ANALOGUE QUANTUM BLACK HOLES AYAN MUKHOPADHYAY IIT MADRAS YITP - - PowerPoint PPT Presentation

AYAN MUKHOPADHYAY IIT MADRAS

ANALOGUE QUANTUM BLACK HOLES

YITP WORKSHOP “STRINGS AND FIELDS 2020”, Nov 16-20, 2020

PLAN

• 1. Introduction: Why a physical model and not just circuits?
• 2. A simple and successful model
• 3. Discussion: A deeper QIT analysis of the model
• 4. Conclusions and Outlook

BASED ON

KIBE, AM, SOLOVIEV, SWAIN; 2006.08644 (PHYS. REV. D 102 (2020) 8, 086008) ONGOING WORKS WITH NIRANJAN KAMATH , TANAY KIBE AND PRABHA MANDAYAM

INTRODUCTION: THE NEED FOR A MODEL OF A QUANTUM BLACK HOLE

COMPUTING THE PAGE CURVE

A remarkable question has been answered: how to compute the Page curve via AdS/CFT! The semi-classical geometry itself reproduces the answer via the Quantum Extremal Surface! However, we do not yet understand the mechanism which leads to the resolution of the information paradox.

Srad t tPage

CENTRAL QUESTIONS - 1

The AMPS Paradox: Strong subadditivity of entanglement entropy (considering three subsystems : early radiation + late radiation + interior) implies that semiclassical horizon and the equivalence principle cannot be valid simultaneously. Resolution 1: The physical Hilbert spaces are not factorizable (wormholes?). Yet such a factorization should work for simple observables in the semiclassical limit. How does this distinction between interior and exterior emerge in quantum gravity operationally? Resolution 2: Following Hayden and Harlow (2013) we can argue that mining information of the interior from the radiation system is exponentially complex in entropy. Given that Hawking evaporation time is polynomial in entropy, no practical paradox arises. However, this leads to its

• wn set of questions.

We need both most likely.

Combining both that black hole is a fast scrambler and that enough Hawking quanta have been emitted, it follows that the infalling qubits should be rapidly encoded in time in the Hawking quanta. [Hayden and Preskill 2007] So the old black hole is an information mirror! It reveals infalling information fast and its own formation much later.

𝒫(rs ln rs)

BLACK HOLE AS A QUANTUM ERASER

• 1. How does the factorization of interior and exterior emerge?
• 2. How come there are observables that does information

mirroring without revealing “interior”?

• 3. What makes the complex encoding of interior into the

radiation system possible?

• 4. How do we find the decoding algorithms for the mirroring and

the interior from the dual CFT?

• 5. How can we ask all these and more questions precisely?

CENTRAL QUESTIONS - 2

WE DO HAVE ONE PHYSICAL MODEL

• 1. Lots and lots of microstates with and without hair
• 2. Energy absorbing and relaxation property (quick transition between microstates post infalling

perturbations with almost complete energy absorption) realized. So unless we resolve microstates it looks like a classical black hole.

• 3. The hair on the horizon provides the key to decode the infalling quanta (mirroring). There does exist
• bservables blind to interior through which we can decode the infalling information after the fast

relaxation.

• 4. The dynamics is pseudorandom (needs to be quantified). So the information of interior is indeed

encoded in a complex way in the “decoupled” hair.

• 5. Many many questions can be posed precisely via QIT.

A SUCCESSFUL AND SIMPLE MODEL

REF: KIBE, AM, SOLOVIEV, SWAIN; 2006.08644 (PHYS. REV. D 102 (2020) 8, 086008)

The near-extremal horizon can fragment into multiple throats via instantons [Brill (1992); Maldacena, Michelson, Strominger (1999)] This inspires a lattice model of fragmented throats with lattice dimensionality same as that of horizon. These “hard” degrees of freedom crystallize and interact via “soft” gravitational mobile charges

A CARTOON

CONCRETELY

INGREDIENTS OF A ONE-DIMENSIONAL HORIZON

• 1. SYK type quantum dots dual to NAdS2 throats — degrees of freedom

where is lattice site and is the experimentalist’s time. The reparametrized time carries all information about the quantum dot’s state.

• 2. The mobile gravitational hair which carries mobile

gravitational charges which take values in SL(2,R).

• 3. Crucially the full system must have only one overall SL(2,R) symmetry corresponding to the original

horizon geometry.

ti(u) i u ti(u) 𝗥i

𝓡+

i = t′

′

i

t′

i 2 − t′

′

i 2

t′

i 3

𝓡0

i = ti (

t′ ′ ′

i

t′

i 2 − t′

′

i 2

t′

i 3 ) − t′

′

i

t′

i

𝓡−

i = t2 i (

t′ ′ ′

i

t′

i 2 − t′

′

i 2

t′

i 3 ) − 2ti (

t′ ′

i

t′

i

− t′

i

ti )

The quantum dots do carry SL(2,R) charges too! Conserved when decoupled.

Mi = − 𝓡i ⋅ 𝓡i = − 2Sch(ti(u), u) = − 2 ( t′ ′ ′

i

t′

i

− 3 2 t′ ′

i 2

t′

i 2 )

The mass of the dual black hole is simply the Casimir of these charges

𝓡i ⋅ 𝓡j := − 𝓡0

i 𝓡0 j + 1

2 (𝓡+

i 𝓡− j + 𝓡− i 𝓡+ j )

SL(2,R) INVARIANT DOT PRODUCT

OUR EQUATIONS

The first equation will turn out to be a diffusion type equation responsible for relaxation and decoupling for The second equation will ensure information mirroring. Non-trivially we will need to know only which is conserved (since ) analogous to primordial information in early radiation for the decoding of infalling qubits and not at all anything about the interior. It is most likely possible to add higher derivative terms in hair and also higher order interactions retaining phenomenological features. The hair must be quantized but we will consider coherent states admitting classical description.

λ > 0 ∑

i

𝗥′

i

∑

i

𝗥′ ′

i = 0

M′

i = − λ (𝓡i−1 + 𝓡i+1 − 2𝓡i) ⋅ 𝗥′ i

𝗥′ ′

i = 1

σ2 (𝗥i−1 + 𝗥i+1 − 2𝗥i) + 1 λ2 (𝓡i−1 + 𝓡i+1 − 2𝓡i)

The total conserved energy of the system is simply the sum of the black hole masses, the kinetic energy and gradient energy of the hair. For a nice split between interior and exterior, crucially there is NO interaction energy.

ℰ = ℰ𝒭 + ℰ𝖱

ℰ𝒭 = ∑

i

Mi = M

ℰ𝖱 = λ3 2 ∑

i

𝗥i

′ ⋅ 𝗥i ′ + λ3

2σ2 ∑

i

(𝗥i+1 − 𝗥i) ⋅ (𝗥i+1 − 𝗥i)

MICROSTATES

Microstates are just the static/stationary solutions of our model. One can readily obtain all such solutions. We choose a global SL(2,R) frame.

THE QUANTUM DOTS (NADS2 THROATS)

𝓡0

i = Q,

𝓡±

i random and constant giving different Mi THE HAIR

𝗥±

i = − σ2

λ2 𝓡±

i + 𝓛± = 𝗥± i loc

Locked with the interior

𝗥0

i = 𝗥0 i mon + 𝗥0 i rad

𝗥0

i mon = αu

q′ ′

i − 1

σ2 (qi+1 + qi−1 − 2qi) = 0

𝗥0

i rad = qi(u),

∑

i

qi = ∑

i

q′

i = 0

Monopole term remains unaffected by perturbations

α = ∑

i

Q0

i ′

Radiation that can propagate

• n top of any microstate

without affecting it

Recall that generally the total energy can be split into two parts But in the hairy microstates we have non-trivially (as a consequence of dynamics) neat split between interior and exterior components of energy.

ℰ = ℰ𝒭 + ℰ𝖱

ℰ𝒭 = ∑

i

Mi = M

ℰ𝖱 = λ3 2 ∑

i

𝗥i

′ ⋅ 𝗥i ′ + λ3

2σ2 ∑

i

(𝗥i+1 − 𝗥i) ⋅ (𝗥i+1 − 𝗥i)

ℰ𝖱 = ℰpot

𝖱 + ℰmon 𝖱

+ ℰrad

𝖱

ℰpot

𝖱 = − σ2

2λ ∑

i

(𝓡i+1 − 𝓡i) ⋅ (𝓡i+1 − 𝓡i)

ℰmon

𝖱

= − 1 2 λ3α2

ℰrad

𝖱 = λ3

2 ∑

i

qi′

2 + λ3

2σ2 ∑

i

(qi+1 − qi)

2

INTERIOR (LOCKED) EXTERIOR (PRIMODIAL) EXTERIOR (DECOUPLED RADIATION)

Also, need to be continuous for and therefore

ti, t′

i, t′

′

i

−∞ < u < ∞

Remarkably these inequalities imply that

• r

for all .

t′

i ≥ 0

t′

i ≤ 0

i

• r

The uniform arrow of time emerges from our model. We choose the future direction. We choose this.

Q ≤ − Mi, 𝒭±

i ≤ 0,

𝒭+

i + 𝒭− i ≥ 2Q

Q ≥ − Mi, 𝒭±

i ≥ 0,

𝒭+

i + 𝒭− i ≤ 2Q

Ensemble of microstate solutions: Fix total mass and . Allocate subject to inequalities discussed. Adding hair on top: Each microstate solution supports hair oscillations that can propagate freely over the lattice without affecting it

M, Q α Mi, 𝒭±

i

𝗥rad

i

SHOCK AND VERIFY

Do our model reproduce energy absorption and relaxation properties of classical black hole? If yes, when we perturb a random initial microstate with one or more sequence of shocks, then it will QUICKLY relax to another microstate absorbing all energy in the continuum limit. In presence of shocks, the equations of motion are:

M′

i = − λ (𝓡i−1 + 𝓡i+1 − 2𝓡i) ⋅ 𝗥′ i + ∑ A

ei,Aδ(u − ui,A)

𝗥′ ′

i = 1

σ2 (𝗥i−1 + 𝗥i+1 − 2𝗥i) + 1 λ2 (𝓡i−1 + 𝓡i+1 − 2𝓡i)

Results for a single shock

We shock a 5-site periodic chain at site 1. The initial microstate is always randomly chosen. Note that the shock energy is almost totally absorbed in the black hole mass

Crucially the relaxes to a new homogeneous value in the microstate. The conservation of the monopole charge implies that the final homogenous component should also be in same direction

𝒭0

i

Also relax to constants as they should in a (final) microstate.

𝒭±

i

The dynamics is pseudorandom (necessary for Harlow-Hayden scenario) The final microstate is hairy with decoupled oscillations

qi

The same happens with multiple shocks. The dynamics has energy-absorption (exact in continuum limit) and quick relaxation properties. It is pseudorandom (needs quantification). Remarkably the decoupling of exterior and interior happens dynamically. The energy of the hair is conserved on its own in the continuum limit also. The final microstate always has decoupled hair oscillations.

Alice throws in her secret information into our BH in the form of an ordered list — time-sequence and locations of shocks. Bob can decode the classical information from as soon as it decouples from the final microstate

𝗥rad

i

INFORMATION MIRRORING

Let us consider a 5-site chain that has two positive frequency normal modes. The amplitudes and phases of the oscillations are very complex and carry information about

• interior. Due to pseudorandom dynamics, it is very hard to decode the interior.

However, crucially the phase differences of the normal modes mirror the infalling qubits. No information of interior is necessary but we do need to know the “early radiation” — the primordial monopole frame.

𝗥rad

i

Note we explicitly see the two positive frequency normal modes in the 5 site model after decoupling — the amplitudes (and phases) look random and encodes the interior in a complex way. But wait for the phase differences! They mirror!

Decoding a single shock

The symmetry in phase differences reveals which site was shocked A highly non-trivial result because we start from highly asymmetric random initial conditions. Also not all features of has this symmetry

𝗥rad

i

Decoding two shocks: 1

The maximum and minimum phase differences are the positions of the shocks The minimum phase difference site was shocked first if the shocked sites are nearest neighbors

The maximum and minimum phase differences are the positions of the shocks The minimum phase difference site was shocked later if the shocked sites are not nearest neighbors

Decoding two shocks: 2

DISCUSSION: QIT ANALYSIS AND A BIGGER PICTURE

ONGOING WORKS WITH NIRANJAN KAMATH , TANAY KIBE AND PRABHA MANDAYAM

WHAT ABOUT QUANTUM INFORMATION

• 1. The entire analysis can be repeated by quantizing the hair and employing the

coherent state path integral

• 2. Presently we are investigating the details of how infalling QUBITS are mirrored

in this coherent state representation of the hair.

• 3. Furthermore, the effective “decoupling” of the interior and exterior can be

precisely formulated in the language of QIT. Simple measurements will see the product of the marginals of the state of the hair system (locked +decoupled)

ADDING THE HAWKING RADIATION

• 1. The Hawking evaporation can be readily incorporated in our model

since it is explicitly computable if the matter in each NAdS2 throat is a CFT.

• 2. It is however a very slow process when each site has sufficiently

large mass. It will not change our basic story.

• 3. Note we can also compute the (fragmented) QES and hence thePage

curve explicitly.

BUT THE HAWKING QUANTA……

• 1. Nature of entanglement between hair, Hawking quanta and interior?

Conjecture: GHZ type? If yes, then any two together is maximally entangled with the third (tracing out any one produces a fully disentangled state of the other two).

• 2. Concretely what feature in the Hawking quanta has maximal mutual

information with the phase differences of decoupled hair oscillations? The hair provides the key to the information mirroring. For the interior the Hawking quanta provides the key to decode information from hair.

BUT WE AVOID PARADOXES?

• 1. The Hawking quanta has access to the Quantum Extremal Surface (QES)

and thus the interior. Is the dynamics of the full system sufficiently chaotic so that the encoding of the interior in the Hawking quanta has enough complexity? The deviation of the QES from the semiclassical geometry has

• nly very very fine substructure (fractal-like)? Should have an answer in
• ur model.

2.What is this code subspace?

FINALLY ELEMENTARY BUT PROFOUND

• 1. Quantum chaos (pseudorandomness) as the universal mechanism for

decoupling of interior and exterior even in the presence of Hawking radiation?

• 2. Is the decoding protocol universal? Why? May be — phase differences

could be always doing the mirroring. Some kind of “complementarity” could determine the observables for interior decoding.

INTERIOR RECONSRTRUCTION IN ADS/CFT

• 1. The hair should be described in dual CFT as infrared degrees of freedom
• 2. The Hawking quanta are more perturbative in nature.
• 3. Together they are important to capture the interior which essentially

couples/entangles perturbative and non-perturbative degrees of freedom.

The black hole interior is the invisible artist! She composes a “perfect tensor state” out of the perturbative and non-perturbative dofs.

CONCLUSIONS AND OUTLOOK

We have a reasonably good phenomenological model of a quantum black hole that relaxes and absorbs like a classical black hole while mirroring information and encoding information of interior in a complex way. Applications of QIT allows us to answer deep questions such as effective decoupling of interior and exterior, reconstruction of the interior and also understanding of information recovery protocols. We do not know how to glue this model to the asymptotic region away from the horizon yet. Perhaps we can benefit by an improved model that can account for the entropy and which can be derived from fuzzball-like picture? Exploring realistic models with Avik Banerjee, Tanay Kibe, Arnab Kundu and Giuseppe Policastro.

THANK YOU

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