In a quantum circuit, it is believed that complexity itself reaches a maximum of order exponential in the number of q-bits or equivalently exponential in entropy of the black hole. However, the current holographic proposals do not meet this criterion. The holographic proposals find the complexity of the very late times to be linear in the entropy, while in the quantum circuit, it is expected that complexity meets within a finite time its maximum value in an exponential in the entropy. These points are required to be altered in holographic proposals of complexity. This paper introduces a new holographic proposal that meets this criterion and consolidates the Lloyd bound.
This paper proves that if eternal neutral black holes satisfy some general conditions and matter fields only appear in the outside of the Killing horizon, the strong energy condition is a sufficient condition to insure that the vacuum Schwarzschild black hole has the fastest action growth of the same total energy. This result is consistent with the bound of computational complexity growth rate and gives a strong evidence for the holographic complexity-action conjecture.
This thesis develops recent work on the so called Volume-Complexity and Action-Complexity conjectures. According to this family of proposals, geometric quantities can be defined in some holographic gravitational theories that can be mapped with the concept of quantum complexity for states in a dual quantum-mechanical theory. In this work, we review the original motivations for the use of quantum-information theory in the search of a theory of quantum gravity, and argue in favour of holographic complexity as a promising new tool that could play a key role in the elucidation of the properties of…
Built on our observation that entangling surfaces of the boundary field theory are great co-dimension one spheres in the context of DS/dS correspondence, we study some information theoretic quantities of the field theory dual intensively using holographic proposals. We will focus on entanglement entropy (EE), entanglement of purification (EoP) and complexity. Several fundamental observations and analysis are provided. For EE, we focus on its scaling behavior, which indicates the nature of the relevant degrees of freedom. Moreover, we find that EE provides us with important information of the e…
The AdS/CFT correspondence often motivates research on questions in gravitational physics whose relevance might not be immediately clear from a purely GR-perspective, but which are nevertheless interesting. In these proceedings, we summarise two such results recently obtained by the author. One concerns, broadly speaking, the possible isometry-groups of a spacetime sourced by physical matter. The other one provides a possible argument against the recently proposed complexity=action conjecture.
The volume of the region inside the bulk Ryu-Takayanagi surface is a codimension-one object, and a natural generalization of holographic complexity to the case of subregions in the boundary QFT. We focus on time-independent geometries, and study the properties of this volume in various circumstances. We derive a formula for computing the volume for a strip entangling surface and a general asymptotically AdS bulk geometry. For an AdS black hole geometry, the volume exhibits non-monotonic behaviour as a function of the size of the entangling region (unlike the behaviour of the entanglement entro…
Motivated by recent studies of quantum computational complexity in quantum field theory and holography, we discuss how weighting certain classes of gates building up a quantum circuit more heavily than others does affect the complexity. Utilizing Nielsen's geometric approach to circuit complexity, we investigate the effects for a regulated field theory for which the optimal circuit is a representation of $GL(N,\mathbb{R})$. More precisely, we work out how a uniformly chosen weighting factor acting on the entangling gates affects the complexity and, particularly, its divergent behavior. We show…
This is the contribution to Quarks'2018 conference proceedings. This contribution is devoted to the holographic description of chaos and quantum complexity in the strongly interacting systems out of equilibrium. In the first part of the talk we present different holographic complexity proposals in out-of-equilibrium CFT following the local perturbation. The second part is devoted to the chaotic growth of the local operator size at a finite chemical potential. There are numerous results stating that the chemical potential may lead to the chaos disappearance, and we confirm these results from ho…
We study circuit complexity for spatial regions in holographic field theories. We study analogues based on the entanglement wedge of the bulk quantities appearing in the "complexity = volume" and "complexity = action" conjectures. We calculate these quantities for one exterior region of an eternal static neutral or charged black hole in general dimensions, dual to a thermal state on one boundary with or without chemical potential respectively, as well as for a shock wave geometry. We then define several analogues of circuit complexity for mixed states, and use tensor networks to gain intuition…
Does gravity constrain computation? We study this question using the AdS/CFT correspondence, where computation in the presence of gravity can be related to non-gravitational physics in the boundary theory. In AdS/CFT, computations which happen locally in the bulk are implemented in a particular non-local form in the boundary, which in general requires distributed entanglement. In more detail, we recall that for a large class of bulk subregions the area of a surface called the ridge is equal to the mutual information available in the boundary to perform the computation non-locally. We then argu…
It was recently conjectured that the quantum complexity of a holographic boundary state can be computed by evaluating the gravitational action on a bulk region known as the Wheeler-DeWitt patch. We apply this complexity=action duality to evaluate the `complexity of formation' (arXiv:1509.07876, arXiv:1512.04993), i.e., the additional complexity arising in preparing the entangled thermofield double state with two copies of the boundary CFT compared to preparing the individual vacuum states of the two copies. We find that for boundary dimensions $d>2$, the difference in the complexities grows li…
Entanglement, chaos, and complexity are as important for de Sitter space as for AdS and for black holes. There are similarities and great differences between AdS and dS in how these concepts are manifested in the space-time geometry. In the first part of this paper the Ryu-Takayanagi prescription, the theory of fast scrambling, and the holographic complexity correspondence are reformulated for de Sitter space. Criteria are proposed for a holographic model to describe de Sitter space. The criteria can be summarized by the requirement that scrambling and complexity growth must be "hyperfast." In…
We propose that in time dependent backgrounds the holographic principle should be replaced by the generalized second law of thermodynamics. For isotropic open and flat universes with a fixed equation of state, the generalized second law agrees with the cosmological holographic principle proposed by Fischler and Susskind. However, in more complicated spacetimes the two proposals disagree. A modified form of the holographic bound that applies to a post-inflationary universe follows from the generalized second law. However, in a spatially closed universe, or inside a black hole event horizon, the…
We study the complexity of Gaussian mixed states in a free scalar field theory using the 'purification complexity'. The latter is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state. We argue that the optimal purifications only contain the essential number of ancillary degrees of freedom necessary in order to purify the mixed state. We also introduce the concept of 'mode-by-mode purifications' where each mode in the mixed state is purified separately and examine the extent to which such purifications are optimal. We explore th…
We estimate the net information exchange between adjacent quantum subsystems holographically living on the boundary of $AdS$ spacetime. The information exchange is a real time phenomenon and only after long time interval it may get saturated. Normally we prepare systems for small time intervals and measure the information exchange over finite interval only. We find that the information flow between entangled subsystems gets reduced if systems are in excited state whereas the ground state allows maximum information flow at any given time. Especially for $CFT_2$ we exactly show that a rise in th…
We propose a new example of entanglement knitting spacetime together, satisfying a series of checks of the corresponding von Neumann and Renyi entropies. The conjectured dual of de Sitter in d+1 dimensions involves two coupled CFT sectors constrained by residual d-dimensional gravity. In the d=2 case, the gravitational constraints and the CFT spectrum are relatively tractable. We identify a finite portion of each CFT Hilbert space relevant for de Sitter. Its maximum energy level coincides with the transition to the universal Cardy behavior for theories with a large central charge and a sparse …
Lanzhou University（兰州大学）；Research Center of Gravitation, Lanzhou University（兰州大学引力研究中心）；University of Waterloo（滑铁卢大学）；Department of Applied Mathematics, University of Waterloo（滑铁卢大学应用数学系）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Perimeter Institute For Theoretical Physics（理论物理前沿研究所）；Department of Physics and Astronomy, University of Waterloo（滑铁卢大学物理与天文系）；School of Physics, Jiangxi Normal University（江西师范大学物理学院）
Over the last twenty five years, holographic duality has revolutionised our understanding of gauge theories, quantum many-body systems and also quantum black holes. This topical issue is a collection of review articles on recent advances in fundamentals of holographic duality and its applications with special focus on a few areas where it is inter-disciplinary to a large measure. The aim is to provide a sufficient background on relevant phenomenology and other theoretical areas such as quantum information theory to researchers whose primary expertise is in quantum fields, strings and gravity, …
We prove that consistency of the holographic dictionary implies a hallmark prediction of the weak cosmic censorship conjecture: that in classical gravity, trapped surfaces lie behind event horizons. In particular, the existence of a trapped surface implies the existence of an event horizon, and that furthermore this event horizon must be outside of the trapped surface. More precisely, we show that the formation of event horizons outside of a strong gravity region is a direct consequence of causal wedge inclusion, which is required by entanglement wedge reconstruction. We make few assumptions b…
Environment effects on a $n$-dimensional mirror from the strongly coupled d-dimensional quantum critical fields with a dynamic exponent $z$ in weakly squeezed states are studied by the holographic approach. The dual description is a $n+1$-dimensional probe brane moving in the $d+1$-dimensional asymptotic Lifshitz geometry with gravitational wave perturbations. Using the holographic influence functional method, we find that the large coupling constant of the fields reduces the position uncertainty of the mirror, but enhances the momentum uncertainty. As such, the product of the position and mom…
Toward Krylov-based holography in double-scaled SYK
Gwangju Institute of Science and Technology（光州科学技术院）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科研中心理论物理研究所）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（浦项科技大学）；Research Center for Photon Science Technology, Gwangju Institute of Science and Technology（光州科学技术院光子科学与技术研究中心）
Recently, a novel formula for computing entropy in theories coupled to semi-classical gravity has been devised. Using this so-called island formula the entropy of semi-classical black holes follows a Page curve. Here, we study the relation between this novel entropy and semi-classical entropy in the context of doubly-holographic models. Double holography allows for two different $d$-dimensional descriptions of a black hole coupled to a non-gravitational bath, both of which allow a holographic computation of von Neumann entropy in bath subregions. We argue that the correct homology constraint f…
The quantum null energy condition (QNEC) is the only known consistent local energy condition in quantum theories. Contrary to the classical energy condition which are known to be violated in QFT, QNEC is a consequence of the quantum focussing conjecture and has been proven for several special cases and in general for QFTs in three or more spacetime dimensions. QNEC involves an intrinsically quantum property of the theory under consideration, the entanglement entropy (EE). While EE is notoriously hard to calculate in QFT, the holographic principle provides a simple geometric description. In gen…
We examine the multifold complexity and Loschmidt echo for an inverted harmonic oscillator. We give analytic expressions for any number of precursors, implementing multiple backward and forward time evolutions of the quantum state, at the leading order in the perturbation. We prove that complexity is dominated by the longest permutation of the given time combination in an alternating ``zig-zag'' order, the exact same result obtained with holography. We conjecture that the general structure for multifold complexity should hold true universally for generic quantum systems, in the limit of a larg…
The holographic description of the three-dimensional Kerr-de Sitter space with a gravitational Chern-Simons term is studied, in the context of dS/CFT correspondence. The space has only one (cosmological) event horizon and its mass and angular momentum are identified from the holographic energy-momentum tensor at the asymptotic infinity. The thermodynamic entropy of the cosmological horizon is computed directly from the first law of thermodynamics, with the usual Hawking temperature, and it is found that the usual Gibbons-Hawking entropy is modified. It is remarked that, due to the gravitationa…
Holographic superconductor is an important arena for holography, as it allows concrete calculations to further understand the dictionary between bulk physics and boundary physics. An important quantity of recent interest is the holographic complexity. Conflicting claims had been made in the literature concerning the behavior of holographic complexity during phase transition. We clarify this issue by performing a numerical study on one-dimensional holographic superconductor. Our investigation shows that holographic complexity does not behave in the same way as holographic entanglement entropy. …
Warped conformal field theories in two dimensions are exotic nonlocal, Lorentz violating field theories characterized by Virasoro-Kac-Moody symmetries and have attracted a lot of attention as candidate boundary duals to warped AdS$_3$ spacetimes, thereby expanding the scope of holography beyond asymptotically AdS spacetimes. Here we investigate WCFT$_2$\,s using \emph{circuit complexity} as a tool. First we compute the holographic volume complexity (CV) which displays a linear UV divergence structure, more akin to that of a local CFT$_2$ and has a very complicated dependence on the Virasoro ce…
We consider a recent generalisation by Bousso of an earlier holography proposal by Fischler and Susskind. We demonstrate that in general inhomogeneous universes such a proposal would involve extremely complicated - possibly fractal - light sheets. Furthermore, in general such a light sheet cannot be known a priori on the basis of theory and moreover, the evolution of the universe makes it clear that in general such bounds cannot remain invariant under time reversal and will change with epoch. We propose a modified version of this proposal in which the light sheets end on the boundary of the pa…
Using the ``complexity equals action''(CA) conjecture, for an ordinary charged system, it has been shown that the late-time complexity growth rate is given by a difference between the value of $Φ_{H}Q+Ω_H J$ on the inner and outer horizons. In this paper, we investigate the complexity of the boundary quantum system with conical deficits. From the perspective of holography, we consider a charged accelerating black holes which contain conical deficits on the north and south poles in the bulk gravitational theory and evaluate the complexity growth rate using the CA conjecture. As a result, the la…
In this thesis I present numerical studies of entanglement entropy and the quantum null energy condition in strongly coupled far-from-equilibrium quantum states using holography. I give a careful introduction into the numerical methods used and provide the computer codes to compute entanglement entropy and the quantum null energy condition. These methods are then applied to systems of various degrees of complexity, including homogeneous and isotropic far- from-equilibrium quantum quenches dual to Vaidya spacetimes, to homogeneous and anisotropic finite temperature states dual to anisotropic bl…
We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state wit…
An effective quantum field theory (QFT) with a manifest UV/IR connection, so as to be valid for arbitrarily large volumes, can successfully be applied to the cosmological dark energy problem as well as the cosmological constant (CC) problem. Motivated by recent approaches to the hierarchy problem, we develop such a framework with a large number of particle species. When applying to systems on the brink of experiencing a sudden collapse to a black hole, we find that the entropy, unlike the total energy, now becomes an increasing function of the number of field species. An internal consistency o…
Quantum circuit complexity has played a central role in recent advances in holography and many-body physics. Within quantum field theory, it has typically been studied in a Lorentzian (real-time) framework. In a departure from standard treatments, we aim to quantify the complexity of the Euclidean path integral. In this setting, there is no clear separation between space and time, and the notion of unitary evolution on a fixed Hilbert space no longer applies. As a proof of concept, we argue that the pants decomposition provides a natural notion of circuit complexity within the category of 2-di…
Recent developments on holography and quantum information physics suggest that quantum information theory come to play a fundamental role in understanding quantum gravity. Cosmology, on the other hand, plays a significant role in testing quantum gravity effects. How to apply this idea to a realistic universe is still missing. Here we show some concepts in quantum information theory have their cosmological descriptions. Particularly, we show complexity of a tensor network can be regarded as a Fisher information measure(FIM) of a dS universe, followed by several observations: (i) the holographic…
