How is information localized in quantum gravity?
Suvrat Raju
International Centre for Theoretical Sciences
Extreme-Universe colloquium
Kyoto
14 March 2024

Collaborators and References

Joydeep Chakravarty

Priyadarshi Paul

Tuneer Chakraborty

Victor Godet▸ 2303.16316 and 2303.16315, Joydeep Chakravarty,
Tuneer Chakraborty, Victor Godet, Priyadarshi Paul, S.R.

Collaborators and References

Alok Laddha

Siddharth Prabhu

Pushkal Shrivastava

Chandramouli Chowdhury

Olga Papadoulaki▸ 2002.02448, Alok Laddha, Siddharth Prabhu, Pushkal
Shrivastava, S.R.
▸ 2008.01740, Chandramouli Chowdhury, OlgaPapadoulaki, S.R.
▸ 2107.14802, Chandramouli Chowdhury, Victor Godet,
Olga Papadoulaki, S.R.

Collaborators and References

Hao Geng

Andreas Karch

Carlos Perez

Lisa Randall

Marcos Riojas

Sanjit Shashi▸ 2107.03390, Geng, Karch, Perez-Pardavila, S.R.,
Randall, Riojas, Shashi▸ 2012.04671, Geng, Karch, Perez-Pardavila, S.R.,
Randall, Riojas, Shashi

Collaborators

Kyriakos Papadodimas▸ 1310.6335, Papadodimas, S.R▸ 1211.6767, Papadodimas, S.R.
▸ 2012.05770, S.R.
▸ 2110.05470, S.R.

The split property in nongravitational theories▸ In a nongravitational theory, the state inside and outside abounded region can be specified independently. (Clear in alattice regularization.)Ordinary localization of information in QFT

The split property is key to our usual idea that information islocalized “inside” some region and is not available in its
complement.

Complement-of-bounded-region has info about region
Nongravitational QFT:

Gravity:Overview▸ This unusual localization of information can be inferredfrom a gravitational analysis.
▸ For simple states, this effect is visible perturbatively.
▸ This effect is important for the black-hole information
problem.
▸ In some regimes, this effect becomes unimportant andgravity behaves like an ordinary QFT.
▸ Results pertain to information and not dynamical
holographic dualities. So we term this “holography ofinformation.”From a physical point of view, why would one believe thatgravity localizes information differently?

Uncertainty principle

Position space

Energy spectrum

Localization via superposition+++ ...

=Gauss law

E=12∫ (gjk ,k − gkk ,j ) d Ωj16πG

Gravitational tails⊗

In the quantum theory, each excitation must be dressed with agravitational tail.

Failure of localization in gravity

⊗++

⊗

⊗
Tails prevent destructive interference outside a bounded region.
Correlators of the energy and other observables carryinformation about the state.

Precise results.

Holography of information in asymptotically flat space

In a nongravitational theory,
asymptotic algebra at I +comprises independent
operatorsspan{Y(u1 )Y(u2 ) . . . Y(un )}In gravity, any operator on I +can be approximated arbitrarily
well fromspan{Y(u1 ) . . . Y(un )},1ui ∈ [−∞, − ]Laddha, Prabhu, S.R., Shrivastava (2020); Marolf (2006–13)

Holography of information in asymptotically AdS

In a nongravitational theory,
asymptotic algebra lives on theentire boundaryspan{Y(t1 )Y(t2 ) . . . Y(tn )}In gravity, any asymptoticoperator can be approximatedarbitrarily well fromspan{Y(t1 ) . . . Y(tn )},ti ∈ [0, ]Laddha, Prabhu, S.R., Shrivastava (2020); R.de Mello Koch, Kemp (2022)

Holography of information in asymptotically dS

In a nongravitational theory,complete set of observablesare⟨Y(x1 ) . . . Y(xn )⟩In gravity, complete set ofobservables are⟨Y(x1 ) . . . Y(xn )⟩, xi ∈ R[Chakraborty, Chakravarty, Godet, Paul, S.R., 2023]

Physical interpretation: picture
Precise results are in terms of asymptotic algebras.
But physically, they imply that information about the whole stateis available in a subregion.

dS
Flat spaceAdSPhysical interpretation: picture

AdS and flat spacedS

The complement of a bounded region has all information aboutthe state.

Assumptions: flat space
These results rely on weak assumptions about the full theory.
Asymptotic algebra
is well defined in fulltheory

Vacua are identifiableby charges at I−+

All operators on I + canbe represented near I−+

Hamiltonian is bounded
below in full theory

Elements of proof: algebra
Assumption: Asymptotic algebra makes sense in the UVtheory.

▸ Study Hilbert space obtained by asymptotic quantization
H(0) = span of{Y(u1 )Y(u2 ) . . . Y(un )}∣0⟩,ui ∈ (−∞, ∞)[Ashtekar, 1981]▸ Define
A−∞, = span of{Y(u1 )Y(u2 ) . . . Y(un )},1ui ∈ (−∞, − ]Elements of proof: entanglement
Assumption: Hamiltonian is bounded below.

E0Ô⇒ any state ∣n⟩ ∈ H(0) is approximated arbitrarily well by∣n⟩ ≐ Xn ∣0⟩
where Xn ∈ A−∞, .

Elements of proof: Gauss law
Assumption: Vacuum can be identified by a boundary
operator.

P0 = ∣0⟩⟨0∣ ∈ A−∞,
A unique property of gravity is that the vacuum can be selectedfrom the asymptotic region.

Proof of holography of information
A = ∑ c(n, m)∣n⟩⟨m∣by step 1n,m≐ ∑ c(n, m)Xn ∣0⟩⟨0∣Xm†by step 2n,m

= ∑ c(n, m)Xn P0 Xm†by step 3n,m

So any operator, H(0) → H(0) , can be approximated arbitrarily
well by an operator near I−+ .

Massive particles

Previous analysis does notaccount for massive particles
that go from i − to i + .

No satisfactory asymptotic description of massive fields at I ± .
Some progress possible by studying massive particles at i 0 .
[Laddha, Prabhu, S.R, Shrivastava, 2022][Anupam, Athira, Paul, S.R., (in progress)]

Holography of Information in AdS
Similar arguments imply that all operators on the boundary ofAdS can be represented in a small time band [0, ]. (Nosubtlety with massive particles.)This is consistent with AdS/CFT but does not assume it.

dS analysis
I+SdHΨ[g, φ] = 0;Hi Ψ[g, φ] = 0.

At late times, WDW equation forcesΨÐ→eiS[g,χ] Z [g, χ][Chakraborty, Chakravarty, Godet, Paul, S.R., 2023]
Related analysis: Freidel 2008

Cosmological correlators
Cosmological correlators must
be gauge fixed. Labelled by
local points but secretly
nonlocal.

⟪Ψ∣χ(x1 ) . . . χ(xn )∣Ψ⟫CC = ∫ ∣Ψ∣2 χ(x1 ) . . . χ(xn )δ(g.f)∆′FP DgDχConstraints ⇒ form of the wavefunctional ⇒ symmetries ofcosmological correlators.

Regimes

Holography of information

Perturbative regime

QFT regime

Black HolesPerturbative evidence
I+I+
I−+
I−

I−

▸ Excitation that hits null infinity between u ∈ [0, 1].
21∣f ⟩ = e−iλ ∫ d Ω ∫0 dx f (x,Ω)O(x,Ω) ∣0⟩.
▸ Challenge: Using perturbative quantum correlators nearI−+ determine f (x, Ω).
[Laddha, Prabhu, S.R., Shrivastava, 2020][Chowdhury, Papadoulaki, S.R., 2020]

Perturbative evidence

I+

I+
I−+
I−

I−

1f (x, Ω′ )dx + O (λ2 )0 (x − u − i+ )∞11= −Gλ ∑ n+1 ∫ x n f (x, Ω′ )dx.
0n=0 u⟨f ∣M(−∞)O(u, Ω′ )∣f ⟩ = Gλ ∫Significance of gravity
I+

I+
I−+
I−

I−

▸ This idea cannot work without gravity.
▸ Nongravitational gauge theories contain local
gauge-invariant bulk operators.
∣0⟩and2eiTr(F )(u=0) ∣0⟩

cannot be distinguished by any measurement near I−+
without gravity.

Global symmetry
I+

I+

I+

I+

I−+
I−

I−+
I−

I−

I−

Say O1 and O2 are identical.
2121∣f1 ⟩ = e−iλ ∫ d Ω ∫0 dx f (x,Ω)O1 (x,Ω) ∣0⟩.
∣f2 ⟩ = e−iλ ∫ d Ω ∫0 dx f (x,Ω)O2 (x,Ω) ∣0⟩.
Then (up to O (λ))⟨f1 ∣M(−∞)O1 (u, Ω′ )∣f1 ⟩ ≠ 0⟨f1 ∣M(−∞)O2 (u, Ω′ )∣f1 ⟩ = 0⟨f2 ∣M(−∞)O1 (u, Ω′ )∣f2 ⟩ = 0⟨f2 ∣M(−∞)O2 (u, Ω′ )∣f2 ⟩ ≠ 0.

Locality

I+

I+

I+

I+

I−+
I−

I−

I−+
I−

I−

▸ Phenomenon visible in linearized theory; no complicatednonlocal dynamics.
▸ In gravity, after sending the excitation, it is impossible toerase information from infinity. (In nongravitational
theories, this is possible).

Regimes

Holography of information

Perturbative regime

QFT regime

Black HolesTwo regions extending to infinity

When R and R both extend to infinity, the split property isobvious.

Classical limit

I+I+
I−+I−

⟨f ∣M(−∞)O(u, Ω′ )∣f ⟩ =I−̵hGf (x, Ω′ )λdx + O (λ2 ) .
∫3+c(x − u − i )Protocol fails when G → 0 or h̵ → 0.

Ordinary systems

Distinguishing similar states from infinity requires precision
O (e−S ) where S controls the density of states.
[Bahiru, Belin, Papadodimas, Sarosi, Vardian, 2022]

By coarse-graining observations, and introducing a classical“observer” we recover ordinary physics to good approximation.
[Jensen, Sorce, Speranza, 2023][Chandrasekaran, Longo, Penington, Witten, 2022][Jensen, Speranza, S.R., in progress]

Regimes

Holography of information

Perturbative regime

QFT regime

Black HolesPair creation

horizonHawking found that spontaneous pair creation leads to radiationthat is seemingly uncorrelated with details of the initial state.

Information paradox

Star 1

Black Hole 1Radiation

Star 2Black Hole 2

Hawking argued that since radiation is uncorrelated with theinitial state, the time-evolution arrows cannot be reversed. This
contradicts unitarity.

Significance of exponentially suppressed terms

Typical pure states are
exponentially close to mixedstatesS⟨Ψ∣A∣Ψ⟩ = Tr(ρmicro A) + O (e− 2 )[Lloyd, 1988]

So to address the information paradox, we must allow
discussion of exponentially precise observations at infinity.

Black-hole information

Holography of information implies that information about theblack-hole microstate is always available outside with the rightmeasurements.
[ Related perspectives: Balasubramanian, Marolf, Rozali, 2006, Jacobson 2012, Jacobson, Nguyen 2019]

Entropy of Hawking radiation

ρ(u)

Su

Hawking’s arguments imply thatsince unitarity implies∂S(u)∂u > 0. This is a paradox

S(I − ) = S(I + )and if S(I − ) = 0 then S(I + ) = 0.

von Neumann entropy on I +ρ(u)SuWe can prove that the von Neumann entropy of a segment(−∞, u) of I + is independent of u.
[Laddha, Prabhu, S.R., Shrivastava, 2020]

Error in Hawking’s argument?

Low point correlators⟨aω aω† ⟩ ≈11 − e−βωare insufficient to argue for information loss.

Where is the error in Hawking’s argument?

▸ Hawking assumed that theHilbert space factorizes up
to a few global constraints
imposed by the no-hairtheorem.
▸ Our previous discussion
shows that this fails inquantum gravity.

Comparison with Page curve

SmaxSS≠uu

The idea that the von Neumann entropy of the region outside
follows the Page curve also assumes factorization. (Otherentropies do follow a Page curve (more below).)Relationship to Page-curve computations

Recent computations of the Page curve do not contradict ourresults.
[Pennington, Almheiri, Engelhardt, Marolf, Mahajan,, ][Maxfield, Maldacena, Hartman, Shaghoulian, ][Tajdini, Stanford, Shenker, Yang..., 2019]

Gravity not dynamical throughout space. Models necessarily
involve a nongravitational bath.

Nature of information transfer→Page curve computes information transfer between
nongravitating subregions. (Phrase: “Page curve of Hawkingradiation” is misleading.)[Geng, Karch, Perez-Pardavila, S.R., Randall, Riojas, Shashi, 2020]

Nature of information transfer
Questions have been raised about how information “emerges”
from the black holes.
[Martinec, 2022][Mathur, 2020–23]

Holography of information Ô⇒ information is present near theboundary; merely transferred to the nongravitational region.

Summary

Standard theories of gravity
store quantum information very
differently from local quantum
field theories.

▸ This is a nonperturbative result that relies on weak
assumptions about the UV theory.
▸ But for simple states, it can be verified in perturbation
theory.
▸ Sheds light on why gravitational theories are holographic.

Summary
Accounting for this unusual
localization of information
resolves several paradoxes
about black holes.

▸ Forgetting about this property of gravity, and assuming
factorization of the Hilbert space often leads to paradoxes!
▸ Recent computations of the Page curve involve a
nongravitational bath where the Hilbert space does
factorize.
▸ Holography of information helps to understand how
information enters the bath.