shaped boundary subregion in $AdS_4/CFT_3$. Due to existing no-go theorems in topological quantum error correction regarding fractal noise, this gives holographic codes a specific advantage over topological codes. We then further argue that a boundary subregion in the shape of the Sierpinski gasket in $AdS_5/CFT_4$ does not possess these holographic quantum error correction properties.
We use flat-space holography to calculate the mutual information and the 3-partite information of a two-dimensional BMS-invariant field theory (BMSFT$_2$). This theory is the putative holographic dual of the three-dimensional asymptotically flat spacetimes. We find a bound in which entangling transition occurs for zero and finite temperature BMSFTs. We also show that the holographic 3-partite information is always non-positive which indicates that the holographic mutual information is monogamous.
We present a line of research aimed at investigating holographic dualities in the context of three dimensional quantum gravity within finite bounded regions. The bulk quantum geometrodynamics is provided by the Ponzano-Regge state-sum model, which defines 3d quantum gravity as a discrete topological quantum field theory (TQFT). This formulation provides an explicit and detailed definition of the quantum boundary states, which allows a rich correspondence between quantum boundary conditions and boundary theories, thereby leading to holographic dualities between 3d quantum gravity and 2d statist…
We identify a special information-theoretic property of quantum field theories with holographic duals: the mutual informations among arbitrary disjoint spatial regions A,B,C obey the inequality I(A:BC) >= I(A:B)+I(A:C), provided entanglement entropies are given by the Ryu-Takayanagi formula. Inequalities of this type are known as monogamy relations and are characteristic of measures of quantum entanglement. This suggests that correlations in holographic theories arise primarily from entanglement rather than classical correlations. We also show that the Ryu-Takayanagi formula is consistent with…
The entanglement exhibits extremal or singular behavior near quantum critical points (QCPs) in many condensed matter models. These intriguing phenomena, however, still call for a widely accepted understanding. In this letter we study this issue in holographic framework. We investigate the connection between the holographic entanglement entropy (HEE) and the quantum phase transition (QPT) in a lattice-deformed Einstein-Maxwell-Dilaton theory. Novel backgrounds exhibiting metal-insulator transitions (MIT) have been constructed in which both metallic phase and insulating phase have vanishing entr…
We discuss, from a quantum information perspective, recent proposals of Maldacena, Ryu, Takayanagi, van Raamsdonk, Swingle, and Susskind that spacetime is an emergent property of the quantum entanglement of an associated boundary quantum system. We review the idea that the informational principle of minimal complexity determines a dual holographic bulk spacetime from a minimal quantum circuit U preparing a given boundary state from a trivial reference state. We describe how this idea may be extended to determine the relationship between the fluctuations of the bulk holographic geometry and the…
Quantum correlation beyond entanglement: Holographic discord and multipartite generalizations
Rikkyo University（立教大学）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
When the Lyapunov exponent $λ_L$ in a quantum chaotic system saturates the bound $λ_L\leqslant 2πk_BT$, it is proposed that this system has a holographic dual described by a gravity theory. In particular, the butterfly effect as a prominent phenomenon of chaos can ubiquitously exist in a black hole system characterized by a shockwave solution near the horizon. In this paper we propose that the butterfly velocity can be used to diagnose quantum phase transition (QPT) in holographic theories. We provide evidences for this proposal with an anisotropic holographic model exhibiting metal-insulator …
In quantum mechanics without application of any superselection rule to the set of the observables, a closed quantum system temporally evolves unitarily, and this Lorentzian regime is characterized by von Neumann entropy of exactly zero. In the holographic theory in the classicalized ground state, we argue that the unitary real-time evolution of a non-relativistic free particle with complex-valued quantum probability amplitude in this Lorentzian regime can be temporally analytically continued to an imaginary-time classical stochastic process with real-valued conditional probability density in t…
In earlier work, we studied holographic entanglement entropy in QCD phases using a dynamical Einstein-Maxwell-dilaton gravity model whose dual boundary theory mimics essential features of QCD above and below deconfinement. The model although displays subtle differences compared to the standard QCD phases, however, it introduces a notion of temperature in the phase below the deconfinement critical temperature and captures quite well the entanglement and thermodynamic properties of QCD phases. Here we extend our analysis to study the mutual and $n$-partite information by considering $n$ strips w…
We investigate the holographic entanglement entropy (HEE) of a strip geometry in four dimensional Q-lattice backgrounds, which exhibit metal-insulator transitions in the dual field theory. Remarkably, we find that the HEE always displays a peak in the vicinity of the quantum critical points. Our model provides the first direct evidence that the HEE can be used to characterize the quantum phase transition (QPT). We also conjecture that the maximization behavior of HEE at quantum critical points would be universal in general holographic models.
We study corner contributions to holographic mutual information for entangling regions composed of a set of disjoint sectors of a single infinite circle in three-dimensional conformal field theories. In spite of the UV divergence of holographic mutual information, it exhibits a first order phase transition. We show that tripartite information is also divergent for disjoint sectors, which is in contrast with the well-known feature of tripartite information being finite even when entangling regions share boundaries. We also verify the locality of corner effects by studying mutual information bet…
Recent progress in studies of holographic dualities, originally motivated by insights from string theory, has led to a confluence with concepts and techniques from quantum information theory. A particularly successful approach has involved capturing holographic properties by means of tensor networks which not only give rise to physically meaningful correlations of holographic boundary states, but also reproduce and refine features of quantum error correction in holography. This topical review provides an overview over recent successful realizations of such models. It does so by building on an …
In this paper, we briefly review the holographic dark energy model and introduce the idea that dark energy is a kind of thermal energy related to the quantum entanglement of the vacuum across a cosmic future event horizon. The holographic dark energy model comes from a theoretical attempt to apply the holographic principle to the dark energy problem and follows the idea that the short distance cut-off or ultraviolet (UV) cut-off is related to the long distance cut-off or infrared (IR) cut-off. The IR cut-off relevant to dark energy is the size of the future event horizon. This model gives a ho…
We investigate the monogamy of holographic $n$-partite information for a system consisting of $n$ disjoint parallel strips with the same width and separation in AdS and AdS black brane geometries. More precisely, we study the sign of this quantity, \emph{e.g.} for $n=4, 5$, in various dimensions and for different parameters. Our results show that for quantum field theories with holographic duals, the holographic 4-partite information is always positive and the sign of holographic 5-partite information is found to be negative in the dual strongly coupled $1+1$ dimensional CFT. This latter resul…
Holographic principle states that the maximum entropy of a system is its boundary area in Planck units. However, such a holographic entropy cannot be realized by the conventional quantum field theory. We need a new microscopic theory which naturally possesses all the holographic degrees of freedom. In this paper, we provide some preliminary thoughts on how to construct a theory with holographic degrees of freedom. It may shed light on the understanding of quantum properties of gravity and the early stage of the universe.
We study $1/N$ corrections to a Wilson loop in holographic duality. Extending the AdS/CFT correspondence beyond the large $N$ limit is an important but a subtle issue, as it needs quantum gravity corrections in the gravity side. To find a physical property of the quantum corrected geometry of near-horizon black 0-branes previously obtained by Hyakutake, we evaluate a Euclidean string worldsheet hanging down in the geometry, which corresponds to a rectangular Wilson loop in the $SU(N)$ quantum mechanics with 16 supercharges at a finite temperature with finite $N$. We find that the potential ene…
In AdS/CFT, the fine grained entropy of a boundary region is dual to the area of an extremal surface X in the bulk. It has been proposed that the area of a certain 'causal surface' C - i.e. the 'causal holographic information' (CHI) - corresponds to some coarse-grained entropy in the boundary theory. We construct two kinds of counterexamples that rule out various possible duals, using (1) vacuum rigidity and (2) thermal quenches. This includes the 'one-point entropy' proposed by Kelly and Wall, and a large class of related procedures. Also, any coarse-graining that fixes the geometry of the bu…
The formation of frozen classical perturbations from vacuum quantum fluctuations during inflation is described as a unitary quantum process with apparent "decoherence" caused by the expanding spacetime. It is argued that the maximum observable information content per comoving volume in classical modes is subject to the covariant entropy bound at the time those modes decohere, leading to a new quantitative bound on the information contained in frozen field modes in phase space. This bound implies holographic correlations of large-scale cosmological perturbations that may be observable.
In this paper, we study $(2+1)$-dimensional quantum anomalous Hall states, i.e. band insulators with quantized Hall conductance, using the exact holographic mapping. The exact holographic mapping is an approach to holographic duality which maps the quantum anomalous Hall state to a different state living in $(3+1)$-dimensional hyperbolic space. By studying topological response properties and the entanglement spectrum, we demonstrate that the holographic dual theory of a quantum anomalous Hall state is a $(3+1)$-dimensional topological insulator. The dual description enables a new characterizat…
We consider the path-sum of Ponzano-Regge with additional boundary contributions in the context of the holographic principle of Quantum Gravity. We calculate an holographic projection in which the bulk partition function goes to a semi-classical limit while the boundary state functional remains quantum-mechanical. The properties of the resulting boundary theory are discussed.
We reply to the comments by P.Midodashvili about our previous paper [1]. We argue that, contrary to the conclusions in Refs. [2,3], the Generalized Uncertainty Principle proposed by Ng and van Dam in Ref. [4] is compatible with the Holographic Principle in spacetimes with extra dimensions only for a very special (and somehow unrealistic) choice of the relation between the size and mass of the clock.
The recent proposal of Almheiri et al.http://arxiv.org/abs/1411.7041, together with the Ryu-Takayanagi formula, implies the entanglement wedge hypothesis for certain choices of boundary subregions. This fact is derived in the pure AdS space. A similar conclusion holds in the presence of quantum corrections, but in a more restricted domain of applicability. We also comment on http://arxiv.org/abs/1601.05416 and some similarities and differences with this work
In this paper, we have investigated the holographic entanglement entropy for a linear subsystem in a $3+1$-dimensional Lifshitz black hole. The entanglement entropy has been analysed in both the infra-red and ultra-violet limits, and has also been computed in the near horizon approximation. The notion of a generalized temperature in terms of the renormalized entanglement entropy has been introduced. This also leads to a generalized thermodynamics like law $E=T_g S_{REE}$. The generalized temperature has been defined in such a way that it reduces to the Hawking temperature in the infra-red limi…
