Our earlier paper "Complexity Equals Action" conjectured that the quantum computational complexity of a holographic state is given by the classical action of a region in the bulk (the 'Wheeler-DeWitt' patch). We provide calculations for the results quoted in that paper; explain how it fits into a broader (tensor) network of ideas; and elaborate on the hypothesis that black holes are fastest computers in nature.

We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in anti-de Sitter spacetime, as well as black holes perturbed with static shells and with shock waves. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature.

I show how recent progress in real space renormalization group methods can be used to define a generalized notion of holography inspired by holographic dualities in quantum gravity. The generalization is based upon organizing information in a quantum state in terms of scale and defining a higher dimensional geometry from this structure. While states with a finite correlation length typically give simple geometries, the state at a quantum critical point gives a discrete version of anti de Sitter space. Some finite temperature quantum states include black hole-like objects. The gross features of equal time correlation functions are also reproduced in this geometric framework. The relationship between this framework and better understood versions of holography is discussed. I. INTRODUCTIONHilbert space, the mathematical representation of possible states of a quantum system, is exponentially large when the system is a macroscopic piece of matter. The traditional theory of symmetry breaking reduces this overwhelming amount of information to three key quantities: the energy (or Hamiltonian), the symmetry of the Hamiltonian, and the pattern of symmetry breaking.However, the existence of exotic phases of matter not characterized by broken symmetry, as in the fractional quantum hall effect [1], demonstrates the need for a more general theory. Fractional quantum hall systems are distinguished by the presence of long range entanglement in the ground state, suggesting that important information is encoded in the spatial structure of entanglement.Here I show how such a "pattern" of entanglement can be defined and visualized using the geometry of an emergent holographic dimension. This picture connects two new tools in many body physics: entanglement renormalization and holographic gauge/gravity duality.Entanglement renormalization [2] is a combination real space renormalization group techniques and ideas from quantum information theory that grew out of attempts to describe quantum critical points. The key message of entanglement renormalization is that the removal of local entanglement is essential for defining a proper real space renormalization group transformation for quantum * Electronic address: bswingle@mit.edu states. This realization has permitted a compact description of some quantum critical points [3,4].Holographic gauge/gravity duality [5,6,7] is the proposal that certain quantum field theories without gravity are dual to theories of quantum gravity in a curved higher dimensional "bulk" geometry. Holography provides a way to compute field theory observables from a completely different point of view using a small amount of information encoded geometrically. Real space renormalization is also important in the holographic framework [8,9,10,11], thus hinting at a possible connection between holography and entanglement renormalization. We will begin with entanglement renormalization and build up to the full holographic picture. II. MANY BODY ENTANGLEMENTWe are interested in quantifying entanglement in many body systems...

General scaling arguments, and the behavior of the thermal entropy density, are shown to lead to an infrared metric holographically representing a compressible state with hidden Fermi surfaces. This metric is characterized by a general dynamic critical exponent, z, and a specific hyperscaling violation exponent, θ. The same metric exhibits a logarithmic violation of the area law of entanglement entropy, as shown recently by Ogawa et al. (arXiv:1111.1023). We study the dependence of the entanglement entropy on the shape of the entangling region(s), on the total charge density, on temperature, and on the presence of additional visible Fermi surfaces of gauge-neutral fermions;for the latter computations, we realize the needed metric in an Einstein-Maxwell-dilaton theory.All our results support the proposal that the holographic theory describes a metallic state with hidden Fermi surfaces of fermions carrying gauge charges of deconfined gauge fields.1 arXiv:1112.0573v3 [cond-mat.str-el]

We provide a protocol to measure out-of-time-order correlation functions. These correlation functions are of theoretical interest for diagnosing the scrambling of quantum information in black holes and strongly interacting quantum systems generally. Measuring them requires an echo-type sequence in which the sign of a many-body Hamiltonian is reversed. We detail an implementation employing cold atoms and cavity quantum electrodynamics to realize the chaotic kicked top model, and we analyze effects of dissipation to verify its feasibility with current technology. Finally, we propose in broad strokes a number of other experimental platforms where similar out-of-time-order correlation functions can be measured.Comment: 12 pages, 5 figures; v3: introduction revised for greater clarity and accessibilit

Purification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all these models, and we support our conjectures with a variety of numerical and analytical results. We find that such minimally entangled purifications have a number of applications, from enhancing entanglement-based tensor network methods for describing mixed states to elucidating novel aspects of the emergence of geometry from entanglement in the AdS/CFT correspondence.

As experiments are increasingly able to probe the quantum dynamics of systems with many degrees of freedom, it is interesting to probe fundamental bounds on the dynamics of quantum information. We elaborate on the relationship between one such bound-the Lieb-Robinson bound-and the butterfly effect in strongly coupled quantum systems. The butterfly effect implies the ballistic growth of local operators in time, which can be quantified with the "butterfly" velocity v B . Similarly, the Lieb-Robinson velocity places a state-independent ballistic upper bound on the size of time evolved operators in nonrelativistic lattice models. Here, we argue that v B is a state-dependent effective Lieb-Robinson velocity. We study the butterfly velocity in a wide variety of quantum field theories using holography and compare with freeparticle computations to understand the role of strong coupling. We find that v B remains constant or decreases with decreasing temperature. We also comment on experimental prospects and on the relationship between the butterfly velocity and signaling. DOI: 10.1103/PhysRevLett.117.091602 In relativistic systems with exact Lorentz symmetry, causality requires that spacelike-separated operators commute. In nonrelativistic systems, there is no analogous notion: a local operator Vð0Þ at the origin need not commute with another local operator Wðx; tÞ at position x at a later time t, even if the separation is much larger than the elapsed time jxj ≫ t. This can be understood by considering the Baker-Campbell-Hausdorff formula for the expansion of Wðx; tÞ ¼ e iHt WðxÞe −iHt ,where H is the Hamiltonian which is assumed to consist of bounded local terms. As long as there is some sequence of terms in H that connect the origin and point x (and absent any special cancellations), the operator Wðx; tÞ will generically fail to commute with Vð0Þ. This does not necessarily imply that the magnitude commutator ½Wðx; tÞ; Vð0Þ between distance operators must be large. A bound of Lieb and Robinson [1], along with many subsequent improvements [2][3][4], limits the size of commutators of local operators separated in space and time, even in nonrelativistic systems. In terms of the Heisenberg operator Wðx; tÞ at position x and time t and an operator V at the origin of space and time, the bound reads ∥½Wðx; tÞ; Vð0Þ∥where K 0 and ξ 0 are constants, ∥ · ∥ indicates the operator norm, and v LR is the Lieb-Robinson velocity. The growth of the commutator is controlled by v LR , which is a function of the parameters of the Hamiltonian. Hence, although operators separated by a distance x may cease to exactly commute for any t > 0, the Lieb-Robinson bound implies that their commutator cannot be Oð1Þ until t ≳ x=v LR . Thus, the Lieb-Robinson velocity provides a natural notion of a "light" cone for nonrelativistic systems. Even for relativistic systems, where causality implies that the commutator of local operators must be exactly zero for t < x (in this Letter we have set the speed of light to unity, c ¼ 1), the Lieb-Robinson cone, ...

We apply and extend the theory of universal recovery channels from quantum information theory to address the problem of entanglement wedge reconstruction in AdS/CFT. It has recently been proposed that any low-energy local bulk operators in a CFT boundary region's entanglement wedge can be reconstructed on that boundary region itself. Existing work arguing for this proposal relies on algebraic consequences of the exact equivalence between bulk and boundary relative entropies, namely the theory of operator algebra quantum error correction. However, bulk and boundary relative entropies are only approximately equal in bulk effective field theory, and in similar situations it is known that predictions from exact entropic equalities can be qualitatively incorrect. The framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture in addition to new physical insights. Most notably, we find that a bulk operator acting in a given boundary region's entanglement wedge can be expressed as the response of the boundary region's modular Hamiltonian to a perturbation of the bulk state in the direction of the bulk operator. This formula can be interpreted as a noncommutative version of Bayes' rule that attempts to undo the noise induced by restricting to only a portion of the boundary, and has an integral representation in terms of modular flows. To reach these conclusions, we extend the theory of universal recovery channels to finite-dimensional operator algebras and demonstrate that recovery channels approximately preserve the multiplicative structure of the operator algebra. arXiv:1704.05839v4 [hep-th]