# Condensed matter physics, area laws & LQG?

1. May 18, 2015

### atyy

It depends, and I don't know exactly which is the case in the MERA. The typical MERA does lose information. On the other hand, the MERA is best suited for describing self-similar systems, where the renormalization typically need not lose information, so I don't know whether there is a MERA that does not lose information.

Looking at Swingle's http://arxiv.org/abs/0905.1317, he writes on p5: "The goal is to reach the ultraviolet by following the renormalization group flow backwards. This is possible because we have recorded the entire renormalization “history” of the state in the network, but subtleties remain because of the possible loss of information. In practice, the truncation error may be quite small with the proper use of disentanglers. More properly, the tensor network defines a large variational class of states for which the entanglement entropy can be computed by reversing the flow [15]".

2. May 18, 2015

### Jimster41

I've been working on reading that paper. And I definitely got hung up on why he was worried about information loss. Seems it's partly dependent on whether or not the fundamental limit on information is considered to be discrete and bounded, or continuous and infinite. Seems like that kind-of comes around full circle to the question at hand.

Thanks for the clarification.

3. May 18, 2015

### Physics Monkey

On the question of information loss I can say one thing.

I believe that any finite bond dimension MERA (meaning all the lines in the tensor network are finite dimensional) will not be able to exactly capture a conformal field theory (CFT) ground state. This is true even if the CFT is regulated on a lattice with a finite dimensional local Hilbert space. In this sense, then, information is lost - say about high dimension operators in the CFT.

For example, consider the so-called transverse field Ising model with Hamiltonian
$$H = -\sum_r Z_r Z_{r+1} - g \sum_r X_r$$
where g an adjustable parameter. This model has a spin 1/2 on every site of a one dimensional chain and g plays the role of the coupling. When g=1 the Hamiltonian possesses long-range correlations in its ground state and is in fact described by the so-called Ising CFT. Vidal, Evenbly, and friends have shown that many features of this CFT can be captured using a finite bond dimension MERA, but nevertheless the exact wavefunction is not reproduced.

Recently John McGreevy and I introduced a generalization of MERA (and some other tensor networks) which we dubbed "s sourcery". We conjecture that the "s source" ansatz can exactly capture the wavefunction of a lattice regulated CFT (like the above model). One replaces the quantum circuit picture in MERA with a more general local unitary transformation (thus allowing long-distance exponentially decaying tails) which maps the (ground state of the) system at size L to the system at size L/2 times some unentangled degrees of freedom. Since the transformation is unitary and the mapping is exact, no information is lost.

More generally, I would just comment that there are many notions of renormalization, it being too useful a concept to limit to just one incarnation. So in some forms perhaps information is lost while in other versions the "history" of the flow may be preserved. In the same way, there are many kinds of tensor networks and depending on the application one may want a version where information is lost or a version where information is preserved. Bottom line: I think we ought to opt for diversity in which case maybe there isn't one right answer the question of whether information is lost.

4. May 18, 2015

### Jimster41

Physics Monkey, Swear to god, I forgot you were on here... I am really enjoying trying to understand your paper.

that...just sends me off on a cartoon comet, on which I get to pretend I understand the things you are saying...

$H=-J\sum _{ <ik> }{ { \sigma }_{ i }^{ z }{ \sigma }_{ k(\lambda ) }^{ z } } -g(\lambda )J\sum _{ i }{ { \sigma }_{ i }^{ x } }$
?
I just picked $\lambda$ to represent an unknown variable.

Look forward to hearing more.

Last edited: May 18, 2015
5. May 19, 2015

### marcus

http://arxiv.org/abs/1505.04753
Entanglement equilibrium and the Einstein equation
Ted Jacobson
(Submitted on 18 May 2015)
We show that the semiclassical Einstein equation holds if and only if the entanglement entropy in small causal diamonds is stationary at constant volume, when varied from a maximally symmetric vacuum state of geometry and quantum fields. The argument hinges on a conjecture about the variation of the conformal boost energy of quantum fields in small diamonds.
7 pages

6. May 19, 2015

### atyy

The new paper by Jacobson seems very interesting! I was hoping he'd talk about Chirco, Haggard, Riello and Rovelli http://arxiv.org/abs/1401.5262, but he only mentions Rovelli's earlier paper.

Would anyone like to guess whether Hadamard states have anything to do with quantum expanders http://arxiv.org/abs/1209.3304?

7. May 23, 2015

### atyy

http://arxiv.org/abs/1505.05515
Integral Geometry and Holography
Bartlomiej Czech, Lampros Lamprou, Samuel McCandlish, James Sully
(Submitted on 20 May 2015)
We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS3/CFT2 correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS3 whose kinematic space is
two-dimensional de Sitter space.

8. May 24, 2015

### Jimster41

That is some wild bussiness. Very interesting. And I was able to follow more of it than I expected.

It occurs to me I had the label "bulk" flipped at the outset, wrong from the holographic point of view.

But I'm a bit confused as to why the model is one where interval relationships on the rigid, geometrically simple boundary are assigned to curves, points and shapes in the bulk, rather the other way around. Where uniform/rigid geometric objects in the bulk express variation in information content (conditional probability?) that lives on the information rich '"shape" of the boundary.

In other words what if all the geodesics in the bulk are the same (geometrically simple, or at least somehow stiff or constrained) and bulk geometry emerges as encoded-interval-relations on the boundary are passed, through them, to the bulk.

Sort of a dual made of Planck-ish strings on a "Brane" contained in a "Bulk" (where I got the inverted "Bulk" labeling).

Edit] It occurs to me that this is maybe the point, but that formulating the Integration scheme might have been a lot harder from that point of view.

Anyway, mind bending stuff. And I see they ref B. Swingle! Pretty cool.

Last edited: May 24, 2015
9. Jun 3, 2015

### atyy

http://arxiv.org/abs/1506.01353
Hawking Radiation Energy and Entropy from a Bianchi-Smerlak Semiclassical Black Hole
Shohreh Abdolrahimi, Don N. Page
(Submitted on 2 Jun 2015)
Eugenio Bianchi and Matteo Smerlak have found a relationship between the Hawking radiation energy and von Neumann entropy in a conformal field emitted by a semiclassical two-dimensional black hole. We compare this relationship with what might be expected for unitary evolution of a quantum black hole in four and higher dimensions. If one neglects the expected increase in the radiation entropy over the decrease in the black hole Bekenstein-Hawking A/4 entropy that arises from the scattering of the radiation by the barrier near the black hole, the relation works very well, except near the peak of the radiation von Neumann entropy and near the final evaporation. These discrepancies are calculated and discussed as tiny differences between a semiclassical treatment and a quantum gravity treatment.

http://arxiv.org/abs/1506.01353
cMERA as Surface/State Correspondence in AdS/CFT
Masamichi Miyaji, Tokiro Numasawa, Noburo Shiba, Tadashi Takayanagi, Kento Watanabe
(Submitted on 3 Jun 2015)
We present how the surface/state correspondence, conjectured in arXiv:1503.03542, works in the setup of AdS3/CFT2 by generalizing the formulation of cMERA. The boundary states in conformal field theories play a crucial role in our formulation and the bulk diffeomorphism is naturally taken into account. We give an identification of bulk local operators which reproduces correct scalar field solutions on AdS3. We also calculate the information metric for a locally excited state and show that it is given by that of 2d hyperbolic manifold, which is argued to describe the time slice of AdS3.

http://arxiv.org/abs/1506.01366
The BFSS model on the lattice
Veselin G. Filev, Denjoe O'Connor
(Submitted on 3 Jun 2015)
We study the maximally supersymmetric BFFS model at finite temperature and its bosonic relative. For the bosonic model in p+1 dimensions, we find that it effectively reduces to a system of gauged Gaussian matrix models. The effective model captures the low temperature regime of the model including the phase transition. The mass becomes p1/3λ1/3 for large p, with λ the 'tHooft coupling. For p=9 simulations of the model give m=(1.90±.01)λ1/3, which is also the mass gap of the Hamiltonian. We argue that there is no sign' problem in the maximally supersymmetric BFSS model and perform detailed simulations of several observables finding excellent agreement with AdS/CFT predictions when 1/α′ corrections are included.

http://arxiv.org/abs/1506.01337
Violations of the Born rule in cool state-dependent horizons
Donald Marolf, Joseph Polchinski
(Submitted on 3 Jun 2015)
The black hole information problem has motivated many proposals for new physics. One idea, known as state-dependence, is that quantum mechanics must be generalized to describe the physics of black holes, and that fixed linear operators do not provide the fundamental description of experiences for infalling observers. Instead, such experiences are to be described by operators with an extra dependence on the global quantum state. We show that any implementation of this idea strong enough to remove firewalls from generic states requires massive violations of the Born rule. We also demonstrate a sense in which such violations are visible to infalling observers involved in preparing the initial state of the black hole. We emphasize the generality of our results; no details of any specific proposal for state-dependence are required.

Last edited: Jun 3, 2015
10. Jun 6, 2015

### atyy

http://arxiv.org/abs/1506.01623
Area Law from Loop Quantum Gravity
Alioscia Hamma, Ling-Yan Hung, Antonino Marciano, Mingyi Zhang
(Submitted on 4 Jun 2015)
We explore the constraints following from requiring the Area Law in the entanglement entropy in the context of loop quantum gravity. We find a unique solution to the single link wave-function in the large j limit, believed to be appropriate in the semi-classical limit. We then generalize our considerations to multi-link coherent states, and find that the area law is preserved very generically using our single link wave-function as a building block. Finally, we develop the framework that generates families of multi-link states that preserve the area law while avoiding macroscopic entanglement, the space-time analogue of "Schroedinger cat". We note that these states, defined on a given set of graphs, are the ground states of some local Hamiltonian that can be constructed explicitly. This can pot
entially shed light on the construction of the appropriate Hamiltonian constraints in the LQG framework.

11. Jun 19, 2015

### atyy

http://arxiv.org/abs/1506.05792
Geometric entropy and edge modes of the electromagnetic field
William Donnelly, Aron C. Wall
(Submitted on 18 Jun 2015)
We calculate the vacuum entanglement entropy of Maxwell theory in a class of curved spacetimes by Kaluza-Klein reduction of the theory onto a two-dimensional base manifold. Using two-dimensional duality, we express the geometric entropy of the electromagnetic field as the entropy of a tower of scalar fields, constant electric and magnetic fluxes, and a contact term, whose leading order divergence was discovered by Kabat. The complete contact term takes the form of one negative scalar degree of freedom confined to the entangling surface. We show that the geometric entropy agrees with a statistical definition of entanglement entropy that includes edge modes: classical solutions determined by their boundary values on the entangling surface. This resolves a longstanding puzzle about the statistical interpretation of the contact term in the entanglement entropy. We discuss the
implications of this negative term for black hole thermodynamics and the renormalization of Newton's constant.

12. Jun 19, 2015

### Berlin

Wrong post. Sorry.

13. Jun 19, 2015

### atyy

http://arxiv.org/abs/1409.6017
The Cheshire Cap
Emil J. Martinec
(Submitted on 21 Sep 2014 (v1), last revised 3 Oct 2014 (this version, v2))
A key role in black hole dynamics is played by the inner horizon; most of the entropy of a slightly nonextremal charged or rotating black hole is carried there, and the covariant entropy bound suggests that the rest lies in the region between the inner and outer horizon. An attempt to match this onto results of the microstate geometries program suggests that a Higgs branch' of underlying long string states of the configuration space realizes the degrees of freedom on the inner horizon, while the Coulomb branch' describes the inter-horizon region and beyond. Support for this proposal comes from an analysis of the way singularities develop in microstate geometries, and their close analogy to corresponding structures in fivebrane dynamics. These singularities signal the opening up of the long string degrees of freedom of the theory, which are partly visible from the geometry side. A conjectural picture of the black hole interior is proposed, wherein the long string degrees of freedom resolve the geometrical singularity on the inner horizon, yet are sufficiently nonlocal to communicate information to the outer horizon and beyond.

Fractionated Branes and Black Hole Interiors
Emil J. Martinec
(Submitted on 20 May 2015)
Combining a variety of results in string theory and general relativity, a picture of the black hole interior is developed wherein spacetime caps off at an inner horizon, and the inter-horizon region is occupied by a Hagedorn gas of a very low tension state of fractionated branes. This picture leads to natural resolutions of a variety of puzzles concerning quantum black holes. Gravity Research Foundation 2015 Fourth Prize Award for Essays on Gravitation.

A model with no firewall
Samir D. Mathur
(Submitted on 14 Jun 2015)
We construct a model which illustrates the conjecture of fuzzball complementarity. In the fuzzball paradigm, the black hole microstates have no interior, and radiate unitarily from their surface through quanta of energy E∼T. But quanta with E≫T impinging on the fuzzball create large collective excitations of the fuzzball surface. The dynamics of such excitations must be studied as an evolution in superspace, the space of all fuzzball solution |Fi⟩. The states in this superspace are arranged in a hierarchy of complexity'. We argue that evolution towards higher complexity maps, through a duality analogous to AdS/CFT, to infall inside the horizon of the traditional hole. We explain how the large degeneracy of fuzzball states leads to a breakdown of the principle of equivalence at the threshold of horizon formation. We recall that the firewall argument did not invoke the limit E≫T when considering a complementary picture; on the contrary it focused on the dynamics of the E∼T modes which contribute to Hawking radiation. This loophole allows the dual description conjectured in fuzzball complementarity.

Last edited: Jun 19, 2015
14. Jun 29, 2015

### atyy

http://arxiv.org/abs/1403.2048
Era of Big Data Processing: A New Approach via Tensor Networks and Tensor Decompositions
Andrzej Cichocki

http://www.unige.ch/math/vandereycken/bibtexbrowser.php?key=Uschmajew_V_2013&bib=my_pubs.bib
The geometry of algorithms using hierarchical tensors
A. Uschmajew, B. Vandereycken
In this paper, the differential geometry of the novel hierarchical Tucker format for tensors is derived. The set HT_k of tensors with fixed tree T and hierarchical rank k is shown to be a smooth quotient manifold, namely the set of orbits of a Lie group action corresponding to the non-unique basis representation of these hierarchical tensors. Explicit characterizations of the quotient manifold, its tangent space and the tangent space of HT_k are derived, suitable for high-dimensional problems. The usefulness of a complete geometric description is demonstrated by two typical applications. First, new convergence results for the nonlinear Gauss--Seidel method on HT_k are given. Notably and in contrast to earlier works on this subject, the task of minimizing the Rayleigh quotient is also addressed. Second, evolution equations for dynamic tensor approximation are formulated in terms of an explicit projection operator onto the tangent space of HT_k. In addition, a numerical comparison is made between this dynamical approach and the standard one based on truncated singular value decompositions.

15. Jun 30, 2015

### atyy

16. Jul 3, 2015

### atyy

http://arxiv.org/abs/1507.00354
Covariant Constraints on Hole-ography
Netta Engelhardt, Sebastian Fischetti
(Submitted on 1 Jul 2015)
Hole-ography is a prescription relating the areas of surfaces in an AdS bulk to the differential entropy of a family of intervals in the dual CFT. In (2+1) bulk dimensions, or in higher dimensions when the bulk features a sufficient degree of symmetry, we prove that there are surfaces in the bulk that cannot be completely reconstructed using known hole-ographic approaches, even if extremal surfaces reach them. Such surfaces lie in easily identifiable regions: the interiors of holographic screens. These screens admit a holographic interpretation in terms of the Bousso bound. We speculate that this incompleteness of the reconstruction is a form of coarse-graining, with the missing information associated to the holographic screen. We comment on perturbative quantum extensions of our classical results.

http://arxiv.org/abs/1507.00591

AdS/CFT without holography: A hidden dimension on the CFT side and implications for black-hole entropy
H. Nikolic
(Submitted on 2 Jul 2015)
We propose a new non-holographic formulation of AdS/CFT correspondence, according to which quantum gravity on AdS and its dual non-gravitational field theory both live in the same number D of dimensions. The field theory, however, appears (D-1)-dimensional because the interactions do not propagate in one of the dimensions. The D-dimensional action for the field theory can be identified with the sum over (D-1)-dimensional actions with all possible values Λ of the UV cutoff, so that the extra hidden dimension can be identified with Λ. Since there are no interactions in the extra dimension, most of the practical results of standard holographic AdS/CFT correspondence transcribe to non-holographic AdS/CFT without any changes. However, the implications on black-hole entropy change significantly. The maximal black-hole entropy now scales with volume, while the Bekenstein-Hawking entropy is interpreted as the minimal possible black-hole entropy. In this way, the non-holographic AdS/CFT correspondence offers a simple resolution of the black-hole information paradox, consistent with a recently proposed gravitational crystal.

17. Jul 17, 2015

### atyy

http://arxiv.org/abs/1507.03836
Perturbative entanglement thermodynamics for AdS spacetime: Renormalization
Rohit Mishra, Harvendra Singh
(Submitted on 14 Jul 2015)
We study the effect of charged excitations in the AdS spacetime on the first law of entanglement thermodynamics. It is found that boosted' AdS black holes give rise to a more general form of first law which includes chemical potential and charge density. To obtain this result we have to resort to a second order perturbative calculation of entanglement entropy for small size subsystems. At first order the form of entanglement law remains unchanged even in the presence of charged excitations. But the thermodynamic quantities have to be appropriately renormalized' at the second order due to the corrections. We work in the perturbative regime where Tthermal≪TE.

http://arxiv.org/abs/1507.04130
Bulk Locality and Boundary Creating Operators
Yu Nakayama, Hirosi Ooguri
(Submitted on 15 Jul 2015)
We formulate a minimum requirement for CFT operators to be localized in the dual AdS. In any spacetime dimensions, we show that a general solution to the requirement is a linear superposition of operators creating spherical boundaries in CFT, with the dilatation by the imaginary unit from their centers. This generalizes the recent proposal by Miyaji et al. for bulk local operators in the three dimensional AdS. We show that Ishibashi states for the global conformal symmetry in any dimensions and with the imaginary dilatation obey free field equations in AdS and that incorporating bulk interactions require their superpositions. We also comment on the recent proposals by Kabat et al., and by H. Verlinde.

http://arxiv.org/abs/1507.04633
Entanglement renormalization and integral geometry
Xing Huang, Feng-Li Lin
(Submitted on 16 Jul 2015)
We revisit the applications of integral geometry in AdS3 and argue that the volume form of the kinematic space can be understood as a measure of entanglement between the end points of a geodesic. We explain how this idea naturally fits into the picture of entanglement renormalization of an entangled pair, from which we can holographically understand the operations of disentangler and isometry in multi-scale entanglement renormalization ansatz (MERA). A renormalization group (RG) equation of the long-distance entanglement is then derived, which indicates how the entanglement is reshuffled by holographic isometry operation. We then generalize this integral geometric construction to higher dimensional bulk space of homogeneity and isotropy.

Last edited: Jul 17, 2015
18. Jul 23, 2015

### atyy

http://arxiv.org/abs/1507.06410
Generalized entanglement entropy
Marika Taylor
(Submitted on 23 Jul 2015)
We discuss two measures of entanglement in quantum field theory and their holographic realizations. For field theories admitting a global symmetry, we introduce a global symmetry entanglement entropy, associated with the partitioning of the symmetry group. This quantity is proposed to be related to the generalized holographic entanglement entropy defined via the partitioning of the internal space of the bulk geometry. The second measure of quantum field theory entanglement is the field space entanglement entropy, obtained by integrating out a subset of the quantum fields. We argue that field space entanglement entropy cannot be precisely realised geometrically in a holographic dual. However, for holographic geometries with interior decoupling regions, the differential entropy provides a close analogue to the field space entanglement entropy. We derive generic descriptions of such inner throat regions in terms of gravity coupled to massive scalars and show how the differential entropy in the throat captures features of the field space entanglement entropy.

19. Jul 29, 2015

### atyy

http://arxiv.org/abs/1507.07555
Gravity Dual of Quantum Information Metric
Masamichi MIyaji, Tokiro Numasawa, Noburo Shiba, Tadashi Takayanagi, Kento Watanabe
(Submitted on 27 Jul 2015)
We study a quantum information metric (or fidelity susceptibility) in conformal field theories with respect to a small perturbation by a primary operator. We argue that its gravity dual is approximately given by a volume of maximal time slice in an AdS spacetime when the perturbation is exactly marginal. We confirm our claim in several examples.

20. Aug 6, 2015

### atyy

http://arxiv.org/abs/1508.00897
Canonical Energy is Quantum Fisher Information
Nima Lashkari, Mark Van Raamsdonk
(Submitted on 4 Aug 2015)
In quantum information theory, Fisher Information is a natural metric on the space of perturbations to a density matrix, defined by calculating the relative entropy with the unperturbed state at quadratic order in perturbations. In gravitational physics, Canonical Energy defines a natural metric on the space of perturbations to spacetimes with a Killing horizon. In this paper, we show that the Fisher information metric for perturbations to the vacuum density matrix of a ball-shaped region B in a holographic CFT is dual to the canonical energy metric for perturbations to a corresponding Rindler wedge R_B of Anti-de-Sitter space. Positivity of relative entropy at second order implies that the Fisher information metric is positive definite. Thus, for physical perturbations to anti-de-Sitter spacetime, the canonical energy associated to any Rindler wedge must be positive. This second-order constraint on the metric extends the first order result from relative entropy positivity that physical perturbations must satisfy the linearized Einstein's equations.