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 CFT
2 states dual to global AdS
3, 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 consider entanglement entropy in the context of gauge/gravity duality for conformal field theories in even dimensions. The holographic prescription due to Ryu and Takayanagi (RT) leads to an equation describing how the entangling surface extends into the bulk geometry. We show that setting to zero, the timetime component of the Brown-York stress tensor evaluated on the co-dimension 1 entangling surface, leads to the same equation. By considering a spherical entangling surface as an example, we observe that the Euclidean actionmethods in AdS/CFT will lead to the RT area functional arising as a counterterm needed to regularize the stress tensor. We present arguments leading to a justification for the minimal area prescription.
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.
Using the volume of the space enclosed by the Ryu-Takayanagi (RT) surface, we study the complexity of the disk-shape subregion (with radius R) in various (
2+1)-dimensional gapped systems with gravity dual. These systems include a class of toy models with singular IR and the bottom-up models for quantum chromodynamics and fractional quantum Hall effects. Two main results are: (i) in the large-R expansion of the complexity, the R-linear term is always absent, similar to the absence of topological entanglement entropy, and (ii) when the entanglement entropy exhibits the classic ``swallowtail'' phase transition, the complexity is sensitive but reacts differently.
We propose a renormalization scheme for entanglement entropy of three-dimensional CFTs with a four-dimensional asymptotically AdS gravity dual in the context of the gauge/gravity correspondence. The procedure consists in adding the Chern form as a boundary term to the area functional of the Ryu-Takayanagi minimal surface. We provide an explicit prescription for the renormalized entanglement entropy, which is derived via the replica trick. This is achieved by considering a Euclidean gravitational action renormalized by the addition of the Chern form at the spacetime boundary, evaluated in the conically-singular replica manifold. We show that the addition of this boundary term cancels the divergent part of the entanglement entropy, recovering the results obtained by Taylor and Woodhead. We comment on how this prescription for renormalizing the entanglement entropy is in line with the general program of topological renormalization in asymptotically AdS gravity.
We derive a generalized version of the Ryu-Takayanagi formula for the entanglement entropy in arbitrary diffeomorphism invariant field theories. We use a recent framework which expresses the measurable quantities of a quantum theory as a weighted sum over paths in the theory's phase space. If this framework is applied to a field theory on a spacetime foliated by a hypersurface
Σ, the choice of a codimension-2 surface
B without boundary contained in
Σ specifies a submanifold in the phase space. For diffeomorphism invariant field theories, a functional integral expression for their density matrices was recently given and then used to derive bounds on phase space volumes in the considered submanifold associated to
B. These bounds formalize the gravitational entropy bound. Here, we present an implication of this derivation in that we show the obtained functional integral expression for density matrices to be naturally suited for the replica trick. Correspondingly, we prove a functional integral expression for the associated entanglement entropies and derive a practical prescription for their evaluation to leading order and beyond. An important novelty of our approach is the contact to phase space. This allows us both to obtain a prescription for entanglement entropies in arbitrary diffeomorphism invariant field theories not necessarily possessing a holographic dual as well as to use entanglement entropies to study their phase space structure. In the case of the bulk-boundary correspondence, our prescription consistently reproduces and hence provides a natural and independent proof of the Ryu-Takayanagi formula as well as its various generalizations. These include the covariant holographic entanglement entropy proposal, Dong's proposal for higher-derivative gravity as well as the quantum extremal surface prescription.