COVARIANT ENTANGLEMENTCONSTRUCTS
Veronika Hubeny
Physics Department & center for Quantum Mathematics and Physics
June 17, 2016

YKIS conference:“Quantum Matter, Spacetime and Information”

based on earlier works w/ {M. Headrick, A. Lawrence, H. Maxfield, M. Rangamani, T. Takayanagi, E. Tonni}& on work in progress w/ M. Headrick

Motivation
Elucidate holography
Fundamental nature of spacetime & its relation to entanglement
Structure/characterization of CFTs (& states) w/ gravity dual

Start w/ situations with large amount of symmetry (e.g. pure AdS)
Explicit calculations possible, can obtain analytical expressions
Use these to guess duality relations ↝ entry in gauge/gravity dictionary

But this has limitations
How to generalize? (e.g. time dependence)
Often symmetry brings degeneracy between logically distinct concepts

Need to “covariantize”
Define a quantity which is purely geometrical (e.g. independent of any
choice of coordinate systems) and fully general

Utility of covariant constructs
Gives a general prescription
Definition of a quantity is equally robust on both sides of duality
Once beyond analytically tractable cases, might as well go for fullgenerality (within the class of systems we want to consider)Time dependence interesting in its own rightNovel phenomena in out-of-equilibrium systems
New insight into the structure of the theory

Breaks degeneracy between distinct constructs
Allows us to identify the true dual ↝ underlying nature of the map

Natural covariant constructs motivate new relations
Even if a given construct is not the sought dual, it eventually finds its use

Example: Holographic EE
Proposal [RT=Ryu & Takayanagi, ‘06] for static configurations:
In the bulk, entanglement entropy SA fora boundary region A is captured by thearea of a minimal co-dimension-2 bulk
surface m at constant t anchored on
entangling surface @A & homologous to A
Area(m)SA = min@m=@A 4 GN

boundary@AAbulkm

Covariant Holographic EE
But the RT prescription is not well-defined outside the context ofstatic configurations:
In Lorentzian geometry, we can decrease
the area arbitrarily by timelike deformations
In time-dependent context, no natural
notion of “const. t” slice…

mAIn time-dependent situations, RT prescription must be covariantized:
Simplest candidate:[HRT = VH, Rangamani, Takayanagi ‘07]

minimal surface mat constant time

extremal surface Ein the full bulk

→This gives a well-defined quantity in any (arbitrarilytime-dependent asymptotically AdS) spacetime
equally robust as in CFTEA

Covariant Holographic EE
In fact, [Hubeny, Rangamani, Takayanagi ’07] identified 4 natural candidates:(all co-dim.2 surfaces ending on @A , and coincident for ball regions A in pure AdS)
• E = Extremal surface

E=is correct= Minimal-area surface on maximal-volume slice= ‘HRT prescription’•= Surface with zero null expansions• ⌅ = Causal wedge rim
Later known as Causal Information Surface;[Hubeny, Rangamani ’12]
w/ area = causal holographic information

•

EAAA

⌅

A

Power of covariant constructs‘Natural’ geometrical constructs (defined for general bulk spacetimes, independent ofcoordinates) provide useful candidates for dual of ‘natural’ quantities in CFTe.g. dual of ⇢A ? [Bousso, Leichenauer, Rosenhaus; Czech, Karczmarek, Nogueira, Van Raamsdonk;…]In generic Lorentzian spacetime, null congruences which define a causal set
provide useful characterization of ‘natural’ bulk regions.
2 options:…starting from bulk:…starting from bdy:

D[A] ⤳ Causal Wedge:

E ⤳ Entanglement Wedge:= future and pastcausally-separated
from bdy region
determined by ⇢A[VH & Rangamani]= spacelike-separated(toward A) from Et[Headrick, VH, Lawrence, Rangamani]xzE =⌅in AdS3A

NB: in pure AdS,& for spherical A ,these coincide,but not in general.

Causal wedge vs. Entanglement wedge
crucial difference: in which direction can null generators cross…
D[A] ⤳ Causal Wedge:

E ⤳ Entanglement Wedge:

BHcrossover
seamcrossover
seam⌅

A⌅

EA

Power of covariant constructs
D[A] ⤳ Causal Wedge:

E ⤳ Entanglement Wedge:…continued past ⌅ : ⤳ Causal Shadow Q@A

We can prove the inclusion property[Headrick, VH, Lawrence, Rangamani; Wall]

CW ⇢ EWor equivalently, E ⇢ Q@AQ@A
D[A]Consequences:
HRT is consistent with CFT causality(= non-trivial check of HRT)

Entanglement plateaux
Entanglement wedge can reachdeep inside a black hole!

⌅ Ac.. . .

⌅A

EA = EAc

ACovariant re-formulations
Covariance is pre-requisite to construct being physicallymeaningful, but it need not be unique
Distinct geometrical formulations can turn out equivalent (cf. E = )This redundancy is useful
Each formulation can have its own advantages
e.g. different properties may be manifest in different formulations(cf. gauge / coordinate choice)Re-formulation can reveal deeper relations (cf. ER=EPR [Maldacena, Susskind])Covariant re-formulations of HEE• E = Extremal surface(relatively) easy to findminimal set of ingredients required in specificationneed to include homology constraint as extra requirement

EA•
A= Surface with zero null expansions(cf. light sheet construction & covariant entropy bound [Bousso, ‘99]:
Bulk entropy through light sheet of surface σ ≤ Area(σ)/4= surface admitting a light sheet closest to bdy• Maximin surface [Wall, ‘12]

maximize over minimal-area surface on a spacelike slice
requires the entire collection of slices & surfaces
implements homology constraint automaticallyuseful for proofs (e.g. SSA)

But none of these elucidate the relation to quantum information

Bit thread picture of (static) EE
Reformulate EE in terms of flux of flow lines [Freedman & Headrick, ’16]
let v be a vector field satisfying r · v = 0 and |v|  1 . Then EE is given by

SA = maxv

ZAv
A

By Max Flow - Min Cut theorem, equivalent to RT:(bottleneck for flow = minimal surface)

mv

Useful reformulation of holographic EEflow continuous under varying region (cf. minimal surfaces can jump discontinuously)
implements QI meaning of EE and its inequalities more naturally
provides more intuition: think of each bit thread as connecting an EPR pair

How does this extend to time-dependent settings?

Covariantizing bit threads
1. Identify the correct geometrical quantities of interestA

Analogous to flow lines (vector field v ) inm

2. Identify the constraints they must satisfyv

Analogous to r · v = 0 and |v|  1

3. Identify the expression for EE obtained from theseZAnalogous to SA = maxvv

4. Test that it fulfills all requisite requirements
5. Extract lessons / implications

ATwo natural possibilities
Step 1:extend threads in timeflow sheets

keep 1-d threads
flow lines

Requirements on constructs
Imperative:
Reduces to bit threads in static case
Equivalent to HRT (when null energy condition (NEC) is obeyed)
Depends only on D[A] (i.e. @A + orientation), not on A itself

Useful:Manifests CFT causality (directly rather than via equivalence to HRT)
Manifests area law, positivity, subadditivity, SSA, etc.
Elucidates role of NEC
Elucidates role of homology constraint w/ time-dependenceFlow sheets
Require:A

flow sheets are timelike everywhere
flow sheets cannot end in the bulk
Pdensity is boundedi Area[Fi \ R]⇢=Vol[R]1

EE = sheets through D[A]FiMost “obvious” generalization of bit threads
entanglement lasts in time & cannot be changed a-causally ✔

Danger:Potentially too global (e.g. future singularity may prevent sheets in past)
Too many sheets through D[A] by local boost

Flow lines
Require:flow lines are spacelike everywhere
flow lines don’t end: i.e. keep v s.t. r · v = 0
but use integrated norm bound:
ZFor a unit normal vector w on any worldline,w·v 1∴ Over full lifetime, any observer sees at most 1 thread / 4 Planck areas

EE counts bit threads in D[A] :SA = maxv

Covariant construct which works…
reduces to bit threads at const time in static case ✔
threads must all pass through extremal surface (for max flow)
endpoints are floppy and can lie anywhere within D[A]Bonus: naturally picks out the entanglement wedge
does not depend on spacetime in the far future ✔

Zv
D[A]

Flow lines

But what is the QI interpretation ?

Entanglement entropy counted by events ?
e.g. # of indep. measurements that can be performed within D[A]novel interpretation…

Why are 1-d structures natural?
why is a specific measurement connected to another instantaneous
event somewhere in Ac ?

Thank you