Bit Threads and Holographic Entanglement
|
|
- Dale Hubbard
- 5 years ago
- Views:
Transcription
1 it Threads and Holographic Entanglement Matthew Headrick randeis Uniersity ased on arxi: [hep-th], with Michael Freedman Entropy and area In semiclassical graity, entropies are related to surface areas General surface: S area 4G N Special surfaces (horizon, minimal surface, extremal surface): S = area 4G N Ryu-Takayanagi [ 06]: S() = 4G N area(m()) m() = minimal surface homologous to (From now on, we set 4G N = ln = ) m()
2 Interpretation Why? Naie answer: Microstate bits of ρ lie on m(), one bit per 4 Planck areas onfusions: Under continuous changes in boundary region, minimal surface can jump, for example: m() =m() [ m() m() 6= m() [ m() (Note: ρ does not jump; not a conentional exchange-of-dominance phase transition [Headrick 3]) Important quantities, like condiitonal entropy H( ) = S() S(), mutual information conditional mutual information I( : ) = S() + S() S(), I( : ) = S() + S() S() S(), are gien by differences of areas of surfaces passing through different regions of bulk Let s recall their information-theoretic meaning lassical: H( ) = # of (independent) bits belonging purely to I( : ) = # shared with
3 H( ) conditional S() entropies S() S() H( ) S() I( S() : ) Quantum: Entangled pair H( ) of bits contributes I( : to ) I( : ), H( ) to H( ) an lead to H( ) < 0 p ( i + i) 0 0 = p ( i + 0 i) I( : ) S() I( : ) H( ) What do differences between areas of surfaces, passing through different parts of bulk, hae to do with redundancy, entanglement, etc between bits of and? What does holographic proof of strong subadditiity hae to do with monotonicity of correlations? To answer these questions, I will present a new formulation of RT Does not refer to minimal = p surfaces; ( i + these i) are demoted to a calculational deice S() S() S() H( ) I( : ) H( ) 0 0 Suggests a new way to think about the connection between spacetime geometry and information Max flow-min cut (Originally on graphs, in context of network theory; continuous ersion [Federer 74, Strang 83, Nozawa 90]) onsider a Riemannian manifold with boundary Define a flow as a ector field st = 0, Let be a subset of boundary For any surface m homologous to, Strongest bound is min cut: = m area(m) max min area(m) m 3
4 Max flow-min cut theorem says there are no other obstructions to increasing flux: max = min area(m) m () := maximizer Notes: () m() On m(), () = unit normal ector; elsewhere, () is highly non-unique Unlike min cut, max flow is a linear programming problem flow can be thought of as a set of oriented threads (flow lines) with transerse density = RT ersion 0: S() = max = max # of threads coming out of m() () Each thread has cross section of 4 Planck areas utomatically incorporates homology condition & global minimization Threads can end on c or horizon Each thread carries one independent bit of ρ, either entangled with c or in a mixed state Threads can also return to, but those don t count Minimal surface does not play a fundamental role, but acts as bottleneck limiting number of threads Naturally implements holographic principle: entropy is area because bits are carried by one-dimensional objects Threads & information Now we address conceptual puzzles with RT raised before First, it can be shown that () changes continuously with, een when m() jumps Now consider two regions, We can maximize flux through or If S() < S() + S(), then we cannot simultaneously maximize through both ut we can always maximize through and (nesting property) all such a flow (, ) () (, ) () 4
5 Example : S() = S() =, S() = 3 I( : ) =, H( ) = Maximizing on, we can also maximize on either or (, ) (,) Lesson : Threads that are stuck on represent bits unique to Threads that can be moed between & represent correlated pairs of bits Example : S() = S() =, S() = I( : ) = 3, H( ) = entanglement! One thread leaing must go to, and ice ersa (, ) (,) Lesson : Threads connecting & (and switch orientation) represent entangled pairs of bits onditional entropy: H( ) = S() S() = (, ) = (, ) (, ) Mutual information: I( : ) = S() H( ) = (, ) (, ) Subadditiity is clear Max flows can be defined without regulator, as flows that cannot be augmented 5
6 Regulator-free definition of mutual information: I( : ) = Define flow ( : ) = ((, ) (, )) ((, ) (, )) which goes from to through entanglement wedge r() Implies I( : ) area (r() bottleneck) ( : ) Gien lessons aboe, freedom to moe threads around on indicates correlations with ; freedom to add loops that begin and end on indicates entanglement within onditional mutual information: I( : ) = H( ) H( ) = (,, ) (,, ) = (max on ) (min on ), while maximizing on & = moeable between &, while maximizing on & = (moeable between & ) (moeable between & ) = I( : ) I( : ) Strong subadditiity is clear Exercise for reader: Find flow-based proof of monogamy of mutual information inequality [Hayden-Hedarick- Maloney ], I( : ) I( : ) + I( : ) and higher inequalities [ao et al 5] annot be proed using just nesting property (see eg 4-party GHZ); indicates some new property of flows Do bit threads indicate that bipartite entanglement is priileged? Possible to represent 3-party GHZ state 6
7 oariant flows To appear (with Veronika Hubeny) Flow ersion of Hubeny-Rangamani-Takayanagi [ 07] coariant entanglement entropy formula: Define a flow as a ector field (in the full spacetime) st = 0 and integrated norm bound: timelike cure and unit normal ector field u on it, ds u In other words, any obserer sees oer their lifetime a total of at most thread per 4 Planck areas HRT ersion 0: S() = max (D() = boundary causal domain of ) D() Max flow () stays inside entanglement wedge, squeezes out through extremal surface it threads can lie on a common auchy slice (equialent to maximin [Wall ]), or spread out in time 7
Bit Threads and Holographic Entanglement
it Threads and Holographic Entanglement Matthew Headrick randeis University Strings 2016, eijing ased on arxiv:1604.00354 [hep-th], with Michael Freedman Contents 1 How should one think about the minimal
More informationBit Threads & Holographic Monogamy
Bit Threads & Holographic Monogamy Matthew Headrick Brandeis University Entanglement in Quantum Systems GGI Florence, June 2018 Based on arxiv:1604.00354 [hep-th] with M. Freedman & forthcoming work with
More information21 Holographic Entanglement Entropy
21 Holographic Entanglement Entropy 21.1 The formula We now turn to entanglement entropy in CFTs with a semiclassical holographic dual. That is, we assume the CFT has a large number of degrees of freedom
More informationProperties of entropy in holographic theories
Properties of entropy in holographic theories Matthew Headrick randeis University Contents 0 Definitions 1 Properties of entropy Entanglement entropy in QFT 3 Ryu-Takayanagi formula 6 Monogamy 8 5 SS of
More informationQuantum Entanglement and the Geometry of Spacetime
Quantum Entanglement and the Geometry of Spacetime Matthew Headrick Brandeis University UMass-Boston Physics Colloquium October 26, 2017 It from Qubit Simons Foundation Entropy and area Bekenstein-Hawking
More informationBit Threads and Holographic Monogamy
Bit Threads and Holographic Monogamy Shawn X. Cui, 1 Patrick Hayden, 1 Temple He, 2 Matthew Headrick, 3,4 Bogdan Stoica, 3,5 and Michael Walter 6 1 Stanford Institute for Theoretical Physics, Stanford
More informationQuantum Information and Entanglement in Holographic Theories
Quantum Information and Entanglement in Holographic Theories Matthew Headrick randeis University Contents 1 asic notions 2 1.1 Entanglement entropy & mutual information............................ 2 1.2
More informationarxiv: v1 [hep-th] 23 Dec 2013
Prepared for submission to JHEP RX-TH673 arxiv:1312.6717v1 [hep-th] 23 Dec 2013 General properties of holographic entanglement entropy Matthew Headrick Martin Fisher School of Physics, randeis University,
More informationOverview: Entanglement Entropy
Overview: Entanglement Entropy Matthew Headrick Brandeis University January 27, 2014 Quantum Fields beyond Perturbation Theory KITP 0 Intro & disclaimer Over past 10 years, explosion of activity in entanglement
More informationCAUSAL WEDGES in AdS/CFT
CUSL WEDGES in ds/cft Veronika Hubeny Durham University Gauge/Gravity Duality 2013 Max Planck Institute for Physics, 29 July 2013 to 2 ugust 2013 Based on: VH & M.Rangamani: 1204.1698, VH, M.Rangamani,
More informationBlack Holes are Alpha-Bit Sup
Black Holes are Alpha-Bit Sup arxiv:1706.09434, arxiv:1805.????? Patrick Hayden and Geoffrey Penington, Stanford University Preview Qubits are composite resources Another resource (that you have never
More informationEntanglement, geometry and the Ryu Takayanagi formula
Entanglement, geometry and the Ryu Takayanagi formula Juan Maldacena Kyoto, 2013 Aitor Lewkowycz Lewkowycz, JM ArXiv:1304.4926 & Faulkner, Lewkowycz, JM, to appear Tom Faulkner Previously argued by Fursaev
More informationHOLOGRAPHIC PROBES! COLLAPSING BLACK HOLES OF! Veronika Hubeny! Durham University & Institute for Advanced Study
HOLOGRAPHIC PROBES! OF! COLLAPSING BLACK HOLES Veronika Hubeny! Durham University & Institute for Advanced Study New frontiers in dynamical gravity workshop Cambridge, March 26, 2014 Based on work w/ H.
More informationEmergent Causality in Holography
Emergent Causality in Holography Netta Engelhardt Princeton University 20.6.18 Based mostly on: NE, Horowitz 16; NE 16; NE, Fischetti 17 Spacetime Emergence Holography: the equivalence of a higher-dim
More informationarxiv: v2 [hep-th] 13 Sep 2015
Non-linear Holographic Entanglement Entropy Inequalities for Single Boundary D CFT Emory Brown, 1, Ning Bao, 1, and Sepehr Nezami 3 1 Institute for Quantum Information and Matter Walter Burke Institute
More informationEntanglement Inequalities
Entanglement Inequalities Hirosi Ooguri Walter Burke Institute for Theoretical Physics, Caltech Kavli IPMU, University of Tokyo Nafplion, Greece, 5 11 July 2015 1/44 Which CFT's have Gravity Duals? Which
More informationHolographic Entanglement Entropy
Motivation Time-dependent Multi-region Summary Holographic entanglement entropy for time dependent states and disconnected regions Durham University INT08: From Strings to Things, April 3, 2008 VH, M.
More informationQuantum mechanics and the geometry of spacetime
Quantum mechanics and the geometry of spacetime Juan Maldacena PPCM Conference May 2014 Outline Brief review of the gauge/gravity duality Role of strong coupling in the emergence of the interior Role of
More information31st Jerusalem Winter School in Theoretical Physics: Problem Set 2
31st Jerusalem Winter School in Theoretical Physics: Problem Set Contents Frank Verstraete: Quantum Information and Quantum Matter : 3 : Solution to Problem 9 7 Daniel Harlow: Black Holes and Quantum Information
More informationHolographic Entanglement Entropy. (with H. Casini, M. Huerta, J. Hung, M. Smolkin & A. Yale) (arxiv: , arxiv: )
v Holographic Entanglement Entropy (with H. Casini, M. Huerta, J. Hung, M. Smolkin & A. Yale) (arxiv:1102.0440, arxiv:1110.1084) Entanglement Entropy what is entanglement entropy? general tool; divide
More informationReview of Holographic (and other) computations of Entanglement Entropy, and its role in gravity
Review of Holographic (and other) computations of Entanglement Entropy, and its role in gravity Entanglement Entropy what is entanglement entropy? general tool; divide quantum system into two parts and
More informationLecture 6: Graphical Models
Lecture 6: Graphical Models Kai-Wei Chang CS @ Uniersity of Virginia kw@kwchang.net Some slides are adapted from Viek Skirmar s course on Structured Prediction 1 So far We discussed sequence labeling tasks:
More informationarxiv: v1 [hep-th] 16 Jan 2018
Prepared for submission to JHEP Space-time random tensor networks and holographic duality arxiv:1801.05289v1 [hep-th] 16 Jan 2018 Xiao-Liang Qi, and Zhao Yang Stanford Institute for Theoretical Physics,
More informationLECTURE 2: CROSS PRODUCTS, MULTILINEARITY, AND AREAS OF PARALLELOGRAMS
LECTURE : CROSS PRODUCTS, MULTILINEARITY, AND AREAS OF PARALLELOGRAMS MA1111: LINEAR ALGEBRA I, MICHAELMAS 016 1. Finishing up dot products Last time we stated the following theorem, for which I owe you
More informationHolographic Entanglement of Purification
20 Years Later: The Many Faces of ds/cft @ Princeton, Oct.31-Nov.3, 2017 Holograhic ntanglement of Purification Tadashi Takayanagi Yukawa Institute for Theoretical Physics (YITP, Kyoto Univ. ased on arxiv:1708.09393
More informationReversal in time order of interactive events: Collision of inclined rods
Reersal in time order of interactie eents: Collision of inclined rods Published in The European Journal of Physics Eur. J. Phys. 27 819-824 http://www.iop.org/ej/abstract/0143-0807/27/4/013 Chandru Iyer
More informationIs spacetime a quantum errorcorrecting
Is spacetime a quantum errorcorrecting code? Fernando Pastawski, Beni Yoshida, Daniel Harlow, John Preskill = HaPPY J. of High Energy Physics 06 (2015) 149 John Preskill STOC in Montreal 20 June 2017 Frontiers
More informationReconstructing Bulk from Boundary: clues and challenges
Reconstructing Bulk from Boundary: clues and challenges Ben Freivogel GRAPPA and ITFA Universiteit van Amsterdam Ben Freivogel () Reconstructing Bulk from Boundary May 24, 2013 1 / 28 Need quantum gravity
More informationCFT PROBES OF BULK GEOMETRY
CFT PROBES OF BULK GEOMETRY Veronika Hubeny Durham University May 24, 2012 Based on: VH: 1203.1044 VH & M.Rangamani: 1204.1698 OUTLINE Motivation & Background Features of Extremal Surfaces Probing Horizons
More information재규격화군흐름, 녹채 & Holography 과홀로그래피
RG Flows, Entanglement 재규격화군흐름, 녹채 & Holography 과홀로그래피 Strings 2013 Sogang Univesity, Seoul Korea, 26-29 June 2013 New Dialogues in Theoretical Physics: Particle Physics Statistical Mechanics Renormalization
More informationMcEliece and Niederreiter Cryptosystems That Resist Quantum Fourier Sampling Attacks
McEliece and Niederreiter Cryptosystems That Resist Quantum Fourier Sampling Attacks Hang Dinh Indiana Uniersity South Bend joint work with Cristopher Moore Uniersity of New Mexico Alexander Russell Uniersity
More informationScalar multiplication and algebraic direction of a vector
Roberto s Notes on Linear Algebra Chapter 1: Geometric ectors Section 5 Scalar multiplication and algebraic direction of a ector What you need to know already: of a geometric ectors. Length and geometric
More informationThermalization after holographic bilocal quench
Prepared for submission to JHEP Thermalization after holographic bilocal quench arxiv:1706.07390v3 [hep-th] 10 Oct 017 Irina Ya. Aref eva, Mikhail A. Khramtsov, Maria D. Tikhanovskaya Steklov Mathematical
More informationQuantum mechanics and the geometry of space4me
Quantum mechanics and the geometry of space4me Juan Maldacena PPCM Conference May 2014 Outline Brief review of the gauge/gravity duality Role of strong coupling in the emergence of the interior Role of
More informationChapter 14 Thermal Physics: A Microscopic View
Chapter 14 Thermal Physics: A Microscopic View The main focus of this chapter is the application of some of the basic principles we learned earlier to thermal physics. This will gie us some important insights
More informationEntanglement entropy and the F theorem
Entanglement entropy and the F theorem Mathematical Sciences and research centre, Southampton June 9, 2016 H RESEARH ENT Introduction This talk will be about: 1. Entanglement entropy 2. The F theorem for
More informationVISUAL PHYSICS ONLINE RECTLINEAR MOTION: UNIFORM ACCELERATION
VISUAL PHYSICS ONLINE RECTLINEAR MOTION: UNIFORM ACCELERATION Predict Obsere Explain Exercise 1 Take an A4 sheet of paper and a heay object (cricket ball, basketball, brick, book, etc). Predict what will
More informationHolographic Entanglement entropy and second law of Black holes. Sudipta Sarkar
Holographic Entanglement entropy and second law of Black holes Sudipta Sarkar Indian Institute of Technology Gandhinagar Based on: arxiv:1504.04706, arxiv:1306.1623 In collaboration with Aron Wall (IAS),
More informationBlack Holes, Holography, and Quantum Information
Black Holes, Holography, and Quantum Information Daniel Harlow Massachusetts Institute of Technology August 31, 2017 1 Black Holes Black holes are the most extreme objects we see in nature! Classically
More informationAdS/CFT Correspondence and Entanglement Entropy
AdS/CFT Correspondence and Entanglement Entropy Tadashi Takayanagi (Kyoto U.) Based on hep-th/0603001 [Phys.Rev.Lett.96(2006)181602] hep-th/0605073 [JHEP 0608(2006)045] with Shinsei Ryu (KITP) hep-th/0608213
More informationQuantum Metric and Entanglement on Spin Networks
Quantum Metric and Entanglement on Spin Networks Fabio Maria Mele Dipartimento di Fisica Ettore Pancini Universitá degli Studi di Napoli Federico II COST Training School Quantum Spacetime and Physics Models
More informationSPACETIME FROM ENTANGLEMENT - journal club notes -
SPACETIME FROM ENTANGLEMENT - journal club notes - Chris Heinrich 1 Outline 1. Introduction Big picture: Want a quantum theory of gravity Best understanding of quantum gravity so far arises through AdS/CFT
More informationTowards a 2nd Law for Lovelock Theory
Towards a 2nd Law for Lovelock Theory Nilay Kundu YITP, Kyoto University This talk is based on the following preprint arxiv:1612.04024 [hep-th] Towards a second law for Lovelock theories Sayantani Bhattacharyya,
More informationRelativistic Energy Derivation
Relatiistic Energy Deriation Flamenco Chuck Keyser //4 ass Deriation (The ass Creation Equation ρ, ρ as the initial condition, C the mass creation rate, T the time, ρ a density. Let V be a second mass
More informationDon Marolf 7/17/14 UCSB
Don Marolf 7/17/14 UCSB D=4: F = GM 2 /r 2, dimensionless coupling GE 2 /ħc 5 grows with E. Quantum fluctuations are a problem, growing at short distances. Growth must(?) stop somewhere (non trivial fixed
More informationHot multiboundary wormholes from bipartite entanglement
Classical and Quantum Gravity PAPER Hot multiboundary wormholes from bipartite entanglement To cite this article: Donald Marolf et al 0 Class. Quantum Grav. 00 Manuscript version: Accepted Manuscript Accepted
More informationYour Thoughts. What is the difference between elastic collision and inelastic collision?
Your Thoughts This seemed pretty easy...before we got the checkpoint questions What is the difference between elastic collision and inelastic collision? The most confusing part of the pre lecture was the
More informationHolographic Entanglement Beyond Classical Gravity
Holographic Entanglement Beyond Classical Gravity Xi Dong Stanford University August 2, 2013 Based on arxiv:1306.4682 with Taylor Barrella, Sean Hartnoll, and Victoria Martin See also [Faulkner (1303.7221)]
More informationOptimization Problems in Multiple Subtree Graphs
Optimization Problems in Multiple Subtree Graphs Danny Hermelin Dror Rawitz February 28, 2010 Abstract We study arious optimization problems in t-subtree graphs, the intersection graphs of t- subtrees,
More informationA holographic entanglement triptych
holographic entanglement triptych Mukund Rangamani Kavli IPMU, Tokyo January 8, 2015 Causality Density Topology 1 Q?@ y D[c ]! E c D[] 2 mple of a causally trivial spacetime and a boundary region whose
More informationarxiv: v1 [hep-th] 4 May 2017
Butterfly velocity and bulk causal structure arxiv:1705.01728v1 [hep-th] 4 May 2017 Xiao-Liang Qi 1, Zhao Yang 1 1 Department of Physics, Stanford University, Stanford, CA 94305, USA Abstract The butterfly
More informationReview of Matrices and Vectors 1/45
Reiew of Matrices and Vectors /45 /45 Definition of Vector: A collection of comple or real numbers, generally put in a column [ ] T "! Transpose + + + b a b a b b a a " " " b a b a Definition of Vector
More informationarxiv: v2 [hep-th] 13 Jul 2018
Complexity and Boost Symmetry arxi:1702.039572 [hep-th] 13 Jul 2018 Ying Zhao Stanford Institute for Theoretical Physics and Department of Physics, Stanford Uniersity, Stanford, CA 94305-4060, USA zhaoying@stanford.edu
More informationThe Thomas Precession Factor in Spin-Orbit Interaction
p. The Thomas Preession Fator in Spin-Orbit Interation Herbert Kroemer * Department of Eletrial and Computer Engineering, Uniersity of California, Santa Barbara, CA 9306 The origin of the Thomas fator
More informationGeneral Lorentz Boost Transformations, Acting on Some Important Physical Quantities
General Lorentz Boost Transformations, Acting on Some Important Physical Quantities We are interested in transforming measurements made in a reference frame O into measurements of the same quantities as
More informationdv dv 2 2 dt dt dv dt
1/3/14 hapter 8 Second order circuits A circuit with two energy storage elements: One inductor, one capacitor Two capacitors, or Two inductors i Such a circuit will be described with second order differential
More informationEntanglement Entropy and AdS/CFT
Entanglement Entropy and AdS/CFT Christian Ecker 2 nd DK Colloquium January 19, 2015 The main messages of this talk Entanglement entropy is a measure for entanglement in quantum systems. (Other measures
More informationGravity Actions from Tensor Networks
Gravity Actions from Tensor Networks [hep-th/1609.04645] with T. Takayanagi and K. Watanabe and ongoing work with P. Caputa, N. Kundu, T. Takayanagi, K. Watanabe Masamichi Miyaji Yukawa Institute for Theoretical
More informationPurpose of the experiment
Impulse and Momentum PES 116 Adanced Physics Lab I Purpose of the experiment Measure a cart s momentum change and compare to the impulse it receies. Compare aerage and peak forces in impulses. To put the
More informationPerturbative Proof of the Covariant Entropy Bound
Perturbative Proof of the Covariant Entropy Bound Work in progress, with H. Casini, Z. Fisher, and J. Maldacena Raphael Bousso Berkeley Center for Theoretical Physics University of California, Berkeley
More informationDoppler shifts in astronomy
7.4 Doppler shift 253 Diide the transformation (3.4) by as follows: = g 1 bck. (Lorentz transformation) (7.43) Eliminate in the right-hand term with (41) and then inoke (42) to yield = g (1 b cos u). (7.44)
More informationTHE LANDAUER LIMIT AND THERMODYNAMICS OF BIOLOGICAL SYSTEMS
THE LANDAUER LIMIT AND THERMODYNAMICS OF BIOLOGICAL SYSTEMS Daid H. Wolpert Santa Fe Institute, MIT ttp://daidwolpert.weebly.com Jos Grocow Santa Fe Institute HEAT OF ERASING A BIT a) b) c) R 1 2 2R 1
More informationUNDERSTAND MOTION IN ONE AND TWO DIMENSIONS
SUBAREA I. COMPETENCY 1.0 UNDERSTAND MOTION IN ONE AND TWO DIMENSIONS MECHANICS Skill 1.1 Calculating displacement, aerage elocity, instantaneous elocity, and acceleration in a gien frame of reference
More informationIntroduction to the Ryu-Takayanagi Formula
Introduction to the Ryu-Takayanagi Formula PHYS 48300 String Theory-1, Masaya Fukami {13 March 2018} 1 Introduction The idea of holography has played central roles in recent developments of string theory.
More informationThe Kinetic Theory of Gases
978-1-107-1788-3 Classical and Quantum Thermal Physics The Kinetic Theory of Gases CHAPTER 1 1.0 Kinetic Theory, Classical and Quantum Thermodynamics Two important components of the unierse are: the matter
More informationA Holographic Description of Black Hole Singularities. Gary Horowitz UC Santa Barbara
A Holographic Description of Black Hole Singularities Gary Horowitz UC Santa Barbara Global event horizons do not exist in quantum gravity: String theory predicts that quantum gravity is holographic:
More informationTest of General Relativity Theory by Investigating the Conservation of Energy in a Relativistic Free Fall in the Uniform Gravitational Field
Test of General Relatiity Theory by Inestigating the Conseration of Energy in a Relatiisti Free Fall in the Uniform Graitational Field By Jarosla Hyneek 1 Abstrat: This paper inestigates the General Relatiity
More informationEntanglement and the Bekenstein-Hawking entropy
Entanglement and the Bekenstein-Hawking entropy Eugenio Bianchi relativity.phys.lsu.edu/ilqgs International Loop Quantum Gravity Seminar Black hole entropy Bekenstein-Hawking 1974 Process: matter falling
More informationLiving on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers
Living on the Edge: Toy Model for Holographic Reconstruction of lgebras with Centers arxiv:1611.05841v1 [hep-th] 17 Nov 2016 William Donnelly, Ben Michel, Donald Marolf, and Jason Wien Department of Physics,
More informationarxiv: v2 [hep-th] 3 Oct 2017
Time dependence of entanglement for steady state formation in AdS 3 /CFT 2 arxiv:1709.08614v2 [hep-th] 3 Oct 2017 M Flory 1 J Erdmenger 2 D Fernández 3 E Megías 4 A Straub 5 and P Witkowski 6 1 Institute
More informationQuantum Null Energy Condition A remarkable inequality in physics
Quantum Null Energy Condition A remarkable inequality in physics Daniel Grumiller Institute for Theoretical Physics TU Wien Erwin-Schrödinger Institute, May 2018 1710.09837 Equalities in mathematics and
More informationInitial-Value Problems in General Relativity
Initial-Value Problems in General Relativity Michael Horbatsch March 30, 2006 1 Introduction In this paper the initial-value formulation of general relativity is reviewed. In section (2) domains of dependence,
More informationEverything should be made as simple as possible, but not simpler -A. Einstein
r1 Eerything should be made as simple as possible, but not simpler -A. Einstein r2 SR1... -3-2 -1 0 1 2 3... Synchronizing clocks At the origin, at three o clock, the clock sends out a light signal to
More informationAlgorithms and Data Structures 2014 Exercises and Solutions Week 14
lgorithms and ata tructures 0 xercises and s Week Linear programming. onsider the following linear program. maximize x y 0 x + y 7 x x 0 y 0 x + 3y Plot the feasible region and identify the optimal solution.
More informationQuantum entanglement in the 21 st century. John Preskill The Quantum Century: 100 Years of the Bohr Atom 3 October 2013
Quantum entanglement in the 21 st century John Preskill The Quantum Century: 100 Years of the Bohr Atom 3 October 2013 My well-worn copy, bought in 1966 when I was 13. George Gamow, recalling Bohr s Theoretical
More informationBalanced Partitions of Vector Sequences
Balanced Partitions of Vector Sequences Imre Bárány Benjamin Doerr December 20, 2004 Abstract Let d,r N and be any norm on R d. Let B denote the unit ball with respect to this norm. We show that any sequence
More informationEntanglement entropy in three dimensional gravity
DCPT-14/67 Prepared for submission to JHEP Entanglement entropy in three dimensional gravity arxiv:141.687v3 [hep-th] 8 Apr 15 Henry Maxfield Centre for Particle Theory & Department of Mathematical Sciences,
More informationBlack Hole Entropy and Gauge/Gravity Duality
Tatsuma Nishioka (Kyoto,IPMU) based on PRD 77:064005,2008 with T. Azeyanagi and T. Takayanagi JHEP 0904:019,2009 with T. Hartman, K. Murata and A. Strominger JHEP 0905:077,2009 with G. Compere and K. Murata
More informationLecture 1. 1 Overview. 2 Maximum Flow. COMPSCI 532: Design and Analysis of Algorithms August 26, 2015
COMPSCI 532: Design and Analysis of Algorithms August 26, 205 Lecture Lecturer: Debmalya Panigrahi Scribe: Allen Xiao Oeriew In this lecture, we will define the maximum flow problem and discuss two algorithms
More informationMath 144 Activity #9 Introduction to Vectors
144 p 1 Math 144 ctiity #9 Introduction to Vectors Often times you hear people use the words speed and elocity. Is there a difference between the two? If so, what is the difference? Discuss this with your
More informationarxiv: v2 [hep-th] 1 Aug 2018
Prepared for submission to JHEP Learning the Alpha-bits of Black Holes arxiv:1807.06041v2 [hep-th] 1 Aug 2018 Patrick Hayden and Geoffrey Penington Stanford Institute for Theoretical Physics, Stanford
More informationHolographic entanglement entropy beyond AdS/CFT
Holographic entanglement entropy beyond AdS/CFT Edgar Shaghoulian Kavli Institute for the Physics and Mathematics of the Universe April 8, 014 Dionysios Anninos, Joshua Samani, and ES hep-th:1309.579 Contents
More informationUniformity of Theta- Assignment Hypothesis. Transit i ve s. Unaccusatives. Unergatives. Double object constructions. Previously... Bill lied.
Preiously... Uniformity of Theta- Assignment Hypothesis, daughter of P = Agent, daughter of P = Theme PP, daughter of! = Goal P called P Chris P P books gae P to PP Chris Unaccusaties Transit i e s The
More informationWhy we need quantum gravity and why we don t have it
Why we need quantum gravity and why we don t have it Steve Carlip UC Davis Quantum Gravity: Physics and Philosophy IHES, Bures-sur-Yvette October 2017 The first appearance of quantum gravity Einstein 1916:
More informationA Geometric Review of Linear Algebra
A Geometric Reiew of Linear Algebra The following is a compact reiew of the primary concepts of linear algebra. I assume the reader is familiar with basic (i.e., high school) algebra and trigonometry.
More informationProblems. 66 km/h B km/h 30 A. v A. 1.5 ft
Problems Problem 3.1 2700-lb automobile starts from rest and traels a quarter of a mile. ssume that the coefficient of static friction between the tires and the paement is 0.70, the automobile has frontwheel
More informationarxiv: v2 [hep-th] 21 Dec 2014
arxiv:1412.3514v2 [hep-th] 21 Dec 2014 Inviolable energy conditions from entanglement inequalities Nima Lashkari, Charles Rabideau, Philippe Sabella-Garnier, Mark Van Raamsdonk Department of Physics and
More informationTime Evolution of Holographic Complexity
Time Evolution of Holographic Complexity Sotaro Sugishita (Osaka Univ.) based on arxiv:1709.10184 [JHEP 1711, 188 (2017)] with Dean Carmi, Shira Chapman, Hugo Marrochio, Robert Myers RIKEN-Osaka-OIST Joint
More informationBLACK HOLES IN LOOP QUANTUM GRAVITY. Alejandro Corichi
BLACK HOLES IN LOOP QUANTUM GRAVITY Alejandro Corichi UNAM-Morelia, Mexico ICGC 07, IUCAA, December 20th 2007 1 BLACK HOLES AND QUANTUM GRAVITY? Black Holes are, as Chandrasekhar used to say:... the most
More informationIdeal Classes and Matrix Conjugation over F q [X]
Ideal lasses and Matrix onjugation oer F q [X] hristian Berghoff, Georg-August-Uniersität Göttingen Trais Morrison, Penn State Uniersity Yujia Qiu, Ruprecht-Karls-Uniersität Heidelerg Thomas Sicking, Georg-August-Uniersität
More informationEPR Pairs, Local Projection and Quantum Teleportation in Holography
YITP long term workshop 2016, One Day Conference, 2016/06/08 @ YITP, Kyoto EPR Pairs, Local Projection and Quantum Teleportation in Holography Kento Watanabe (YITP, Kyoto) arxiv: 1604.01772 [hep-th] with
More informationLast Time: Start Rotational Motion (now thru mid Nov) Basics: Angular Speed, Angular Acceleration
Last Time: Start Rotational Motion (now thru mid No) Basics: Angular Speed, Angular Acceleration Today: Reiew, Centripetal Acceleration, Newtonian Graitation i HW #6 due Tuesday, Oct 19, 11:59 p.m. Exam
More informationNonarchimedean Holographic Entropy from Networks of Perfect Tensors
CALT-TH-2018-053 Nonarchimedean Holographic Entropy from Networks of Perfect Tensors arxiv:1812.04057v1 [hep-th] 10 Dec 2018 Matthew Heydeman, 1 Matilde Marcolli, 2,3,4 Sarthak Parikh, 2 & Ingmar Saberi
More informationOn a holographic quantum quench with a finite size effect
On a holographic quantum quench with a finite size effect Tomonori Ugajin (U. Tokyo KITP) Based on work in progress with G.Mandal, R.Sinha Holographic Vistas on gravity and strings YITP, 2014 Introduction
More informationEntanglement Entropy for Disjoint Intervals in AdS/CFT
Entanglement Entropy for Disjoint Intervals in AdS/CFT Thomas Faulkner Institute for Advanced Study based on arxiv:1303.7221 (see also T.Hartman arxiv:1303.6955) Entanglement Entropy : Definitions Vacuum
More informationLecture 20. Phase Transitions. Phase diagrams. Latent heats. Phase-transition fun. Reading for this Lecture: Elements Ch 13.
Lecture 20 Phase ransitions Phase diagrams Latent heats Phase-transition fun Reading for this Lecture: Elements Ch 13 Lecture 20, p 1 Solid-gas equilibrium: vapor pressure Consider solid-gas equilibrium
More informationQuantum entanglement, it s entropy, and why we calculate it
Quantum entanglement, it s entropy, and why we calculate it Piotr Witkowski Max Planck Institute for Physics 14.7 2016 Munich 1 What is entanglement? 2 Quantifying entanglement the entropy 3 The (very)
More informationA. unchanged increased B. unchanged unchanged C. increased increased D. increased unchanged
IB PHYSICS Name: DEVIL PHYSICS Period: Date: BADDEST CLASS ON CAMPUS CHAPTER B TEST REVIEW. A rocket is fired ertically. At its highest point, it explodes. Which one of the following describes what happens
More informationSection 1.7. Linear Independence
Section 1.7 Linear Independence Motiation Sometimes the span of a set of ectors is smaller than you expect from the number of ectors. Span{, w} w Span{u,, w} w u This means you don t need so many ectors
More informationHolographic entanglement entropy
Holographic entanglement entropy Mohsen Alishahiha School of physics, Institute for Research in Fundamental Sciences (IPM) 21th Spring Physics Conference, 1393 1 Plan of the talk Entanglement entropy Holography
More information