Space-time random tensor
networks and holographicduality
Xiao-Liang Qi
Stanford University
YITP Kyoto University, 6/16/2016

Outline• Background: tensor networks and holographic duality• Definition of space-time random tensor networks (RTN)• Entanglement properties of space-time RTN--HRT formula with quantum corrections--Quantum error correction properties of theholographic mapping• Gauge fixing and finite D correction

Reference• Zhao Yang, XLQ, in preparation• Closely related previous work on spatial RTN
Patrick Hayden, Sepehr Nezami, XLQ, Nathaniel Thomas, MichaelWalter, Zhao Yang, arxiv: 1601.01694

Tensor networks• Spatial tensor networks: many-body entangled states
made from few-qubit states. (PEPS, F. Verstraete, J.I. Cirac,04’)

𝑇1𝜇𝜈𝜏 𝜇𝜈𝜏 = 𝑉1

• Space-time tensor network: a discretization of path
integral.
𝑡𝑒 −𝑖𝑡𝐻≃ 𝑒 −𝑖𝜖𝐻

Tensor networks and holographic duality

𝛾𝐴𝛾𝐴A

ATensor networks
𝑆 ≤ 𝛾𝐴 log 𝐷

Ryu&Takayanagi ‘06, Swingle ’09

Holographic duality
Ryu-Takayanagi formula1
𝑆=𝛾𝐴4𝜋𝐺

Tensor networks and holographic duality
boundarybulk

Energyspectrum• The tensor network proposal: geometry emerges from theentanglement structure of quantum states• Various tensor network proposals ( Nozaki et al ’12, XLQ ’13,Pastawski et al ’15, Yang et al ’15, Hayden et al ‘16)

• Entanglement properties can be studied for tensors withgood properties (Pastawski et al ‘15, Yang et al ‘15)

Random tensor networks Hayden et al arxiv: 1601.01694• Tensor networks with a random tensor at each vertex

𝑛-th Renyi entropy

randomaverage Free energy of 𝑆 𝑛 spin model𝑇 −1 = log 𝐷• In 𝐷 → ∞ limit, many interesting properties emerge---RT formula and its quantum correction---Error correction properties of bulk-boundary operator
correspondence---Scaling dimension gap• Problem: Need to pick a time slice. Hard to talk about
dynamics. Solution: going to space-time networks.

Space-time tensor networks• Space-time tensor networks are
discretized version of path
integral.

𝑍=• Or 𝑍 = ∫ 𝐷𝜙𝑒 −𝑆 𝜙
𝑍 = ∫ 𝐷𝜙𝑒 𝑖𝑆 𝜙

1
𝑇𝑂1 𝑂2 𝑂3 … 𝑂𝑛 = ×𝑍𝑂1𝑂2𝑂3

Space-time tensor networks• Example: Ising model 𝑆 = −𝐽𝑥𝑦 𝑠𝑥 𝑠𝑦

𝑍=𝑠4𝑠3𝑠1𝑠2= 𝛿𝑠1𝑠2 𝑒 𝐽𝑠1𝑠4+𝐽𝑠2𝑠3

Space-time tensor network for holography• Wanted:---a tensor network representation of the 𝑑 + 1 dimensional bulk geometry---The tensor network defines 𝑍𝑏𝑢𝑙𝑘 = 𝑍𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 , a 𝑑dimensional QFT.

QFT=

Space-time tensor network for holography• Our proposal:• 1. Introduce a space-time tenor
network in the bulk
• This defines a bulk QFT, which we
called the “parent theory”. (Can bedefined for either Euclidean orLorentzian time)• 2. Introduce a random projection
at each bulk link
• Key result: 𝒏-th Renyi entropy⇒
𝑺𝒏 discrete gauge theory• Subtlety: gauge fixing (will discusslater)

𝜏𝑒 −𝑆𝑃Random projection𝜏𝑒 −𝑆𝑏𝑑𝑟𝑦 ≡ 𝑒 −𝑆𝑔𝑟𝑎𝑣

Holographic duality from space-time𝑖tensor networks𝑖• Random projection at each link
• Correlation function
𝑇𝑂1 𝑂2 𝑂3 … 𝑂𝑛 =𝜏𝑂
1𝑂2𝑂3…
𝑂𝑛𝜙𝑖∗ 𝜙𝑗=𝑗𝑗

𝜏Second Renyi entropy of random
tensor networks• Second Renyi entropy definition. 𝑡𝑟 𝜌𝐴2 = 𝑡𝑟 𝜌 ⊗ 𝜌 ⋅1= 2𝐷 +𝐷+

Second Renyi entropy
𝑋𝐴=𝑋𝐴=or• The random average leads to thepartition function of a 𝑍2 gauge theory.
• 𝑡𝑟 𝜌𝐴2 = 𝑔𝑥𝑦=𝐼 𝑜𝑟 𝑋 𝑒 −𝒜 𝑔𝑥𝑦• The choice of boundary region 𝐴 determines theboundary condition of gauge field 𝑔𝑥𝑦 .
• Gauge invariance = permutation symmertry between tworeplica

Second Renyi entropy• The 𝑍2 gauge field is minimally coupled to two copies ofthe parent theory=,=

Second Renyi entropy• For example, if the parent theory is an Ising model• 𝑆𝑃 = − 𝑥𝑦 𝑠𝑥 𝑠𝑦• The random average gives (𝛼 = 1,2)
𝛼𝛽 𝛽𝛼𝛼• 𝑆𝑏𝑢𝑙𝑘 𝑠𝑥 , 𝑔𝑥𝑦 = − 𝑥𝑦 𝑠𝑥 𝑔𝑥𝑦 𝑠𝑦𝛼𝛽
• [𝑔𝑥𝑦 ] = 𝐼 or 𝜎𝑥 is the 𝑍2 connection• Schematically, if we could take continuum limit,−𝒜𝑒𝑓𝑓 𝑔

1 ,𝜙2 ,𝑔−𝒜𝜙𝑃≡ ∫ 𝐷𝜙𝑒•𝑒(or corresponding form in real time)HRT formula• The dynamics of the gauge field is induced byintegrating out the bulk “parent theory”.
• Consider a particular example: the valence bond solid(VBS) state=• Gauge field action:
1
𝑆 = − log 𝐷 𝑝∈□ 𝐹𝑝2• 𝐹𝑝 = 𝑥𝑦 ∈𝑝 𝑔𝑥𝑦 is the flux inplaquette 𝑝.

HRT formula• For the VBS state, 𝑍 =− log 𝐷 𝛾𝑒𝛾=𝑓𝑙𝑢𝑥 𝑐𝑜𝑛𝑓𝑖𝑔.

• In large 𝐷 limit, 𝑍 ≃ 𝑒 − log 𝐷 𝛾𝐴 ,and 𝑆2 ≃ log 𝐷 𝛾𝐴
• 𝛾𝐴 = min |𝛾| is the area ofminimal co-dimension-2
surface bounding 𝐴.
• In general, when the bulk
parent theory is a massivetheory, the effective actionof gauge theory has a similar
Maxwell form, leading toHRT formula(Hubeny Rangamani Takayanagi ’07).

𝐵𝐴HRT formula for Lorentzian time
• The bulk parent theory can be defined with either
Euclidean or Lorentzian signature.
• For Lorentzian bulk theory, the effective action of gauge
field is also Lorentzian.
• Lorentzian Maxwell action of a 𝑍2 gauge theory(continuum limit)
1• 𝑆 = − 2 𝐹𝜇𝜈 𝐹𝜇𝜈 + 𝜌𝑠 𝜕𝜇 𝜃 − 2𝐴𝜇 𝜕𝜇 𝜃 − 2𝐴𝜇4𝑔
1• In the classical limit ( 2 , 𝜌𝑠 → ∞), 𝑍 ≃ 𝑒 𝑖𝛼 𝛾𝐴𝑔• The area of saddle point surface |𝛾𝐴 | is imaginary (real)
for surface with Euclidean (Lorentzian) signature. 𝛼 ∝log 𝐷HRT formula for Lorentzian time
• In our approach, HRT surface can be defined even fortime-like regions.
• Renyi entropies are generalized to multipoint
functions of twist operators.

• 𝑒 −𝑆𝑛 𝑛−1) = 𝑇𝑋𝑛 𝑥2 , 𝑡2 𝑋𝑛 𝑥1 , 𝑡1
• HRT formula can be generalized
if 𝒜 𝛾 ∝ 𝑛 − 1 |𝛾| and if
the saddle point approximation
applies.

=𝑖𝒜 𝛾𝑒𝛾

𝐵𝐶𝐴Quantum corrections to HRT• Consider a bulk parent theory withVBS⊗low energy field theory=𝐷𝐷𝑏1• Int. out VBS leads to 𝑆 = − log 𝐷2𝑝 𝐹𝑝 for gauge field.

• Gauge field is coupled to the lowenergy theory with dimension 𝐷𝑏 ≪ 𝐷• 𝑆2 𝐴 ≃ 𝒜𝑐𝑙 = log 𝐷 |𝛾𝐴 | + 𝑆2𝑏𝑢𝑙𝑘 𝐸𝐴 )• Both terms are entanglement entropy inthe bulk
Consistent with Faulkner et al ‘13

Generalization to higher Renyientropies• The random average can be generalized to higher
Renyi entropies 𝑡𝑟 𝜌𝐴𝑛 = 𝑡𝑟 𝜌⊗𝑛 𝑋𝐴𝑛1∗∗∗• 𝜙𝑖1 𝜙𝑗1 𝜙𝑖2 𝜙𝑗2 … 𝜙𝑖𝑛 𝜙𝑗𝑛 =𝐶𝑛𝑗 𝑗 …𝑗1 2 𝑛𝑔𝑛𝑔∈𝑆𝑖1 𝑖2 …𝑖𝑛• Summation over all permutation group elements

1=𝐶3++⋯

Generalization to higher Renyientropies• Random average of 𝑛 copies of the network𝑆 𝑛gauge theory minimal coupled to 𝑛 replica of theparent theory−𝒜 𝑔 𝑛𝜙 1 ,𝜙 2 ,…,𝜙 𝑛 ,𝑔 𝑛•𝑒≡ ∫ 𝐷𝜙 𝑖 𝑒 −𝒜𝑃𝐷𝑏• For short-range entangled states such as VBS, in large
𝐷 limit one obtains
1
𝑆𝑛 ≃𝒜𝑐𝑙 = log 𝐷 𝛾𝐴 + 𝑆𝑛 𝑏𝑢𝑙𝑘 𝐸𝐴𝑛−1
with 𝑆𝑛 𝑏𝑢𝑙𝑘 𝐸𝐴 the bulk low energy field
contribution• Leading order Renyi entropies are 𝑛 independent,different from CFT results. Reason: absence of backreaction. (c.f. Xi Dong 1601.06788)

𝐷Operator correspondence• Definition of Bulk local operators:• Bulk operators 𝜙𝑥 acting
on low energy subspace havenontrivial effect to theboundary after random
projection.
𝜙𝑂1
𝑂2…

𝑥…
𝐷𝐷𝑏

• Does each bulk low energy operator correspond to aboundary operator? If yes, can the bulk operator berepresented in a region of the boundary?

𝑂𝑛

Operator correspondence
Randomprojection|𝜓1𝑃 ⟩|𝜓2𝑃 ⟩|𝜓1 ⟩|𝜓2 ⟩• Different operators create different states in the parenttheory 𝜓1𝑃 = 𝜙𝑖 𝑥𝑖 , 𝑡𝑖 )|𝐺⟩• Correspondingly there are different boundary states afterthe random projection.
• Generically each state gives different gauge field action.

Error correction properties• A “code subspace” can be defined as the subspace ofbulk states with the same gauge field action (i.e.
quantum corrections to HRT can be ignored)• For such states in 𝐷 → ∞, 𝑡𝑟𝑃 𝜌1𝑃 𝐸𝐴 𝜌2𝑃 𝐸𝐴 𝑛 =𝑡𝑟 𝜌1 𝐴 𝜌2 𝐴 𝑛 ∀𝑛. ⇒ S 𝜌1𝑃 𝐸𝐴 𝜌2𝑃 𝐸𝐴 =𝑆 𝜌1 𝐸𝐴 𝜌2 𝐸𝐴⇒ (Jaffer et al ’15,Harlow et al ‘16)

Randomprojection• For each bulkoperator 𝜙 acting in𝐸𝐴 , ∃ boundary
operator 𝑂𝐴 , s.t.
𝜓1 𝑂𝐴 𝜓2 = 𝜓1𝑏𝑢𝑙𝑘 𝜙 𝜓2𝑏𝑢𝑙𝑘 for any two states in
the code subspace.

Gauge redundancy and gauge fixing• The random average leads to a gauge field partition
function with gauge redundancy Zn A =−𝒜 𝑔𝑥𝑦𝑒= 𝑍𝑆 𝑛 𝐴) ⋅ Ω𝑔 ∈𝑆 𝑛𝑥𝑦• Ω is the volume of gauge orbit.
• Not a problem if we calculate ratios such as

𝑍𝑛 𝐴𝑍𝑛 ∅• However, problem happens when we consider
fluctuations:• 𝛿𝑍22 = 𝑡𝑟 𝜌2 2 − 𝑡𝑟 𝜌2
Ω4 𝑍𝑆 4 ∅ − Ω22 𝑍𝑆 2 ∅ 22= 𝑍4 ∅ − 𝑍2 ∅ 2 =• The fluctuation is large because Ω4 ≫ Ω22 . We need toremove the gauge redundancy.

Gauge redundancy and gauge fixing• For discrete gauge theories, gauge fixing can be doneby directly fixing 𝑔𝑥𝑦 on some links, without
constraining any gauge flux.
• This corresponds to picking a spanning tree in the bulk,
and only impose random projection on links off thetree.
• Fluctuations are suppressed in large 𝐷 limit after gauge
fixing. (Similar to spatial RTN Hayden et al ‘16)𝜏𝜏Summaryduality⊂⊂

Bulk gravity(𝑄𝐺𝑑+1 )Boundarytheory(𝑄𝐹𝑇𝑑 )

⊂Bulk “parenttheory”(𝑄𝐹𝑇𝑑+1 )Randomprojections

Low energy
statesLow energy
QFT on a fixedbackgroundLow energy
states ofboundary

Bulk locality
HRT formula
Error correction code

• “Low energy states” actually means states that share
a common entanglement structure in large 𝐷 limit.

Open questions• How to start from the boundary and construct thebulk theory?
• How to take into account of the back reaction anddescribe correct entanglement properties of CFTs?
• How to obtain the bulk geometry equation (Einsteinequation)?
• A formalism in the continuum limit?
• Does the space-time RTN helps us to understand of
the Black hole information paradox?
• Does this approach allow us to define holography inflat and positively curved space?

More details about finite D
fluctuations•𝑓 𝐷𝑒 𝜆Ω=gives a very loose bound that can be prooved.
𝐷Ω•𝑓 𝐷 =Δ2𝑛𝐷 2

gives a better bound that’s less rigorous.

• Ω is spacetime volume.