Complexity in the AdS/CFT correspondence

The main goal of this thesis is to carefully analyze aspects of the gravitational quantities conjectured to be dual to quantum complexity in the AdS/CFT correspondence. The two most promising candidates for such holographic proposals are known as the complexity=volume (CV) and complexity=action (CA) proposals, which will be the main objects of study in this thesis. The latter involves the evaluation of the gravitational action in a region of spacetime known as the Wheeler-DeWitt patch, whose boundary includes null hypersurfaces and null codimension-two joints. There are several subtleties when evaluating the action in a region bounded by null surfaces, and a major part of the work presented here is based on a careful treatment of the boundary contributions to the gravitational action. We start by evaluating the complexity of formation in holography, which is the additional complexity required to build the thermofield double state (TFD) in comparison to the complexity of building two copies of the vacuum. From the gravitational perspective, such quantity is interesting as it involves geometries with black holes. We find that for AdS-Schwarzschild black holes, both the CA and CV proposals yield a UV finite complexity of formation, and at large temperatures it becomes proportional to the thermodynamic entropy for boundary spacetime dimensions d>2. In addition, we investigate dynamical properties of the holographic duals of complexity. We study the time evolution of the thermofield double state for AdS-Schwarzschild and AdS-Reissner-Nordstrom black holes. In the AdS/CFT correspondence, this time evolution corresponds to time slices that cover the interior region of the black hole. We find the striking result that the late time rate of change of complexity in the CA proposal is approached from above, which implies that the originally proposed connection to the conjectured Lloyd's bound on computation rate is violated. In contrast, the CV proposal growth rate is approached from below at late times for these geometries. Next, we investigate the time evolution of holographic complexities when the bulk spacetime has non-trivial dynamics. We investigate both one-sided and two-sided Vaidya geometries, which are sourced by the collapse of an infinitesimally thin layer of null dust. In order to evaluate the complexity in the CA proposal, we construct a null fluid action that sources the Vaidya geometry. Our main result is that the inclusion of a surface counterterm that ensures reparametrization invariance to the null normals at the null boundaries of the Wheeler-DeWitt patch is necessary in order to reproduce desired properties of complexity, such as the switchback effect. In addition, we find that for one-sided black holes, the late time rate of change is approached from below in the CA approach, in contrast to what was found in two-sided geometries

Towards Complexity for Quantum Field Theory States

We investigate notions of complexity of states in continuous quantum-many body systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of Multiscale Entanglement Renormalization Ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum preserving quadratic generators which form $\mathfrak{su}(1,1)$ algebras. On the manifold of Gaussian states generated by these operations the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and holographic complexity proposals.Comment: 6+7 pages, 6 appendices, 2 figures; v2: references added; acknowledgments expanded; appendix F added, reviewing similarities and differences with hep-th/1707.08570; v3: version published in PR

On the Time Dependence of Holographic Complexity

We evaluate the full time dependence of holographic complexity in various eternal black hole backgrounds using both the complexity=action (CA) and the complexity=volume (CV) conjectures. We conclude using the CV conjecture that the rate of change of complexity is a monotonically increasing function of time, which saturates from below to a positive constant in the late time limit. Using the CA conjecture for uncharged black holes, the holographic complexity remains constant for an initial period, then briefly decreases but quickly begins to increase. As observed previously, at late times, the rate of growth of the complexity approaches a constant, which may be associated with Lloyd's bound on the rate of computation. However, we find that this late time limit is approached from above, thus violating the bound. Adding a charge to the eternal black holes washes out the early time behaviour, i.e., complexity immediately begins increasing with sufficient charge, but the late time behaviour is essentially the same as in the neutral case. We also evaluate the complexity of formation for charged black holes and find that it is divergent for extremal black holes, implying that the states at finite chemical potential and zero temperature are infinitely more complex than their finite temperature counterparts.Comment: 52+31 pages, 30 figure

On the Time Dependence of Holographic Complexity

We evaluate the full time dependence of holographic complexity in various eternal black hole backgrounds using both the complexity=action (CA) and the complexity=volume (CV) conjectures. We conclude using the CV conjecture that the rate of change of complexity is a monotonically increasing function of time, which saturates from below to a positive constant in the late time limit. Using the CA conjecture for uncharged black holes, the holographic complexity remains constant for an initial period, then briefly decreases but quickly begins to increase. As observed previously, at late times, the rate of growth of the complexity approaches a constant, which may be associated with Lloyd's bound on the rate of computation. However, we find that this late time limit is approached from above, thus violating the bound. Adding a charge to the eternal black holes washes out the early time behaviour, i.e., complexity immediately begins increasing with sufficient charge, but the late time behaviour is essentially the same as in the neutral case. We also evaluate the complexity of formation for charged black holes and find that it is divergent for extremal black holes, implying that the states at finite chemical potential and zero temperature are infinitely more complex than their finite temperature counterparts.Comment: 52+31 pages, 30 figure

Holographic Complexity in Vaidya Spacetimes II

In this second part of the study initiated in arxiv:1804.07410, we investigate holographic complexity for eternal black hole backgrounds perturbed by shock waves, with both the complexity$=$action (CA) and complexity$=$volume (CV) proposals. In particular, we consider Vaidya geometries describing a thin shell of null fluid with arbitrary energy falling in from one of the boundaries of a two-sided AdS-Schwarzschild spacetime. We demonstrate how known properties of complexity, such as the switchback effect for light shocks, as well as analogous properties for heavy ones, are imprinted in the complexity of formation and in the full time evolution of complexity. Following our discussion in arxiv:1804.07410, we find that in order to obtain the expected properties of the complexity, the inclusion of a particular counterterm on the null boundaries of the Wheeler-DeWitt patch is required for the CA proposal.Comment: 83+38 pages, 34 figure

Holographic Complexity in Vaidya Spacetimes I

We examine holographic complexity in time-dependent Vaidya spacetimes with both the complexity$=$volume (CV) and complexity$=$action (CA) proposals. We focus on the evolution of the holographic complexity for a thin shell of null fluid, which collapses into empty AdS space and forms a (one-sided) black hole. In order to apply the CA approach, we introduce an action principle for the null fluid which sources the Vaidya geometries, and we carefully examine the contribution of the null shell to the action. Further, we find that adding a particular counterterm on the null boundaries of the Wheeler-DeWitt patch is essential if the gravitational action is to properly describe the complexity of the boundary state. For both the CV proposal and the CA proposal (with the extra boundary counterterm), the late time limit of the growth rate of the holographic complexity for the one-sided black hole is precisely the same as that found for an eternal black hole.Comment: 55 pages, 8 figure