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If entanglement entropy in a small geodesic ball is maximized at fixed volume in the vacuum, then it should be stationary under variation to a nearby state. I will show that this stationarity condition is equivalent to the semiclassical Einstein equation. If the matter QFT is not conformal, then the derivation requires a further assumption about QFT, whose validity is currently under investigation. [Based on http://arxiv.org/abs/1505.04753]

We discuss area terms in entanglement entropy and show that a recent formula by Rosenhaus and Smolkin is equivalent to the Adler-Zee sum rule for the renormalization of the Newton constant in terms of correlator of traces of the stress tensor. We elaborate on how to fix the ambiguities in these formulas: Improving terms for the stress tensor of free fields, boundary terms in the modular Hamiltonian, and contact terms in the Euclidean correlation functions. We make computations for free fields and show how to apply these calculations to understand some results for interacting theories which have been studied in the literature. We check the sum rule holographicaly. We also discuss an application to the F-theorem.

The fact that a black hole is a fast-scrambler is at the heart of black hole information paradoxes. It has been suggested that chaos can be diagnosed by using an out-of-time correlation function, which is closely related to the commutator of operators separated in time. In this talk I propose that the tripartite information (also known as topological entanglement entropy) can be used as a quantitative information theoretic measure of chaos. By viewing a quantum channel as a state via the Choi-Jamilkowski isomorphism, the tripartite information measures four-party entanglement between the “past” and the “future”, much like an out-of-time correlation function. I will compute the time-evolution of the tripartite information for three systems; (a) non-integrable spin systems on a lattice, (b) planar networks of perfect tensors which mimic the growth of the Einstein-Rosen bridge and (c) a holographic system. This talk is based on an ongoing work with Xiaoliang Qi and Daniel Roberts.

We investigate causality constraints on the time evolution of entanglement entropy after a global quench. We analyze three models for the spread of entanglement: holographic quenches, the free particle streaming model of Calabrese and Cardy generalized to higher dimensions and an arbitrary pattern of entanglement, and a particle model with an infinite scattering rate. In these models we exhibit the intricate interplay of causality, strong subadditivity, and maximally entangled subsystems. Finally, using strong subadditivity we prove that the normalized rate of growth of entanglement entropy is bounded by the speed of light.

We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.

In this talk I will introduce a family of exactly solvable toy models of a holographic correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. The building block for these models are a special type of tensor with maximal entanglement along any bipartition, which gives rise to an exact isometry from bulk operators to boundary operators. The entire tensor network is a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the holographic correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. I will describe how bulk operators may be represented on the boundary regions mimicking the Rindler-wedge reconstruction.

The Multiscale Entanglement Renormalization Ansatz has proven to capture the ground state properties of strongly correlated quantum lattice systems, both in gapped regimes and at critical points, and realizes a lattice version of the holographic principle. In this talk, I will review a construction of entanglement renormalization that applies in the continuum (i.e. to quantum fields) and discuss several aspects such as the renormalization group equation and scaling exponents, illustrated using free field theories as example

I describe a class of non-perturbative renormalization group (RG) transformations which, when applied to the (discrete time) Euclidean path integrals of a quantum systems on the lattice, can give results consistent with conformal transformations of quantum field theories. In particular, this class of transformation, which we call Tensor Network Renormalization (TNR), is shown to generate a scale-invariant RG flow for quantum systems at a critical point. Applications of TNR towards study of quantum critical systems, and its relationship to the multi-scale-entanglement renormalization ansatz (MERA) for ground and thermal states of quantum systems, will be discussed.

Of course not, but if one believes that information cannot be destroyed in a theory of quantum gravity, then we run into apparent contradictions with quantum theory when we consider evaporating black holes. Namely that the no-cloning theorem or the principle of entanglement monogamy is violated. Here, we show that neither violation need hold, since, in arguing that black holes lead to cloning or non-monogamy, one needs to assume a tensor product structure between two points in space-time that could instead be viewed as causally connected. In the latter case, one is violating the semi-classical causal structure of space, which is a strictly weaker implication than cloning or non-monogamy. We show that the lack of monogamy that can emerge in evaporating space times is one that is allowed in quantum mechanics, and is very naturally related to a lack of monogamy of correlations of outputs of measurements performed at subsequent instances of time of a single system. A particular example of this is the Horowitz-Maldacena proposal, and we argue that it needn't lead to cloning or violations of entanglement monogamy. In the case of the AMPS firewall experiment we find that the entanglement structure is modified, and one must have entanglement between the infalling Hawking partners and early time outgoing Hawking radiation which surprisingly tame violation of entanglement monogamy. http://arxiv.org/abs/1506.07133

The Ryu-Takayanagi proposal (and generalizations) for holographic entanglement makes predictions for geometric CFT entanglement entropy (EE) that continue to hold for any CFT, regardless of existence of large-N limit or strong coupling. We establish this using a direct field theory calculation, thus providing a non-trivial check of the holographic proposal. This universality emerges for small perturbations of the EE of a ball shaped region. Einstein’s equations arise from the field theory calculation as a way to efficiently encode this perturbative CFT entanglement holographically in the geometry of a dual space-time.