Bit Threads and Holographic Entanglement

Transcription

1 it Threads and Holographic Entanglement Matthew Headrick randeis University Strings 2016, eijing ased on arxiv: [hep-th], with Michael Freedman

2 Contents 1 How should one think about the minimal surface? 3 2 Reformulation of RT 7 3 Threads & information 9 4 Extensions Emergent geometry Quantum corrections Covariant bit threads

3 1 How should one think about the minimal surface? In semiclassical gravity, surface areas are related to entropies ekenstein-hawking [ 74]: For black hole Why? S = 1 4G N area(horizon) Possible answer: Microstate bits live on horizon, at density of 1 bit per 4 Planck areas m() Ryu-Takayanagi [ 06]: For region in holographic field theory (classical Einstein gravity, static state) S() = 1 4G N area(m()) m() = bulk minimal surface homologous to Do microstate bits of live on m()? Unlike horizon, m() is not a special place; by choosing, we can put m() almost anywhere 3

4 Puzzles: Under continuous changes in boundary region, minimal surface can jump Example: Union of separated regions, m() =m() [ m() m() 6= m() [ m() Information-theoretic quantities are given by differences of areas of surfaces passing through different parts of bulk: Conditional entropy: Mutual information: Conditional mutual information: H( ) = S() S() I( : ) = S() + S() S() I( : C) = S() + S(C) S(C) S(C) 4

5 S() S() H( ) = S() S() I( : ) = S() + S() S() H( ) I( 0 0 1: 0) H( ) Information-theoretic meaning (heuristically): I( : ) conditional entropies H( ) Classical: H( ) = # of (independent) bits belonging purely to I( : ) = # shared with S() S() S() S() H( ) I( 1 1 0: 0) H( ) I( : ) H( ) Quantum: Entangled (ell) pair contributes 2 to I( : ), 1 to H( ); can lead to H( ) < 0 = 1 p ( i + i) p ( i + i) H( ) S() I( : ) I( : C) = S() + S(C) S(C) S(C) = correlation between & conditioned on C What do differences between areas of surfaces, passing through different parts of bulk, have to do with these measures of information? S() S() S()

6 RT obeys strong subadditivity [Headrick-Takayanagi 07] C I( : C) I( : C) + What does proof (by cutting & gluing minimal surfaces) have to do with information-theoretic meaning of SS (monotonicity of correlations)? + To answer these questions, I will present a new formulation of RT Does not refer to minimal surfaces (demoted to a calculational device) Suggests a new way to think about the holographic principle, & about the connection between spacetime geometry and information 6

7 2 Reformulation of RT Consider a Riemannian manifold with boundary Define a flow as a vector field v obeying v = 0, v 1 Think of flow as a set of oriented threads (flow lines) beginning & ending on boundary, with transverse density = v 1 Let be a subset of boundary Max flow-min cut theorem (originally on graphs; Riemannian version: [Federer 74, Strang 83, Nozawa 90]): max v = min area(m) v m v() Note: Max flow is highly non-unique (except on m(), where v = unit normal) Let v() denote any max flow Finding max flow is a linear programming problem m() 7

8 horizon RT version 2.0: S() = max v (4G N = 1) v = max # of threads beginning on H horizon Threads can end H horizon on c or horizon Each thread has cross section of 4 Planck areas & is identified with 1 (independent) bit of H horizon utomatically incorporates homology & global minimization conditions of RT Threads are floppy : lots of freedom to move them around in bulk & move where they attach to lso lots of room near boundary to add extra threads that begin & end on (don t contribute to S()) Role of minimal surface: bottleneck, where threads are maximally packed, hence counted by area Naturally implements holographic principle: entropy area because bits D() are carried by one-dimensional objects ekenstein-hawking: t v() m() entanglement wedge 8

9 3 Threads & information Now we address conceptual puzzles with RT raised before First, v() changes continuously with, even when m() jumps Now consider two regions, We can maximize flux through or, not in general through both ut we can always maximize through and (nesting property) Call such a flow v(, ) Example 1: S() = S() = 2, S() = 3 I( : ) = 1, H( ) = 1 v() v(, ) v(). v(, ) v(,) Lesson 1: Threads that are stuck on represent bits unique to Threads that can be moved between & represent correlated pairs of bits 9

10 Example 2: S() = S() = 2, S() = 1 I( : ) = 3, H( ) = 1 entanglement! One thread leaving must go to, and vice versa v(, ) v(,) Lesson 2: H horizon Threads that connect & (switching orientation) represent entangled pairs of bits H horizon pply lessons to single region: freedom to move beginning points around reflects correlations within freedom to add threads that begin & end on reflects entanglement within 10

11 In equations: Conditional entropy: H( ) = S() S() = v() v() = v(, ) v(, ) = v(, ) = min flux on (maximizing on ) Mutual information: I( : ) = S() H( ) = v(, ) v(, ) = max min flux on (maximizing on ) = flux movable between and (maximizing on ) Max flow can be defined even when flux is infinite: flow that cannot be augmented Regulator-free definition of mutual information: I( : ) = (v(, ) v(, )) 11

12 Conditional mutual information: C I( : C) = H( C) H( C) = v(c,, ) v(c,, ) = max min flux on (maximizing on C & C) = flux movable between & (maximizing on C & C) = (flux movable between & C) (movable between & C) = I( : C) I( : C) Strong subadditivity (I( : C) 0) is clear In each case, clear connection to information-theoretic meaning of quantity/property Open problem: Use flows to prove monogamy of mutual information property of holographic EEs [Hayden- Headrick-Maloney 12] I( : C) I( : ) + I( : C) and generalizations to more parties [ao et al 15] Flow-based proofs may illuminate the information-theoretic meaning of these inequalities 12

13 4 Extensions 4.1 Emergent geometry Metric Set of allowed thread configurations 4.2 Quantum corrections Faulkner-Lewkowycz-Maldacena [ 13]: Quantum (order G 0 N ) corrections to RT come from entanglement of bulk fields m() May be reproduced v() by allowing threads to jump from one point to another (or tunnel through microscopic wormholes, à la ER = EPR [Maldacena-Susskind 13]) m() v() 4.3 Covariant bit threads With Veronika Hubeny (to appear) Hubeny-Rangamani-Takayanagi [ 07] covariant entanglement entropy formula: m() = minimal extremal surface homologous to S() = area(m()) 13

14 Need generalization of max flow-min cut theorem to Lorentzian setting Define a flow as a vector field v (in full Lorentzian spacetime) obeying v = 0 no flux into or out of singularities integrated norm bound: timelike curve C, ds v 1 (v = projection of v orthogonal to C) C ny observer sees over their lifetime a total of at most 1 thread per 4 Planck areas Theorem (assuming NEC, using results of Wall [ 12] & Headrick-Hubeny-Lawrence-Rangamani [ 14]): max v = area(m()) D() = boundary causal domain of v D() Linearizes problem of finding extremal surface area 14

15 t HRT version 2.0: S() = max v v D() To maximize flux, threads seek out m(), automatically confining themselves to entanglement wedge D() v() m() Threads can lie on common Cauchy slice (equivalent to Wall s [ 12] maximin by standard max flow-min cut) or spread out in time entanglement wedge 15