Bit threads, Einstein’s equations and bulk locality
Juan F. Pedraza

June 18, 2021
Based on:C. Agón, J. de Boer, & J.P. [1811.08879]
C. Agón, E. Cáceres, & J.P. [2007.07907]
C. Agón & J.P. [2105.08063]
J.P., A. Svesko, A. Russo & Z. Weller-Davies [2105.12735, 2106.XXXXX]

Juan F. Pedraza (UCL & Brandeis U.)

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Outline

Organization of the talk:
Motivations/advantages of bit threads
Constructive realizations of bit threads & properties [1811.08879]
Einstein’s equations and bulk reconstruction [2007.07907]
Generalizations:II

1/N corrections [2105.08063]
Lorentzian threads [2105.12735, 2106.XXXXX]

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Holographic entanglement entropy
Ryu-Takayanagi (RT):

Geometry emerges fromentanglement
Dynamics of entanglement ⇐⇒Dynamics of geometry
Properties of S(A) ⇐⇒ stateswith holographic duals
Where are the quantum bits?

S(A) = minm∼Aarea(m(A))4GN

S(A) = −Trρ̂A ln ρ̂Aρ̂A = TrAc ρ̂Juan F. Pedraza (UCL & Brandeis U.)

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Some conceptual puzzles[Headrick & Freedman]

Minimal surfaces are discontinuous
QI meanings of quantities such as S(A), I (A : B) are obscure as wellas their properties
In particular SSA and MMI appear in the same footing

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Bit thread re-formulation of RTConsider a v µ such that |v | ≤ 1/4GN and ∇µ v µ = 0Z √S(A) := maxv

h nµ vµ = minAm∼Aarea(m(A))
4GN

Equivalence follows from MFMC theorem / convex optimization[Freedman & Headrick; Headrick & Hubeny]

Integral lines of v , a.k.a. threads, codify local pattern of entanglementm(A) is unique while v is highly non-unique! ∼ different microstates
Oftentimes is convenient to think of threads as having finite thickness (4GN )Juan F. Pedraza (UCL & Brandeis U.)

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Solution to conceptual puzzlesACBD = ABCEntropy ∼ area due to 1d nature of threads
Threads and/or V µ can be continuous
Properties of entropy are aligned with their QI meanings
SA and SSA comes from nesting
MMI comes from “multicommodity ”
Juan F. Pedraza (UCL & Brandeis U.)

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Bulk reconstruction via RT

RT surfaces probe bulk metric and can be used to reconstruct it.
Hole-ography makes this concrete [Balasubramanian,Chowdhury,Czech,deBoer,Heller; Myers,Rao,Sugishita; Czech,Dong,Sully; Czech,Lamprou,McCandlish,Sully;
Headrick,Myers,Wien; etc]

Differential entropy E computes areas:I∂SA (θ− (λ)θ+ (λ̄))
E = dλ∂ λ̄
λ̄=λSurfaces can be shrunk to a point. Distance between points can becomputed, and ultimately gµν [Czech,Lamprou].
Juan F. Pedraza (UCL & Brandeis U.)

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Bulk reconstruction via RT II
Caveat 1: Shadows = regions not reached by RT surfaces.

Caveat 2: Requires an infinite set of RT surfaces.
BTs for one region probe the full bulk, including shadows.
Q1: Given a vector field v (or perhaps a set of v ’s) is gµν fully
determined? If so, how can we recover the metric? (not obvious,opposite problem is multivalued).
Q2: What kind of thread configurations can we construct without theknowledge of the bulk metric? Is this even possible?
Juan F. Pedraza (UCL & Brandeis U.)

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Max flow as a convex program
Common Techniques:
Convex Relaxation, Lagrange Duality, · · ·Proved MaxFlow-MinCut as well as a Lorentzian version MinFlow-MaxCut[Headrick,Hubeny]

Discover and Prove the Multi-comodity of Max multiflows: Leading to theprove of MMI (Monogamy of Mutual Information) [Cui,Hayden,He,Headrick,Stoica,Walter]

Derive an analogue of the Bit-threads, for higher curvature theories ofgravity [Headrick,Harpen,Rolph].
Metric minimization for String Field Theory [Headrick,Zwiebach]
Derive a Bit Thread like description of membrane theory for the dynamics ofholographic entanglement [Agon,Mezei]

Juan F. Pedraza (UCL & Brandeis U.)

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An alternative approach /complementary

Construct and study explicit instantiations of Bit-threads
New properties → Higher party entropic inequalities [Bao, et al.]Role of special constructions in studies of dynamics, bulk emergence etc..
Results:
Two different constructions (integral curves and level set)
Illustration of MMI
Linearized Einstein’s equations from Bit Threads
Bulk reconstruction from Bit Threads

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Integral curves methodAlgorithm

Graphically1

Given a connected region A withknown m(A)2

Proposed a set of integral curves:
V |m(A) = nµ (Non-intersecting)

3

Compute |V |,X (xm , λ)|V (xm , λ)|m(A) •|V (xm , λm )| = 1ph(xm , λm )|V (xm , λ)| = ph(xm , λ)4

Check that |V | ≤ 1 everywhere

Juan F. Pedraza (UCL & Brandeis U.)

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Examples for pure AdSd+1 :NormA sphere (Geodesics)

z

2.0z

|V |1.5

1.00.5

0.0-2-1012xx
Normz

Strips (Effective Geodesics 2D)
|V |
2.0z

1.5

1.00.5x0.0-2-10

Juan F. Pedraza (UCL & Brandeis U.)

12x

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Examples for pure AdSd+1 :Norm

A sphere (Geodesics)

z

2.0z

|V |

1.5

1.0

0.5

0.0-20-12xx!d 2Rz

aV =1p(R 2 + r 2 + z 2 )2 − 4R 2 r 2|V | =Juan F. Pedraza (UCL & Brandeis U.)

rz R 2 − r 2 + z 2,.
R2R!d−12Rzp(R 2 + r 2 + z 2 )2 − 4R 2 r 2

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Level set method
1

They must contain the minimal surface γA as one of its members.

2

They must be continuous and not self-intersecting.

3

They must not include closed bulk surfaces.

4

They must be homologous to A∗ .
va = Υ(ϕ, g )∂a ϕ ,∇·v =0Juan F. Pedraza (UCL & Brandeis U.)→Υ2 (ϕ, g )g ab ∂a ϕ∂b ϕ= 1,γA(∇ϕ) · (∇Υ) + (∇2 ϕ)Υ = 0 .

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Properties of max flows

Nesting
There exist flows that simultaneously maximize the flux through a nested
set of regions

Max multi flow/Max thread configuration
There exist a thread configuration such that for any partition of theboundary manifold: ∂M = ∪ni=1 Ai , the number of threads connecting
each individual region is maxima.
These properties can be used to prove SA, SSA and MMI

Juan F. Pedraza (UCL & Brandeis U.)

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Properties of max flows
Nesting
There exist flows that simultaneously maximize the flux through a nested
set of regions

m(B)m(A)Green region is an example of maximally packed flows
Juan F. Pedraza (UCL & Brandeis U.)

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Properties of max flows
Max multi flow/Max thread configuration
There exist a thread configuration such that for any partition of theboundary manifold: ∂M = ∪ni=1 Ai , the number of threads connecting
each individual region is maxima.

NADNBDNABNAC NBCA
Juan F. Pedraza (UCL & Brandeis U.)

CBit threads and bulk locality

B

D = ABC
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Properties of entanglement
S(A) = max NAĀ
Subadditivity: I (A, B) ≡ S(A) + S(B) − S(AB) > 0
For n = 3, regions A, B, C using S(AB) = S(C )S(A) = NAB + NAC ,S(B) = NBA + NBC ,S(C ) = NAC + NBC

I (A, B) = 2NA,B > 0
Monogamy of MI:I (A : BC ) ≥ I (A : B) + I (A : C )For n = 4 regions A, B, C , D using S(ABC ) = S(D)S(AB) ≥ NAC + NAD + NBC + NBD
S(AB) + S(BC ) + S(AC ) ≥ S(A) + S(B) + S(C ) + S(D)Juan F. Pedraza (UCL & Brandeis U.)

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Dynamical situations
Hubeny-Rangamani-Takayanagi[HRT, 07]

[Headrick, Hubeny]
Covariant formulation

Maximin prescription
S(A) =

1
max min area(m)4GN Σ⊃∂A m∼Am⊂Σ

Maxflow version
1S(A) =max max
4GN Σ v ∈FΣZ √h nµ v µAFΣ ≡ {v µ ∈ X(Σ) | ∇µ v µ = 0,S(A) = minext∗mm∼Aarea(m(A))
4GN

Juan F. Pedraza (UCL & Brandeis U.)

Bit threads and bulk locality|v | ≤ 1} ,June 18, 2021

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Dynamical situations
Hubeny-Rangamani-Takayanagi[HRT, 07]
∂M

[Headrick, Hubeny]
Covariant formulation

Maximin prescriptionΣ

S(A) =

D[A]
γA

Σ0

1max min area(m)4GN Σ⊃∂A m∼Am⊂Σ

Maxflow versionγ̃A

S(A) =1max max
4GN Σ v ∈FΣZ √h nµ v µAµarea(m(A))S(A) = minext∗
m m∼A
4GNJuan F. Pedraza (UCL & Brandeis U.)

FΣ ≡ {v ∈ X(Σ) | ∇µ v µ = 0,

Bit threads and bulk locality|v | ≤ 1} ,June 18, 2021

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Dynamical situations
Hubeny-Rangamani-Takayanagi[HRT, 07]
∂M

[Headrick, Hubeny]
Covariant formulation

Maximin prescription
ΣD[A]

γAγ̃A

S(A) =1max min area(m)4GN Σ⊃∂A m∼Am⊂Σ

Σ0

Maxflow version
S(A) =1max
4GN v ∈FΣ0Z √h nµ v µAFΣ0 ≡ {v µ ∈ X(Σ0 ) | ∇µ v µ = 0,area(m(A))
m∼A
4GN|v | ≤ 1} ,S(A) = minext∗
mJuan F. Pedraza (UCL & Brandeis U.)

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Perturbations in AdS
In Fefferman Graham coordinates1ηµν dx µ dx ν + dz 2 + δgµν (x σ , z)dx µ dx νz2dHµν (x σ , 0) .
δgµν (x σ , z) ≡ z d−2 Hµν (x σ , z) , and δhTµν (x σ )i =16πGds 2 =Q: given δgµν how can we get v = v0 + δv ? Can we invert it? Previous methods
can be adapted to study perturbations of pure AdS via v λ = v + λδv , where v isλa max flow of gµν and v λ a max flow of gµν= gµν + λδgµνThey correctly reproduce the first law on entanglement and hence encode
Einstein’s equations. However, both construction are highly non-local(cannot be inverted to recover δgµν ).
Q: can we get v = v0 + δv without δgµν ? (e.g. exploiting bulk locality?)

Juan F. Pedraza (UCL & Brandeis U.)

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In the language of differential forms

SA =where1max
4GN w∈WZAw.

W is the set closed (d − 1) forms which obeys the norm bound1g a1 b1 · · · g ad−1 bd−1 wa1 ...ad−1 wb1 ...bd−1 ≤ 1(d − 1)!

The mapw

v a = g ab (? )b ,dJuan F. Pedraza (UCL & Brandeis U.)w(? )b ≡√
1g w a1 ...ad−1 εa1 ...ad−1 b .
(d − 1)!

w = (∇ v ), w | = (n v )˜aaΓ

Bit threads and bulk localityaa

June 18, 2021

23 / 33Linear perturbations
λ = g + λδg , then w = w + λδ w
We assume gababab
λd (w + λδ w ) = 0(w + λδ w ) |γA = (˜ + λδ˜)d (δ w ) = 0→→

δ w |γA = δ˜Notice γAλ = γA . Norm bound constraint adopts the form:
hw , w ig + λ [2hw , δ w ig + hw , w iδg ] ≤ 1 ,where:
hw , w̃ ig =hw , w̃ iδg =1g a1 b1 · · · g ad−1 bd−1 wa1 ...ad−1 w̃b1 ...bd−1(d − 1)!

1
δ(g a1 b1 . . . g ad−1 bd−1 )wa1 ...ad−1 w̃b1 ...bd−1(d − 1)!

Juan F. Pedraza (UCL & Brandeis U.)

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[Casini,Huerta,Myers]
Boundary conformal Killing vector(action of modular flow)ξA = −2π(t − t0 ) [(x i − x0i )∂i ]Rπ+ [R 2 − (t − t0 )2 − (~x − ~x0 )2 ]∂tR
Bulk Killing vector

ξ=2π(t0 − t) [z∂z + (x i − x0i )∂i ]Rπ+ [R 2 − z 2 − (t − t0 )2 − (~x − ~x0 )2 ]∂tR

Juan F. Pedraza (UCL & Brandeis U.)

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Iyer-Wald formalism and Einstein’s Equations[Faulkner,Guica,Hartman,Myers,Van Raamsdonk]
Associated to the killing vector ξ A there is a conserved (d − 1) form χχ=−1 δ(∇A ξ B AB ) + ξ B AB (∇c δg AC + ∇A δg CC ) ,16πGN

where1
ABC3 ···Cd+1 dx C3 ∧ · · · ∧ dx Cd+1 ,(d − 1)!

AB =which satisfies:
ZZg bdχ = −2ξ a δEab ,χ = δhHA i ,χ = δSA ,γAATaking χ̃ ≡ χ|Σ and integrating d χ̃ZZΣAZχ̃ −d χ̃ =γAJuan F. Pedraza (UCL & Brandeis U.)

Z
χ̃

A

⇐⇒

−2ξ t δEttg t = δSA − δhHA i

ΣA

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Canonical Bit Thread from Iyer-Waldχ̃ =w

δ |γA = δ˜


1
δ
4GNw.

w

d (δ ) = 0 ⇐⇒ Linearized − EEqs

Norm constraint?
Ifw = geodesic, → hw , w i ≤ 1 + O(λ )λwδ |A =Juan F. Pedraza (UCL & Brandeis U.)

λ gλ2

4πGN
R 2 − |~x − ~x0 |2 hT00 i ¯
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Metric Reconstructionw

Note: δ can be obtained in M from boundary condition at ∂M together
with the closedness condition [Wald]AdS
We also assume gµν(fixed by symmetries).

Canonical tread construction provides a way to locally reconstruct bulk
metrics for perturbative excited states!

w

w

Two ways of metric reconstruction: (i) starting from a family of δ ’s
associated to different regions or (ii) starting from only one (or a few) δ .
(i) Gives a set of algebraic equations that can be inverted (ii) Gives afirst-order differential equation that can be inverted.

w(?δ )a = Fabc δgbc
For d = 2, d = 3 a single δ→

w suffices to solve for δg .
bc

For d ≥ 4 a finite number is needed.
hia−1δgbc = F(i)(?δbc

Juan F. Pedraza (UCL & Brandeis U.)wδgab = [F −1 ]cab (?δ )c .

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Metric Reconstruction (ii) d = 2, 3
We assume δ w is only known for one (R, ~x0 ). For example one finds
for the trace (z∗2 ≡ R 2 − |~x − ~x0 |2 )H ii (z, ~x ) = 4R(z∗2 − z 2 )Z 1+i
λd−1 δwz (λz, ~x )dλ,[z∗2 − (λz)2 ]2iand similarly for other components Hij .

For d ≥ 4, can we solve the inversion with “a few” δ w

Can also obtain the time components of the perturbation Htt and Hti ,specializing to boosted Σ’s
At next (non-linear) orders we expect the same methodology shouldwork, but inverting a higher order operator instead —following[Faulkner,Haehl,Hijano,Parrikar,Rabideau,Van Raamsdonk]

Juan F. Pedraza (UCL & Brandeis U.)

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Generalizations: I. 1/N corrections
Recently, we derive a quantum corrected prescription for bit threads [Agon,JP]Z1
SA =max v ,F ≡ {v | ∇ · v = −4GN s(x), |v | ≤ 1} ,4GN v ∈F AZs(x) = Sbulk [ΣA ] .
ΣAEquivalent to FLM, or rather, to QES at order O(GN0 )Derived via convex optimization and strong duality
Interpretation of quantum corrections as distillation of bulk state

Juan F. Pedraza (UCL & Brandeis U.)

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Generalizations: I. 1/N corrections
Iyer-Wald works but dictionaries are modified:
ZZbulkδSAgrav =δ +ξ µ hTµν(x)iνγAΣZAgrav
δEA =δw

Awithw

w

bulkd(δ ) = −4GN s(x) = −4GN ξ µ hTµν(x)iνSemiclassical Einstein’s equations arise by consistency!
ZZZbulkδSAgrav − δEAgrav =δ − δ +ξ t hT00(x)it = 0γA

Z−2ξt

ΣAw

Aw

ΣA


1 bulkδE00 − hT00(x)i t = 0
2

Bulk reconstruction remains unexplored! Wald’s theorem of uniqueness does
not apply straightforwardly.
Juan F. Pedraza (UCL & Brandeis U.)

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Generalizations: II. Lorentzian threads
The Lorentzian MinFlow-MaxCut theorem [Headrick,Hubeny] was recently
used in the context of CV duality [JP,Svesko,Russo,Weller-Davies]
1max Vol(Σ(A)) = min
C(A) =v ∈FGN ` Σ∼AZno1v , F ≡ v | ∇ · v = 0 , |v | ≥ GN `A

Uncovered new properties/inequalities derived from nesting
Tightly connected with Lorentzian AdS/CFT and state preparation
Makes evident the role of the reference state (unclear in CV and CA)
Interpreted in terms of ‘gatelines’ preparing an optimal tensor network

Juan F. Pedraza (UCL & Brandeis U.)

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Generalizations: II. Lorentzian threads

Bulk symplectic form ωbulk [Belin,Lewkowycz,Sarosi] gives a canonical flow
for linear perturbations over arbitrary states!
dωbulk = 0 for on-shell perturbations, so the linearized Einsten’s
equations can be derived covariantly from complexity!
Lorentzian threads can probe the black hole interior, and the region
near the singularity! Metric reconstruction possible?
Gives an intuitive picture of how time emerges in quantum gravity!

Juan F. Pedraza (UCL & Brandeis U.)

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Bulk symplectic form ωbulk [Belin,Lewkowycz,Sarosi] gives a canonical flow
for linear perturbations over arbitrary states!
dωbulk = 0 for on-shell perturbations, so the linearized Einsten’s
equations can be derived covariantly from complexity!
Lorentzian threads can probe the black hole interior, and the region
near the singularity! Metric reconstruction possible?
Gives an intuitive picture of how time emerges in quantum gravity!

Questions?

Juan F. Pedraza (UCL & Brandeis U.)

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