Download Entanglement, Gravity, and Quantum Error Correction

Entanglement, Gravity,
and
Quantum Error Correction
Xi Dong
Institute for Advanced Study
July 12, 2016
[XD, arXiv: 1601.06788 to appear in Nature Commun.]
[XD, Phys. Rev. Lett. 116, 251602 (2016)]
[XD, Harlow, Wall, Phys. Rev. Lett. 117, 021601 (2016)]
[XD, Miao, JHEP 1512, 100 (2015)]
[Almheiri, XD, Harlow, JHEP 1504, 163 (2015)]
[XD, JHEP 1401, 044 (2014)]
21st International Conference on General Relativity and Gravitation, Columbia University
Gravity
Entanglement
July 12, 2016
Quantum Error
Correction
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Bekenstein-Hawking entropy for
black holes
𝑘𝑐 3 Area Horizon
𝑆=
4𝐺𝑁 ℏ
• Led to much progress in understanding quantum
gravity, e.g. holographic principle.
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Anti-de Sitter/Conformal Field Theory
Correspondence
[Maldacena ’97]
Quantum gravity in
AdSd+1 (bulk)
Holographic CFTs on
𝝏AdSd+1 (boundary)
Isometry group 𝑂(𝑑, 2)
Conformal group 𝑂(𝑑, 2)
Black hole states
Thermal states
Gauge symmetry
Global symmetry
States and operators
States and operators
• Best-understood model of emergent spacetime/gravity
• Framework for generalizing area law beyond black holes
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Holographic Entanglement Entropy
A simple and powerful prescription for entanglement
[Ryu & Takayanagi ’06]
entropy:
Area Minimal Surface
𝑆=
Spacetime
“Spooky action
4𝐺𝑁
geometry
at a distance”
Recall the definition:
𝑆 ≝ −Tr ρ𝐴 ln ρ𝐴
This is only the tip of the iceberg!
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Holographic Renyi Entropy
A simple and powerful prescription for Renyi entropy:
2
𝑛 𝜕𝑛
𝑛−1
Area Cosmic Brane𝑛
𝑆𝑛 =
𝑛
4𝐺𝑁
[XD 1601.06788]
Has tension and
backreacts on ambient
geometry by creating
conical deficit angle
𝑛−1
2𝜋
.
1
𝑆𝑛 ≝
ln Trρ𝑛𝐴
1−𝑛
More general measures
of entanglement
𝑛
• Gravity dual of Renyi entropy is a cosmic brane!
• As 𝑛 → 1: probe brane settles at minimal surface.
• One-parameter generation of Ryu-Takayanagi.
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Example: Renyi entropy for two disks
in holographic CFT
2
𝑛 𝜕𝑛
𝑛−1
Area Cosmic Brane𝑛
𝑆𝑛 =
𝑛
4𝐺𝑁
• For one disk, it was calculated by exploiting a symmetry and finding
hyperbolic black hole solutions.
[Hung, Myers, Smolkin & Yale ’11]
• Area-law prescription is more powerful: applies to arbitrary regions.
• For two disks, we study mutual Renyi information:
𝐼𝑛 𝐴1 , 𝐴2 ≝ 𝑆𝑛 𝐴1 + 𝑆𝑛 𝐴2 − 𝑆𝑛 (𝐴1 ∪ 𝐴2 )
• To linear order in 𝛿𝑛 = 𝑛 − 1, brane backreaction is weak:
23−𝑑 𝜋 𝑑+1 𝐶𝑇 𝛿𝑛
2−𝑥
𝐼𝑛 =
2 𝑥 𝐵
𝑑−1
𝑑 𝑑2 − 1 Γ
2
July 12, 2016
𝑥
2−𝑥
2
𝑑+1 2−𝑑
;
,
+ 𝑂(𝛿𝑛2 )
2
2
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
[XD 1601.06788]
So far:
We used the area law to understand structure of
quantum entanglement and to efficiently study Renyi
entropies.
Rest of the talk:
How to use it to understand quantum gravity?
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
AdS/CFT: our best-understood
model of quantum gravity [Maldacena ’97]
Quantum gravity in
AdSd+1
Holographic CFTs on
𝝏AdSd+1
Isometry group 𝑂(𝑑, 2)
Conformal group 𝑂(𝑑, 2)
Black hole states
Thermal states
Gauge symmetry
Global symmetry
States and operators
States and operators
lim 𝑟 Δ 𝜙 𝑟, 𝑥 = 𝑂(𝑥)
𝑟→∞
𝜙 𝑟, 𝑥 = ?
• What operator in CFT represents a local bulk operator?
• Answering this question helps us reconstruct the bulk.
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Global AdS reconstruction
[Hamilton, Kabat, Lifschytz, Lowe ’06]
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
AdS-Rindler reconstruction for disk 𝐴
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
What region of the dual
spacetime is described by a
general subregion in a
holographic CFT?
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Reconstruction conjecture for
entanglement wedge
• Entanglement wedge is defined as a bulk region bounded
by the Ryu-Takayanagi minimal surface.
• It may change discontinuously.
• Conjecture: Any bulk operator in entanglement wedge of
𝐴 may be represented as a CFT operator on 𝐴.
[Czech, Karczmarek, Nogueira
& Van Raamsdonk ’12]
[Wall ’12]
[Headrick, Hubeny, Lawrence
& Rangamani ’14]
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Conjecture:
Any bulk operator in entanglement wedge of 𝐴
may be represented as a CFT operator on 𝐴.
Ingredients for proving the conjecture:
Holography as a quantum error correcting code
CFT relative entropy = bulk relative entropy
⇒ Reconstruction theorem for entanglement wedge
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Holography as a quantum error
[Almheiri, XD, Harlow ’14]
correcting code
• 𝜙(𝑥) can be represented
on 𝐴 ∪ 𝐵, 𝐵 ∪ 𝐶, or 𝐴 ∪ 𝐶.
• Obviously they cannot be
the same CFT operator.
• This redundancy is a
defining feature for
quantum error correction.
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Relative entropy is bulk relative entropy
• Quantum corrections to the area law:
Area Cosmic Brane𝑛
𝑆𝑛 =
+ 𝑆𝑛 ,bulk
4𝐺𝑁
• Provides (alternative) derivation of similar statements
about modular Hamiltonian 𝐾 ≝ − ln ρ𝐴
Area
𝐾=
+ 𝐾bulk
4𝐺𝑁
• and relative entropy 𝑆 𝜌 𝜎 ≝ Tr 𝜌 ln 𝜌 − Tr 𝜌 ln 𝜎
𝑆(𝜌|𝜎) = 𝑆bulk (𝜌|𝜎)
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
[Jafferis, Lewkowycz,
Maldacena & Suh
1512.06431]
Reconstruction theorem for
[XD, Harlow & Wall
entanglement wedge
1601.05416]
Any bulk operator 𝑂𝑒 in the entanglement wedge 𝑒 of 𝐴
may be reconstructed as a CFT operator 𝑂𝐴 on 𝐴 via
quantum error correction, as long as the relative entropy
of any two bulk states satisfies 𝑆(𝜌𝐴 |𝜎𝐴 ) = 𝑆
bulk (𝜌𝑒 |𝜎𝑒 )
𝑒
Intuitive proof:
• WLG assume 𝑂𝑒 is Hermitian.
• Take any bulk state 𝜌, let 𝜎 be 𝑒 𝑖𝜆𝑂𝑒 acting on 𝜌.
• Use 𝑆 𝜌𝐴 𝜎𝐴 = 𝑆bulk 𝜌𝑒 𝜎𝑒 = 0 to conclude
𝜌𝐴 = 𝜎𝐴 .
• No measurements on 𝐴 distinguish the two states
⇒ 𝑂𝑒 , 𝑋𝐴 = 0 under state 𝜌 for any 𝑋𝐴 .
• 𝑂𝑒 must have a CFT realization as 𝑂𝐴 on 𝐴.
Via theorem in [Almheiri, XD, Harlow ’14]
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
What We Learned
• Area law is universal in quantum gravity and is not
restricted to black hole or entanglement entropy.
• It is a powerful statement for Renyi entropy,
generalizing the Ryu-Takayanagi prescription.
• Quantum information theory enables us to
understand the basic dictionary of quantum gravity.
• Viewing holography as a quantum error correcting
code, we can make progress on how to “build
spacetime from entanglement”.
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)
Thank you!
July 12, 2016
Entanglement, Gravity, and Quantum Error
Correction (Xi Dong)