This thesis investigates quantum complexity in the AdS/CFT duality. We study a recently proposed measure of quantum complexity, the spread complexity. This is a measure of the number of orthonormal basis vectors needed to describe a given quantum state (its “spread” in the Hilbert space), naturally selecting a preferred basis adapted to the dynamics of the state which minimises this quantity. This thesis demonstrates that the spread complexity provides a quantum information- theoretic quantity which directly connects to bulk dynamics in holography. Quantum complexity has long been studied in the context of holography. One result of these investigations is a conjecture relating the rate of change of the complexity of certain states in the CFT to the radial momentum of a dual massive particle falling along a time-like geodesic in the bulk spacetime. We demonstrate that there is an exact match between the rate of change of the spread complexity of an excited 2D CFT state and the dual particle’s radial momentum, measured in appropriate coordinates. As a toy model of the localised bulk particle state, we show that a similar relationship exists between the spread complexity and momentum of coherent states of the quantum harmonic oscillator. The spread complexity of these CFT states can be expressed analytically by relating them to a discrete series representation of SL(2,R). Additionally, we find analytical results for the spread complexity when the excited CFT state is evolved by the modular Hamiltonian of an interval region in the CFT, and by the half-sided modular Hamiltonian associated with a further sub-interval region. Next, we study the spread complexity of CFT states which describe a particle on the BTZ black hole background. We also study the spread complexity of local states in the semi-classical bulk theory. This provides a direct bulk interpretation of the spread complexity to complement our CFT calculations. Finally, we explore the extensivity of the spread complexity of states and of the Krylov complexity of density matrix operators in non-interacting bipartite systems.
This thesis investigates quantum complexity in the AdS/CFT duality. We study a recently proposed measure of quantum complexity, the spread complexity. This is a measure of the number of orthonormal basis vectors needed to describe a given quantum state (its “spread” in the Hilbert space), naturally selecting a preferred basis adapted to the dynamics of the state which minimises this quantity. This thesis demonstrates that the spread complexity provides a quantum information- theoretic quantity which directly connects to bulk dynamics in holography. Quantum complexity has long been studied in the context of holography. One result of these investigations is a conjecture relating the rate of change of the complexity of certain states in the CFT to the radial momentum of a dual massive particle falling along a time-like geodesic in the bulk spacetime. We demonstrate that there is an exact match between the rate of change of the spread complexity of an excited 2D CFT state and the dual particle’s radial momentum, measured in appropriate coordinates. As a toy model of the localised bulk particle state, we show that a similar relationship exists between the spread complexity and momentum of coherent states of the quantum harmonic oscillator. The spread complexity of these CFT states can be expressed analytically by relating them to a discrete series representation of SL(2,R). Additionally, we find analytical results for the spread complexity when the excited CFT state is evolved by the modular Hamiltonian of an interval region in the CFT, and by the half-sided modular Hamiltonian associated with a further sub-interval region. Next, we study the spread complexity of CFT states which describe a particle on the BTZ black hole background. We also study the spread complexity of local states in the semi-classical bulk theory. This provides a direct bulk interpretation of the spread complexity to complement our CFT calculations. Finally, we explore the extensivity of the spread complexity of states and of the Krylov complexity of density matrix operators in non-interacting bipartite systems.
Fast scrambling and exponential growth of complexity are two distinctive features of quantum gravity, offering a unique window into the mechanisms that govern the semiclassical limit of the holographic duality. In this thesis, we explore these information-theoretic phenomena through two key diagnostics --- out-of-time ordered correlation functions (OTOCs) and spread complexity --- computed in the setting of two-dimensional holographic conformal field theories (CFTs). First, we focus on the large-c limit of Liouville theory, where the computation of OTOCs reveals distinctive insights into both the chaotic nature and the structural features of the theory. Remarkably, we find it exhibits a maximal Lyapunov exponent, despite lacking the identity operator in its spectrum, and a scrambling time logarithmic in the central charge. Using path integral techniques, we calculate heavy-light four-point functions, and uncover a conformal block expansion dominated by the exchange of an infinite tower of spinless double-trace operators. We attribute the chaotic behaviour of the OTOC to this exchange of operators, which can be thought of as dressed scramblons. Additionally, we observe that at the onset of scrambling, the dominance of conformal blocks shifts from heavy primaries at early times to light primaries at late times. Second, we turn to the diagnostic of spread complexity in locally excited holographic CFTs and argue that its time derivative is dual to the proper radial momentum of a massive point particle propagating in Anti-de Sitter spacetime (AdS). Exploiting the conformal symmetry and the emergence of an ``effective'' SL(2,R) group, we employ representation-theoretic techniques to compute return amplitudes, Lanczos coefficients, and spread complexities. We then verify our conjectured duality for three settings: the CFT on the line, dual to Poincaré AdS; the CFT on the circle, dual to global AdS; and the CFT at finite temperature, dual to a BTZ black hole. In the gravitational theory, the same SL(2,R) structure emerges as the isometry group of the infalling plane of the bulk particle. Third, we generalise the analysis of spread complexity to modular Hamiltonian evolution. Exploiting the rich algebraic structure of modular theory, we construct the modular Hamiltonians Kₘ and the half-sided modular translations G for the three example states above. Leveraging conformal symmetry, we once again compute modular return amplitudes, Lanczos coefficients, and modular spread complexities. In particular, we emphasise the role of ``conformal regularisation'' --- i.e. regulating with the operator that generates modular evolution - in preserving the effective SL(2,R) structure of the amplitudes. We find that evolution under Kₘ exhibits thermal spread complexity for β = 1 across all examples, while G-evolution universally leads to quadratic growth. Finally, we provide a formal explanation of these results, identifying the role of the Killing norm D of the evolution operator as the key factor governing the behaviour of complexity.
In recent years many discoveries have been made that reveal a close relation between quantum information and geometry in the context of the AdS/CFT correspondence. In this duality between a conformal quantum field theory (CFT) and a theory of gravity on Anti-de Sitter spaces (AdS) quantum information quantities in CFT are associated with geometric objects in AdS. Subject of this thesis is the examination of this intriguing property of AdS/CFT. We study two central elements of quantum information: subregion complexity -- which is a measure for the effort required to construct a given reduced state -- and the modular Hamiltonian -- which is given by the logarithm of a considered reduced state. While a clear definition for subregion complexity in terms of unitary gates exists for discrete systems, a rigorous formulation for quantum field theories is not known. In AdS/CFT, subregion complexity is proposed to be related to certain codimension one regions on the AdS side. The main focus of this thesis lies on the examination of such candidates for gravitational duals of subregion complexity. We introduce the concept of \textit{topological complexity}, which considers subregion complexity to be given by the integral over the Ricci scalar of codimension one regions in AdS. The Gauss-Bonnet theorem provides very general expressions for the topological complexity of CFT2 states dual to global AdS3, BTZ black holes and conical defects. In particular, our calculations show that the topology of the considered codimension one bulk region plays an essential role for topological complexity. Moreover, we study holographic subregion complexity (HSRC), which associates the volume of a particular codimension one bulk region with subregion complexity. We derive an explicit field theory expression for the HSRC of vacuum states. The formulation of HSRC in terms of field theory quantities may allow to investigate whether this bulk object indeed provides a concept of subregion complexity on the CFT side. In particular, if this turns out to be the case, our expression for HSRC may be seen as a field theory definition of subregion complexity. We extend our expression to states dual to BTZ black holes and conical defects. A further focus of this thesis is the modular Hamiltonian of a family of states ρλ depending on a continuous parameter λ. Here λ may be associated with the energy density or the temperature, for instance. The importance of the modular Hamiltonian for quantum information is due to its contribution to relative entropy -- one of the very few objects in quantum information with a rigorous definition for quantum field theories. The first order contribution in λ~=λ−λ0 of the modular Hamiltonian to the relative entropy between ρλ and a reference state ρλ0 is provided by the first law of entanglement. We study under which circumstances higher order contributions in λ~ are to be expected. We show that for states reduced to two entangling regions A, B the modular Hamiltonian of at least one of these regions is expected to provide higher order contributions in λ~ to the relative entropy if A and B saturate the Araki-Lieb inequality. The statement of the Araki-Lieb inequality is that the difference between the entanglement entropies of A and B is always smaller or equal to the entanglement entropy of the union of A and B. Regions for which this inequality is saturated are referred to as entanglement plateaux. In AdS/CFT the relation between geometry and quantum information provides many examples for entanglement plateaux. We apply our result to several of them, including large intervals for states dual to BTZ black holes and annuli for states dual to black brane geometries.
We demonstrate a precise relation between the rate of complexity of quantum states excited by local operators in two-dimensional conformal field theories and the radial momentum of particles in three-dimensional anti–de Sitter spacetimes. Similar relations have been anticipated based on qualitative models for operator growth. Here, we make this correspondence sharp with two key ingredients: the precise definition of quantum complexity given by the spread complexity of states, and the match of its growth rate to the bulk momentum measured in the proper radial distance coordinate.
A bstract Within the AdS/CFT correspondence, computational complexity for reduced density matrices of holographic conformal field theories has been conjectured to be related to certain geometric observables in the dual gravity theory. We study this conjecture from both the gravity and field theory point of view. Specifically, we consider a measure of complexity associated to the Bures metric on the space of density matrices. We compute this complexity measure for mixed states associated to single intervals in descendant states of the vacuum in 2d CFTs. Moreover, we derive from first principles a geometric observable dual to the Bures metric which is localized in the entanglement wedge of the AdS spacetime associated to the quantum circuit on the boundary. Finally, we compare the Bures metric complexity measure with holographic subregion complexity within the “complexity=volume” paradigm for perturbatively small transformations of the vacuum. While there is no exact agreement between these two quantities, we find striking similarities as we vary the target state and interval size, suggesting that these quantities are closely related.
The AdS/CFT correspondence and quantum entanglement (or quantum information). This is one of topics exciting many physicists in this decade. And black holes always play a role of inspiring lab in many cases. In this thesis, we will discuss about scrambling of quantum entanglement and complexity of states in AdS/CFT. In both topics, black holes are expected to be the fastest processors of quantum information. This thesis contains brief reviews of each topics and our own works. One work is an explicit calculation of fast scrambling time in two dimensional holographic CFTs and three dimensional BTZ black holes using the two-sided mutual information. This calculation is a first CFT counterpart of Shenker-Stanford’s work in the gravity dual and an extension of their shock-wave approximation to finite time region in the gravity side. Another work is about socalled optimization procedure of Euclidean path- integrals in CFTs. The optimization procedure gives us a new way to extract dual geometries only from CFT data. And it also gives a new proposal for the definition of complexity of states in CFTs.
In this thesis, we study a couple of different but related questions: holography and quantum error correction, the gravitationally dressed operators and locality in holographic theory, introducing generalized versions of the class of continuous matrix product state as the most famous class of tensor network, and the notion of Krylov complexity of matrix quantum mechanics. In the first part, we use the notion of the Petz map and apply it to reconstruct operators in the entanglement wedge. Moreover, for geometries that contain black holes, we generalize the notion of the Petz map by itself and reconstruct black hole interior modes. In particular, we show that the Petz map reconstruction of the black hole interior is equivalent to the Papadodimas-Raju Proposal. In the second part, within the AdS/CFT correspondence, we identify a class of CFT operators which represent diff-invariant and approximately local observables in the gravitational dual. The interpretation of these observables is that they are not gravitationally dressed with respect to the boundary, but instead to features of the state. We also provide evidence that there are bulk observables whose commutator vanishes to all orders in 1/N with the entire algebra of single-trace operators defined in a space-like separated time-band. In the third part, we defined two new classes of continuous tensor networks by generalizing the class of continuous matrix product states. One is appropriate to describe the ground state of the relativistic field theory at strong coupling and the other is proper for the theories on compact spacetime. We conjecture that the possible bulk dual of it in the AdS/CFT can be an evaporating black hole microstates with the end of the word branes. In the last project, we study the Krylov complexity for 1-matrix quantum mechanics. In the ground state, the Lancsoz coefficients have linear behavior. The bn coefficients have a positive slope while an coefficients have a negative slope. In the thermal states, the an coefficients are zero while bn have two branches of linear growth. Although matrix quantum mechanics is a solvable theory rather than chaotic, we find the linear behavior of the Lancsoz coefficients of this theory.