Quantum Entanglement
inHolography
Xi Dong

Review talkStrings 2018, OIST, Okinawa, Japan
June 27, 2018

Quantum entanglement• Correlation between parts of the full system• Simplest example: |Ψ⟩ = |↑⟩|↓⟩ − |↓⟩|↑⟩

Quantum entanglement
General case:Entanglement is complicated• Entanglement is not just entanglement entropy.
• “Structure” of (many-body) quantum state.
• Entanglement entropy is one of many probes.
• Other probes: correlators, Renyi entropy, modular
Hamiltonian/flow, complexity, quantum error
correction, …

Entanglement is complicated• Structure of entanglement is increasingly importantin the study of many-body physics.
• String theory and quantum field theory are(special) many-body systems.
• Strategy: study patterns of entanglement toaddress interesting questions such as:ØHow does quantum gravity work?
ØHow do we describe the black hole interior?
ØHow do we describe cosmology?

Plan• Entanglement in QFT• Entanglement in holography• Applications and generalizations• Holographic error correction

Entanglement entropy in QFT• Von Neumann entropy: 𝑆 ≝ −Tr ρ, ln ρ,0• Renyi entropy: 𝑆/ ≝ 01/ ln Tr ρ/,• Interesting in QFT and holography

Entanglement entropy in QFTArea• Area-law divergence: 𝑆 = # 6 789 + ⋯ + 𝑆</=> + ⋯• Even 𝑑: 𝑆</=> = ∑= 𝑐= 𝐼= ln 𝜖
ØE.g. CFT4: 𝑆</=> = ∫ 𝑎𝑅 + 𝑎𝐾 H + 𝑐𝑊 ln 𝜖
• Odd 𝑑 : 𝑆</=> is finite and nonlocal.
ØE.g. spherical disk: 𝑆</=> ∼ 𝐹• Can generalize to Renyi entropy• Can generalize to cases with corners[Allais, Balakrishnan, Banerjee, Bhattacharya, Bianchi, Bueno, Carmi, Chapman, Czech, XD,Dowker, Dutta, Elvang, Faulkner, Fonda, Galante, Hadjiantonis, Hubeny, Klebanov, Lamprou,Lee, Leigh, Lewkowycz, McCandish, McGough, Meineri, Mezei, Miao, Myers, Nishioka,Nozaki, Numasawa, Parrikar, Perlmutter, Prudenziai, Pufu, Rangamani, Rosenhaus, Safdi,Seminara, Smolkin, Solodukhin, Sully, Takayanagi, Tonni, Witczak-Krempa, …]

Strong subadditivity (SSA)
𝑆,L + 𝑆LM ≥ 𝑆,LM + 𝑆,
• In Lorentz-invariant QFT, leads to:ØMonotonic C-function in 2d and F-function in 3dØA-theorem in 4d• Basic idea in 2d:• A particular 2nd derivative of 𝑆(𝑙) is non-positive.
• Define 𝐶 𝑙 = 3𝑙 𝑆′(𝑙) ⇒ 𝐶’ 𝑙 ≤ 0[Casini, Huerta, Teste, Torroba]

Modular Hamiltonian𝐾 ≝ − ln 𝜌
• Makes the state look thermal• Nonlocal in general• Exceptions:
1. Thermal state 𝜌 = 𝑒 1[\ /𝑍
2. Half space in QFT vacuum:𝐾 ∼ 2𝜋 a𝑧 𝑇 dd
[Bisognano, Wichmann]Modular Hamiltonian
3. Spherical disk in CFT vacuum

𝑅 H − 𝑟 H dd
𝐾 ∼ 2𝜋 a𝑇𝑅• Generalizes to wiggly cases[Casini, Huerta, Myers, Teste, Torroba]

Modular Hamiltonian• Shape deformation governed by 𝑇fg , leading to:ØAveraged Null Energy Condition (ANEC):a𝑑𝑥 i 𝑇ii ≥ 0• Uses monotonicity of relative entropy (⇔ SSA)
• Relative entropy 𝑆 𝜌 𝜎 ≝ Tr 𝜌 ln 𝜌 − Tr 𝜌 ln 𝜎
• 𝑆 𝜌 𝜎 ≥ 0; 𝑆 𝜌 𝜎 = 0 iff 𝜌 = 𝜎• A measure of distinguishability• Monotonicity: 𝑆 𝜌,L 𝜎,L ≥ 𝑆 𝜌, 𝜎,[Faulkner, Leigh, Parrikar, Wang; Hartman, Kundu, Tajdini; Bousso, Fisher, Leichenauer, Wall;
Balakrishnan, Faulkner, Khandker, Wang]

Modular Hamiltonian• Shape deformation governed by 𝑇fg , leading to:ØQuantum Null Energy Condition (QNEC):
1 mm𝑇ii ≥𝑆2𝜋
• Uses causality (in modular time)[Faulkner, Leigh, Parrikar, Wang; Hartman, Kundu, Tajdini; Bousso, Fisher, Leichenauer, Wall;
Balakrishnan, Faulkner, Khandker, Wang]

Plan• Entanglement in QFT• Entanglement in holography• Applications and generalizations• Holographic error correction

Holographic Entanglement Entropy
In holography, von Neumann entropy is given by the[Ryu & Takayanagi ’06]
area of a dual surface:Area
𝑆 = min4𝐺r• Practically useful for understanding entanglement
in strongly-coupled systems.
[Huijse, Sachdev & Swingle ’11]
[XD, Harrison, Kachru, Torroba & Wang '12; …]• Conceptually important for understanding theemergence of spacetime from entanglement.
[Van Raamsdonk ’10; Maldacena & Susskind ’13; …]

Proof from AdS/CFT
Basic idea:• Use replica trick to calculate Renyi entropy 𝑆/ .
• Given by the action of a bulk solution 𝐵/ with aconical defect.
• Von Neumann entropy: 𝑆 = 𝜕/ 𝐼 𝐵/ |/u0 .
• The RT minimal surface is a relic of the conical0defect as deficit angle 2𝜋 1 − / goes to zero.
• Variation of on-shell action is a boundary term,giving the area.
[Lewkowycz, Maldacena]

HRT in time-dependent states• Strictly speaking, the RT formula applies at a
moment of time-reflection symmetry.
• For time-dependent states, we have a covariant[Hubeny, Rangamani & Takayanagi ’07]
generalization (HRT):Area
𝑆 = ext
4𝐺r• Powerful tool for studying time-dependent physics
such as quantum quenches.
• Can also be derived from AdS/CFT.

[XD, Lewkowycz & Rangamani ’16]

Corrections to RT formulaThe RT formula has been refined by:• Higher derivative corrections• Quantum corrections

Higher derivative corrections to RT• Higher derivative corrections (𝛼′):𝐴gen
𝑆 = ext
4𝐺r[XD ’13; Camps ’13; XD & Lewkowycz,1705.08453; …]• “Generalized area” 𝐴gen = 𝜕/ 𝐼 𝐵/ |/u0
• Has the form: 𝑆Wald + 𝑆extrinsic curvature• Example:
1
𝐿=−
𝑅 + 𝜆𝑅fg 𝑅fg
16𝜋𝐺r

yields𝐴gen = a 1 + 𝜆‡𝑅†† −1𝐾† 𝐾 †2

• Applies to (dynamical) black holes and shown toobey the Second Law.
[Bhattacharjee, Sarkar & Wall ’15; Wall ’15]

Quantum corrections to RT
These corrections (𝐺r ∼ 1/𝑁 H) come from matterfields and gravitons.
• The prescription is surprisingly simple:
𝐴𝑆 = ext+ 𝑆bulk
4𝐺r[Engelhardt & Wall ’14; XD & Lewkowycz,1705.08453]• Quantum extremal surface.
• Valid to all orders in 𝐺r .
• Natural: invariant under bulk RG flow.
• Matches one-loop FLM result.
[Faulkner, Lewkowycz & Maldacena ’13]
• 𝑆bulk defined in entanglement wedge (domain ofdependence of achronal surface between 𝐴 and 𝛾,)Quantum corrections to RT
These corrections (𝐺r ∼ 1/𝑁 H) come from matterfields and gravitons.
• The prescription is surprisingly simple:
𝐴𝑆 = ext+ 𝑆bulk
4𝐺r• Can be derived from AdS/CFT. [XD & Lewkowycz 1705.08453]• Example: 2d dilaton gravity with 𝑁Œ matter fields.
[CGHS; Russo, Susskind & Thorlacius ’92]ØQuantum effects generate nonlocal effective action:
𝑁Œ
1 1H•
1H
𝐿=−𝑒
𝑅 + 4𝜆 +𝑅𝑅
2𝜋96𝜋 𝛻 HØAppears local in conformal gauge. Can check
quantum extremality.

Holographic Renyi entropy
1𝑆/ ≝ln Trρ/,1−𝑛• Given by cosmic branes instead of minimal surface𝑛−1
Area Cosmic Brane/
H𝑛 𝜕/
𝑆/ =𝑛4𝐺r[XD]

𝑛H 𝜕/

𝑛−1
Area Cosmic Brane/
𝑆/ =𝑛4𝐺r• Cosmic brane similar to minimal surface; they areboth codimension-2 and anchored at edge of 𝐴.
/10• But brane is different in having tension 𝑇/ = ”/• .
–• Backreacts on ambient geometry by creating/10
conical deficit angle 2𝜋 / .
• Useful way of getting the geometry: find solution toclassical action 𝐼total = 𝐼bulk +𝐼brane.
• As 𝑛 → 1: probe brane settles at minimal surface.
• One-parameter generation of RT.
[XD]

𝑛H 𝜕/𝑛−1
Area Cosmic Brane/
𝑆/ =𝑛4𝐺r• Why does this area law work?
• Because LHS is a more natural candidate for generalizing
von Neumann entropy:𝑛−11H
H𝑆˜/ ≝ 𝑛 𝜕/
𝑆/ = −𝑛 𝜕/ ln Trρ/,𝑛𝑛• This is standard thermodynamic relation
00/
𝜕𝐹/with 𝐹/ = − / ln Trρ, , 𝑇 = /𝑆˜/ = −𝜕𝑇Trρ/, is partition function w/ modular Hamiltonian − ln ρ,• 𝑆˜/ ≥ 0 generally. [Beck & Schögl ’93] Automatic by area law!
[XD]

JLMS relations™
Area
𝐾=+ 𝐾bulk4𝐺r• CFT modular flow = bulk modular flow
𝑆(𝜌|𝜎) = 𝑆bulk (𝜌|𝜎)• Two states are as distinguishable in the CFT as inthe bulk.
• Both relations hold to one-loop order in 1/N.
[Jafferis, Lewkowycz, Maldacena, Suh]Plan
• Entanglement in QFT• Entanglement in holography• Applications and generalizations• Holographic error correction

Phases of holographic
entanglement entropy• First-order phase transition• Similar to Hawking-Page transition• Mutual information 𝐼 𝐴, 𝐵 ≝ 𝑆, + 𝑆L − 𝑆,Lchanges from zero to nonzero (at order 1/𝐺r ).
• Holographic Renyi entropy has similar transitions.

Phases of holographic Renyi
entropy• Can have second-order phase transition at 𝑛 =𝑛›œ=•• Need a sufficiently light bulk field• Basic idea: increase 𝑛 ~ decrease T ⇒ bulk field
condenses.
• Examples:1. Spherical disk region
2. In 𝑑 = 2: multiple intervals[Belin, Maloney, Matsuura; XD, Maguire, Maloney, Maxfield]

Holographic Entropy Cone
• RT satisfies strong subadditivity:[Headrick & Takayanagi]• Also satisfies other inequalities such as monogamy ofmutual information
and𝑆,L + 𝑆LM + 𝑆M, ≥ 𝑆,LM + 𝑆, + 𝑆L + 𝑆M

[Hayden, Headrick& Maloney]

𝑆,LM + 𝑆LMŸ + 𝑆MŸ + 𝑆Ÿ , + 𝑆 ,L
≥ 𝑆,LMŸ + 𝑆,L + 𝑆LM + 𝑆MŸ + 𝑆Ÿ + 𝑆 ,[Bao, Nezami,Ooguri, Stoica,
Sully & Walter]• Provide nontrivial conditions for a theory to have agravity dual.
• Holographic entropy cone for time-dependent states?
[XD & Czech, to appear]• AdS3/CFT2: same as the static case.

Time evolution of entanglement• Consider the thermofield double (TFD) state:1|Ψ⟩ =¡ 𝑒 [ ¢ /H |𝑛⟩£ |𝑛⟩¤𝑍 /• Dual to eternal AdS black hole:[Maldacena]

Time evolution of entanglement• Consider the union of half of the left CFT and halfof the right CFT at time 𝑡:• Its entanglement entropy grows linearly in 𝑡.
• Related to the growth of the BH interior.
[Hartman, Maldacena]

Holographic complexity• Such growth of the BH interior is conjectured tocorrespond to the linear growth of complexity ofthe state.
• Complexity: minimum # of gates to prepare a stateØComplexity = Volume
ØComplexity = Action[Stanford, Susskind; Brown, Roberts, Susskind, Swingle, Zhao]

Traversable wormhole• TFD is a highly entangled state, but the two CFTs do
not interact with each other.
• Dual to a non-traversable wormhole.

• Can be made traversable by adding interactions.
• Quantum teleportation protocol.
[Gao, Jafferis, Wall; Maldacena, Stanford, Yang; Maldacena, Qi]

Einstein equations fromentanglement• The proof of RT relied on Einstein equations.
• Can go in the other direction: deriving Einsteinequations from RT.
• Basic idea: first law of entanglement entropy𝛿𝑆 = 𝛿 𝐾§

under𝜌 → 𝜌 + 𝛿𝜌• Apply to a spherical disk in the CFT vacuum, so that 𝐾§
is locally determined from the CFT stress tensor.
• Both sides are controlled by the bulk metric: via RT on
the left, and via extrapolate dictionary on the right.
• Gives the (linearized) Einstein equations.
[XD, Faulkner, Guica, Haehl, Hartman, Hijano, Lashkari, Lewkowycz, McDermott, Myers,Parrikar, Rabideau, Swingle, VanRaamsdonk, ...]

Bit threads• Reformulate RT using maximal flows instead ofminimal surfaces.

• Flow: 𝛻 ¨ 𝑣 = 0, 𝑣 ≤ 𝐶• Max flow-min cut theorem:
max a 𝑣 = 𝐶 min Area 𝑚>,ª~,[Freedman, Headrick; Headrick, Hubeny]

Bit threads• One advantage is that max flows change
continuously across phase transitions, unlike mincuts:• This bit thread picture has been generalized to HRT.
[Freedman, Headrick; Headrick, Hubeny]Entanglement of purification• Given a mixed state 𝜌,L , it may be purified as|𝜓⟩,,-LL - .
• Entanglement of purification: min 𝑆 𝜌,,m®

• Conjectured to be holographically given by the°±² :entanglement wedge cross section Σ,L[Takayanagi, Umemoto, ...]

Modular flow as a disentangler• “Modular minimal entropy” is given by the area ofa constrained extremal surface.
• Given two regions 𝑅, 𝐴, define the modular evolved
state: 𝜓 𝑠 ⟩ = 𝜌¤=´ 𝜓⟩• And the modular minimal entropy:
𝑆¤ 𝐴 = min 𝑆, |𝜓(𝑠)⟩´• Holographically given by a constrained extremalsurface (i.e. extremal except at intersections withthe HRT surface 𝛾¤ of 𝑅).
[Chen, XD, Lewkowycz & Qi, appearing today]Modular flow as a disentangler• “Modular minimal entropy” is given by the area ofa constrained extremal surface.
• Obvious when the modular flow is local:• Seems to be generally true (can be proven in d=2).
• A useful diagnostic of whether 𝛾, and 𝛾¤ intersect.
[Chen, XD, Lewkowycz & Qi, appearing today]Entanglement in de Sitterholography• dS/dS correspondence:
HH + sinH 𝑤 𝑑𝑠 H𝑑𝑠µ¶=𝑑𝑤µ¶77·¸• Suggests a dual description as two matter sectors
coupled to each other and to 𝑑-dimensional
gravity.
• Reminiscent of (but different from) the TFD
example discussed previously.
• In particular, the two sides are in a maximallyentangled state (to leading order), as shown byentanglement and Renyi entropies.
• Matches and gives an interpretation of theSilverstein, Torroba]
Gibbons-Hawking [XD,entropy.

Plan• Entanglement in QFT• Entanglement in holography• Applications and generalizations• Holographic error correction

Why quantum error correction?
• 𝜙(𝑥) can be represented
on 𝐴 ∪ 𝐵, 𝐵 ∪ 𝐶, or 𝐴 ∪ 𝐶.
• Obviously they cannot be
the same CFT operator.
• Defining feature forquantum error correction.
• Holography is a quantumerror correcting code.
• Reconstruction works in acode subspace of states.
[Almheiri, XD, Harlow ’14]

AdS/CFT as a Quantum Error
Correcting CodeBulk Gravity

Quantum Error Correction• Low-energy bulk states• States in the code
subspace• Different CFT
representations of abulk operator• Redundant
implementation of thesame logical operation• Algebra of bulkoperators• Algebra of protected
operators acting on thecode subspace• Radial distance• Level of protection[Almheiri, XD, Harlow]

Tensor network toy models[Pastawski, Yoshida, Harlow & Preskill ’15; Hayden, Nezami, Qi,Thomas, Walter & Yang ’16; Donnelly, Michel, Marolf & Wien ’16;…]

Entanglement wedge
reconstruction

𝑂†

Any bulk operator in the entanglement wedge ofregion 𝐴 may be represented as a CFT operator
on 𝐴.
In other words:∀ 𝑂† in the entanglement wedge of 𝐴, ∃ 𝑂, on
𝐴, s.t. 𝑂,|𝜙〉 = 𝑂† |𝜙〉 and 𝑂,½ 𝜙 = 𝑂†½ 𝜙 hold for∀ |𝜙〉 ∈ 𝐻›ÀµÁ .

Entanglement wedge
reconstruction

𝑂†

Any bulk operator in the entanglement wedge ofregion 𝐴 may be represented as a CFT operator
on 𝐴.
This new form of subregion duality goes beyond theold “causal wedge reconstruction”: entanglement
wedge can reach behind black hole horizons.
Valid to all orders in 𝐺r .

Reconstruction theorem• Goal is to prove:

𝜙 [𝑂†, 𝑋,̅] 𝜙 = 0𝑂†• This is necessary and sufficient for∃ 𝑂,, s.t. 𝑂, 𝜙 = 𝑂† 𝜙 and 𝑂,½ 𝜙 = 𝑂†½ 𝜙[Almheiri, XD & Harlow ’14]
• WLG assume 𝑂† is Hermitian.
• Consider two states |𝜙〉, 𝑒 =ÄÅÆ |𝜙〉 in code subspace 𝐻› :𝑆 𝜌,̅ 𝜎,̅ = 𝑆 𝜌†Ë 𝜎† = 0
𝜙 𝑒 1=ÄÅÆ 𝑋,̅𝑒 =ÄÅÆ 𝜙= 𝜙 𝑋,̅ 𝜙

𝜙 [𝑂†, 𝑋,̅] 𝜙 = 0[XD, Harlow & Wall 1601.05416]

RT from QEC• We saw that entanglement wedge reconstruction
arises from JLMS which is in turn a result of thequantum-corrected RT formula.
• Can go in the other direction.
• In any quantum error-correcting code withcomplementary recovery, we have a version of theRT formula with one-loop quantum corrections:𝑆, = 𝐴 + 𝑆†
[Harlow]

Conclusion and Questions• Patterns of entanglement provide a unconventional
window towards understanding QFT and gravity.
ØCan we generalize the entanglement-based proof of c-,
F-, and a-theorems to higher dimensions?
• We discussed 𝛼′ corrections to the RT formula.
ØIs there a stringy derivation that works for finite 𝛼′?
• We saw that RT with one-loop quantum corrections
matches QEC with complementary recovery.
ØDo higher-order corrections mean something in QEC?
ØWhat conditions for a code to have a gravity dual?
ØHow do we fully enjoy all of this in the context of blackhole information problem and cosmology?

Thank you!