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Robert Myers Lecture 5
TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAA
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“It from Qubit”: New Collision of Ideas
![Page 3: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/3.jpg)
“It from Qubit”: New Collision of Ideas
Gravity
Quantum
“Holography”
Information
Theory
Quantum
![Page 4: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/4.jpg)
“It from Qubit”: New Collision of Ideas
Gravity
Quantum
Confluence
“Holography”
Information
Theory
Quantum
![Page 5: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/5.jpg)
“It from Qubit”: New Collision of Ideas
Gravity
Quantum
“Holography”
Information
Theory
Quantum
![Page 6: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/6.jpg)
General Relativity is the geometric arena for physics
on very large scales: planets, stars, galaxies, cosmology
Einstein:
Gravity? It’s all about geometry!
![Page 7: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/7.jpg)
General Relativity is the geometric arena for physics
on very large scales: planets, stars, galaxies, cosmology
Einstein:
Gravity? It’s all about geometry!
Spacetime moves from simply stage for physical phenomena, to being both the stage and an active player in the dynamics
![Page 8: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/8.jpg)
General Relativity + Quantum Theory = Quantum Gravity?
+ ℏ
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gtheory = 2:0023193043070
General Relativity + Quantum Theory = Quantum Gravity?
• quantum fluctuations become manifest at small scales
e.g., magnetic moment of the electron, ,
with 𝑔 ≈ 2 but modified by quantum fluctuations
10
experiment
erimentexptheory10
¹e = g e¹h=4me
+ ℏ
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• spacetime geometry exhibits strong fluctuations
when examined on very short distance scales
• how do we make sense of spacetime framework?
General Relativity + Quantum Theory = Quantum Gravity?
![Page 11: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/11.jpg)
• spacetime geometry exhibits strong fluctuations
when examined on very short distance scales
• how do we make sense of spacetime framework?
modify geometry at short distances
modify spectrum at short distances
General Relativity + Quantum Theory = Quantum Gravity?
![Page 12: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/12.jpg)
“It from Qubit”: New Collision of Ideas
Gravity
Quantum
“Holography”
Information
Theory
Quantum
![Page 13: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/13.jpg)
Quantum Entanglement
Einstein-Podolsky-Rosen Paradox:
“spukhafte Fernwirkung” =
spooky action at a distance
• polarizations of pair of photons connected,
no matter how far apart they travel
• different subsystems are correlated through
global state of full system
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Quantum Entanglement
Einstein-Podolsky-Rosen Paradox:
“spukhafte Fernwirkung” =
spooky action at a distance
• polarizations of pair of photons connected,
no matter how far apart they travel
Quantum Information: entanglement becomes a resource for
(ultra)fast computations and (ultra)secure communications
Condensed Matter: key to “exotic” phases and phenomena,
e.g., quantum Hall fluids, unconventional superconductors,
quantum spin fluids, . . . .
• different subsystems are correlated through
global state of full system
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Quantum Entanglement
Einstein-Podolsky-Rosen Paradox:
“spukhafte Fernwirkung” =
spooky action at a distance
• polarizations of pair of photons connected,
no matter how far apart they travel
Quantum Information: entanglement becomes a resource for
(ultra)fast computations and (ultra)secure communications
Condensed Matter: key to “exotic” phases and phenomena,
e.g., quantum Hall fluids, unconventional superconductors,
quantum spin fluids, . . . .
• different subsystems are correlated through
global state of full system
Quantum Fields & Quantum Gravity
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• in QFT, typically introduce a (smooth) boundary or entangling
surface which divides the space into two separate regions
• integrate out degrees of freedom in “outside” region
• remaining dof are described by a density matrix
A
calculate von Neumann entropy:
• general diagnostic to give a quantitative measure of entanglement
using entropy to detect correlations between two subsystems
A –
Entanglement Entropy in QFT
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Bulk: gravity with negative Λ
in d+1 dimensions
anti-de Sitter
space
Boundary: quantum field theory
without intrinsic scales
in d dimensions
conformal
field theory
“holography”
Holography: AdS/CFT correspondence
energy radius (Maldacena ‘97)
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Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
Bekenstein- Hawking formula
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Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
• conjecture many detailed consistency tests (Ryu, Takayanagi, Hubeny, Rangamani, Headrick, Hung, Smolkin, RM, Faulkner, . . . )
Bekenstein- Hawking formula
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Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
• conjecture many detailed consistency tests (Ryu, Takayanagi, Hubeny, Rangamani, Headrick, Hung, Smolkin, RM, Faulkner, . . . )
(Maldacena & Lewkowycz) • 2013 proof (for static geometries) (Dong, Lewkowycz & Rangamani) • 2016 proof (for general geometries)
Bekenstein- Hawking formula
![Page 21: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/21.jpg)
Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
• conjecture many detailed consistency tests (Ryu, Takayanagi, Hubeny, Rangamani, Headrick, Hung, Smolkin, RM, Faulkner, . . . )
(Maldacena & Lewkowycz) • 2013 proof (for static geometries) (Dong, Lewkowycz & Rangamani) • 2016 proof (for general geometries)
• holographic EE: fruitful forum for bulk-boundary dialogue
Bekenstein- Hawking formula
![Page 22: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/22.jpg)
Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
Bekenstein- Hawking formula
• holographic EE teaches us lessons about QFTs, eg,
diagnostic in RG flows and c-theorms, eg, F-theorem (Sinha & RM, ……)
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●
(F)UV
gravity/holography + EE RG flows in (2+1)-dimensions
●
●
●
(F)UV ¸ (F)IRF-theorem:
C(R) =R@RS(R)¡S(R) A B
R
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Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
Bekenstein- Hawking formula
• holographic EE teaches us lessons about QFTs, eg,
geometric properties of entanglement entropy in QFT’s (Mezei, Perlmutter, Lewkowycz, Bueno, RM, Witczak-Krempa, ….)
diagnostic for quantum quenches/phase transitions
diagnostic in RG flows and c-theorms, eg, F-theorem (Sinha & RM, ……)
(Lopez, Johnson, Balasubramanian, Bernamonti, Craps, Galli, ….)
![Page 25: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/25.jpg)
Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
Bekenstein- Hawking formula
• holographic EE teaches us lessons about (quantum) gravity, eg,
connectivity of spacetime requires entanglement (van Raamsdonk)
BH formula applies beyond black holes/horizons
![Page 26: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/26.jpg)
Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
Bekenstein- Hawking formula
• holographic EE teaches us lessons about (quantum) gravity, eg,
connectivity of spacetime requires entanglement (van Raamsdonk)
BH formula applies beyond black holes/horizons
Spacetime Geometry = Entanglement
![Page 27: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/27.jpg)
black hole entropy is entanglement entropy (Sorkin, ….)
connectivity of spacetime requires entanglement (van Raamsdonk)
AdS spacetime as a tensor network (MERA) (Swingle, Vidal, ….)
spacetime entanglement conjecture (Bianchi & RM)
“ER = EPR” conjecture (Maldacena & Susskind)
(Balasubramanian, Chowdhury, Czech, de Boer & Heller;
RM, Rao & Sugishita; Czech, Dong & Sully; ….)
Spacetime Geometry = Entanglement
A
B
Σ
use BH formula for holographic entanglement entropy (Ryu & Takayanagi; ….)
Bekenstein-Hawking formula: spacetime geometry encodes SBH
hole-ographic spacetime
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black hole entropy is entanglement entropy (Sorkin, ….)
connectivity of spacetime requires entanglement (van Raamsdonk)
AdS spacetime as a tensor network (MERA) (Swingle, Vidal, ….)
spacetime entanglement conjecture (Bianchi & RM)
“ER = EPR” conjecture (Maldacena & Susskind)
(Balasubramanian, Chowdhury, Czech, de Boer & Heller;
RM, Rao & Sugishita; Czech, Dong & Sully; ….)
Spacetime Geometry = Entanglement
A
B
Σ
use BH formula for holographic entanglement entropy (Ryu & Takayanagi; ….)
Bekenstein-Hawking formula: spacetime geometry encodes SBH
hole-ographic spacetime
spacetime provides both the stage for physical phenomena and the agent which manifests gravitational dynamics
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• entanglement entropy:
• make a small perturbation of state:
S(½A) =¡tr(½A log½A)
~½ = ½A+ ±½
Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)
![Page 30: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/30.jpg)
• entanglement entropy:
• make a small perturbation of state:
S(½A) =¡tr(½A log½A)
~½ = ½A+ ±½
Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)
![Page 31: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/31.jpg)
• entanglement entropy:
• make a small perturbation of state:
S(½A) =¡tr(½A log½A)
~½ = ½A+ ±½
Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)
![Page 32: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/32.jpg)
• entanglement entropy:
• make a small perturbation of state:
S(½A) =¡tr(½A log½A)
~½ = ½A+ ±½
Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)
![Page 33: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/33.jpg)
• entanglement entropy:
• make a small perturbation of state:
S(½A) =¡tr(½A log½A)
~½ = ½A+ ±½
½A = exp(¡HA)• modular (or entanglement) Hamiltonian:
Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)
![Page 34: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/34.jpg)
• entanglement entropy:
• make a small perturbation of state:
S(½A) =¡tr(½A log½A)
~½ = ½A+ ±½
±SA = ±hHAi“1st law” of entanglement entropy
½A = exp(¡HA)(Blanco, Casini, Hung & RM)
• modular (or entanglement) Hamiltonian:
Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)
![Page 35: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/35.jpg)
• entanglement entropy:
• make a small perturbation of state:
S(½A) =¡tr(½A log½A)
~½ = ½A+ ±½
±SA = ±hHAi“1st law” of entanglement entropy
½A = exp(¡HA)(Blanco, Casini, Hung & RM)
• modular (or entanglement) Hamiltonian:
½A = exp(¡H=T)• this is the 1st law for thermal state:
Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)
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“1st law” of entanglement entropy:
• generally is “nonlocal mess” and flow is not geometric
HA =
Zdd¡1x°¹º1 (x)T¹º +
Zdd¡1x
Zdd¡1y °¹º;½¾2 (x; y)T¹ºT½¾ + ¢ ¢ ¢
hence usefulness of first law is very limited, in general
![Page 37: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/37.jpg)
• by causality, and describe physics throughout domain of dependence ; eg, generate boost flows
“1st law” of entanglement entropy:
• generally is “nonlocal mess” and flow is not geometric
HA =
Zdd¡1x°¹º1 (x)T¹º +
Zdd¡1x
Zdd¡1y °¹º;½¾2 (x; y)T¹ºT½¾ + ¢ ¢ ¢
hence usefulness of first law is very limited, in general
• famous exception: Rindler wedge
HA = 2¼K
= 2¼
Z
A(x>0)
dd¡2y dx [x Ttt ] + c0
boost generator
A B ●
HA
Σ
Σ = (x = 0, t = 0) • any QFT in Minkowski vacuum; choose
(Bisognano & Wichmann; Unruh)
![Page 38: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/38.jpg)
• by causality, and describe physics throughout domain of dependence ; eg, generate boost flows
“1st law” of entanglement entropy:
• generally is “nonlocal mess” and flow is not geometric
HA =
Zdd¡1x°¹º1 (x)T¹º +
Zdd¡1x
Zdd¡1y °¹º;½¾2 (x; y)T¹ºT½¾ + ¢ ¢ ¢
hence usefulness of first law is very limited, in general
• famous exception: Rindler wedge
HA = 2¼K
= 2¼
Z
A(x>0)
dd¡2y dx [x Ttt ] + c0
boost generator
A B ●
HA
Σ
Σ = (x = 0, t = 0) • any QFT in Minkowski vacuum; choose
(Bisognano & Wichmann; Unruh)
![Page 39: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/39.jpg)
• another exception: CFT in vacuum of d-dim. flat space and entangling surface which is Sd-2 with radius R
HA = 2¼
Z
A
dd¡1yR2 ¡ j~yj2
2RTtt(~y) + c0
(Casini, Huerta & RM; Hislop & Longo)
“1st law” of entanglement entropy:
A
A –
![Page 40: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/40.jpg)
• another exception: CFT in vacuum of d-dim. flat space and entangling surface which is Sd-2 with radius R
HA = 2¼
Z
A
dd¡1yR2 ¡ j~yj2
2RTtt(~y) + c0
(Casini, Huerta & RM; Hislop & Longo)
“1st law” of entanglement entropy:
A
A –
K = 2¼
Z
A
d§¹ T¹ºKº
• construct with conformal transformation:
![Page 41: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/41.jpg)
• holographic realization:
“1st law” of entanglement entropy:
• HA has simply form for CFT and spherical entangling surface:
HA = 2¼
Z
A
dd¡1yR2 ¡ j~yj2
2RTtt(~y) + c0
![Page 42: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/42.jpg)
• holographic realization:
• apply 1st law for spheres of all sizes, positions and in all frames:
1st law of SEE bulk geometry satisfies linearized Einstein eq’s
“1st law” of entanglement entropy:
• HA has simply form for CFT and spherical entangling surface:
HA = 2¼
Z
A
dd¡1yR2 ¡ j~yj2
2RTtt(~y) + c0
![Page 43: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/43.jpg)
• holographic realization:
• apply 1st law for spheres of all sizes, positions and in all frames:
1st law of SEE bulk geometry satisfies linearized Einstein eq’s
spacetime provides both the stage for physical phenomena and the agent which manifests gravitational dynamics
“1st law” of entanglement entropy:
• HA has simply form for CFT and spherical entangling surface:
HA = 2¼
Z
A
dd¡1yR2 ¡ j~yj2
2RTtt(~y) + c0
![Page 44: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/44.jpg)
Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
Bekenstein- Hawking formula
• holographic EE teaches us lessons about (quantum) gravity, eg,
connectivity of spacetime requires entanglement (van Raamsdonk)
BH formula applies beyond black holes/horizons
Spacetime Geometry = Entanglement
![Page 45: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/45.jpg)
Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
Bekenstein- Hawking formula
• holographic EE teaches us lessons about (quantum) gravity, eg,
connectivity of spacetime requires entanglement (van Raamsdonk)
BH formula applies beyond black holes/horizons
Spacetime Geometry = Entanglement
Susskind: Entanglement is not enough!
![Page 46: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/46.jpg)
Holographic Entanglement Entropy: (Ryu & Takayanagi `06)
A
AdS boundary
AdS bulk spacetime
boundary conformal field theory
A –
Bekenstein- Hawking formula
• holographic EE teaches us lessons about (quantum) gravity, eg,
connectivity of spacetime requires entanglement (van Raamsdonk)
BH formula applies beyond black holes/horizons
Spacetime Geometry = Entanglement
Susskind: Entanglement is not enough! V
![Page 47: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/47.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V ds2 = ¡
µr2
L2+ 1¡ ¹
rd¡2
¶dt2 +
dr2
r2
L2+ 1¡ ¹
rd¡2
+ r2d2d¡1
![Page 48: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/48.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
singularity
singularity
asymptotic AdS bndry
asymptotic AdS bndry horizons
ds2 = ¡µr2
L2+ 1¡ ¹
rd¡2
¶dt2 +
dr2
r2
L2+ 1¡ ¹
rd¡2
+ r2d2d¡1
![Page 49: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/49.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
jTFDi 'X
®
e¡E®=(2T )jE®iLjE®iR
• pure state: 𝑆𝐸𝐸 = 0
![Page 50: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/50.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
jTFDi 'X
®
e¡E®=(2T )jE®iLjE®iR
½R =TrLjTFDihTFDj
'X
®
e¡E®=T jE®iRhE®jR
• pure state: 𝑆𝐸𝐸 = 0
• mixed state: 𝑆𝐸𝐸 = 𝐴𝐻4𝐺
![Page 51: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/51.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
jTFDi 'X
®
e¡E®=(2T )jE®iLjE®iR
½R =TrLjTFDihTFDj
'X
®
e¡E®=T jE®iRhE®jR
• pure state: 𝑆𝐸𝐸 = 0
• mixed state: 𝑆𝐸𝐸 = 𝐴𝐻4𝐺
![Page 52: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/52.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
![Page 53: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/53.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
A A’
![Page 54: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/54.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
A A’
![Page 55: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/55.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
A’ A
![Page 56: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/56.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
A’ A
![Page 57: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/57.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
A’ A
![Page 58: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/58.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
A’ A
Holographic Complexity!
![Page 59: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/59.jpg)
Complexity?
• recall 𝑆𝐸𝐸 only probes the eigenvalues of the density matrix
SEE = ¡Tr [½A log ½A] =¡X
¸i log¸i
![Page 60: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/60.jpg)
Complexity?
• recall 𝑆𝐸𝐸 only probes the eigenvalues of the density matrix
SEE = ¡Tr [½A log ½A] =¡X
¸i log¸i
• would like a new probe “sensitive to phases”
jTFDi 'X
®
e¡E®=(2T )¡iE®(tL+tR)jE®iLjE®iR
![Page 61: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/61.jpg)
Complexity:
• computational complexity: how difficult is it to implement a task? eg, how difficult is it to prepare a particular state?
![Page 62: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/62.jpg)
Complexity:
• computational complexity: how difficult is it to implement a task? eg, how difficult is it to prepare a particular state?
• quantum circuit model:
jÃi = U jÃ0i
![Page 63: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/63.jpg)
Complexity:
• computational complexity: how difficult is it to implement a task? eg, how difficult is it to prepare a particular state?
• quantum circuit model:
jÃi = U jÃ0isimple reference state eg, j00000 ¢ ¢ ¢0i
![Page 64: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/64.jpg)
Complexity:
• computational complexity: how difficult is it to implement a task? eg, how difficult is it to prepare a particular state?
• quantum circuit model:
jÃi = U jÃ0isimple reference state eg, j00000 ¢ ¢ ¢0i
unitary operator built from set of
simple gates
![Page 65: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/65.jpg)
Complexity:
• computational complexity: how difficult is it to implement a task? eg, how difficult is it to prepare a particular state?
• quantum circuit model:
jÃi = U jÃ0isimple reference state eg, j00000 ¢ ¢ ¢0i
unitary operator built from set of
simple gates
![Page 66: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/66.jpg)
Complexity:
• computational complexity: how difficult is it to implement a task? eg, how difficult is it to prepare a particular state?
• quantum circuit model:
jÃi = U jÃ0isimple reference state eg, j00000 ¢ ¢ ¢0i
unitary operator built from set of
simple gates
tolerance: j jÃi ¡ jÃiTarget j2 · "
![Page 67: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/67.jpg)
Complexity:
• computational complexity: how difficult is it to implement a task? eg, how difficult is it to prepare a particular state?
• quantum circuit model:
jÃi = U jÃ0isimple reference state eg, j00000 ¢ ¢ ¢0i
unitary operator built from set of
simple gates
tolerance: j jÃi ¡ jÃiTarget j2 · "
• complexity = minimum number of gates required to prepare the desired state
![Page 68: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/68.jpg)
Complexity:
• computational complexity: how difficult is it to implement a task? eg, how difficult is it to prepare a particular state?
• quantum circuit model:
jÃi = U jÃ0isimple reference state eg, j00000 ¢ ¢ ¢0i
unitary operator built from set of
simple gates
tolerance: j jÃi ¡ jÃiTarget j2 · "
• complexity = minimum number of gates required to prepare the desired state
• does the answer depend on the choices?? YES!!
• compare to “circuit depth” for spin chain
![Page 69: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/69.jpg)
Complexity:
• computational complexity: how difficult is it to implement a task? eg, how difficult is it to prepare a particular state?
• quantum circuit model:
jÃi = U jÃ0isimple reference state eg, j00000 ¢ ¢ ¢0i
unitary operator built from set of
simple gates
tolerance: j jÃi ¡ jÃiTarget j2 · "
• complexity = minimum number of gates required to prepare the desired state
• does the answer depend on the choices?? YES!!
• but what does this really mean in quantum field theory? ???
![Page 70: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/70.jpg)
Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind
the horizon and which are predicted by general relativity.”
V
Holographic Complexity!
![Page 71: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/71.jpg)
CV(§) = max§=@B
·V(B)GN `
¸CA(§) =
IWDW
¼ ¹h
A Tale of Two Dualities: Holographic Complexity
Complexity = Volume Complexity = Action
Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao
![Page 72: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/72.jpg)
CA(§) =IWDW
¼ ¹h
A Tale of Two Dualities: Holographic Complexity
Complexity = Volume Complexity = Action
CV(§) = max§=@B
·V(B)GN `
¸
Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao
![Page 73: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/73.jpg)
dCVdt
¯̄¯t!1
=8¼
d¡ 1M CA(§) =
IWDW
¼ ¹h
A Tale of Two Dualities: Holographic Complexity
Complexity = Volume Complexity = Action
Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao
(planar)
(d= boundary dimension) (universal; Lloyd bound)
![Page 74: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/74.jpg)
Wheeler-DeWitt patch: domain of dependence of Cauchy surface ending on boundary time slice
Complexity = Action
null boundaries null boundaries
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
Complexity = Action:
![Page 75: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/75.jpg)
Wheeler-DeWitt patch: domain of dependence of Cauchy surface ending on boundary time slice
Complexity = Action
null boundaries null boundaries
evaluate the action? what action??
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
Complexity = Action:
![Page 76: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/76.jpg)
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 77: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/77.jpg)
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• well-defined action principle requires boundary terms
M
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 78: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/78.jpg)
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• well-defined action principle requires boundary terms
eg,
M
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 79: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/79.jpg)
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• well-defined action principle requires boundary terms
Gibbons-Hawking-York
K K
K = habranb
eg,
M K
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 80: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/80.jpg)
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• well-defined action principle requires boundary terms
Gibbons-Hawking-York
• K K
K = habranb
eg,
M K
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 81: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/81.jpg)
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• well-defined action principle requires boundary terms
Hayward
• • ´
´K K
K = habranbM K
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 82: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/82.jpg)
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• well-defined action principle requires boundary terms
• •
·
·
•
•
• a
a
a
´
´K K
K = habranb
eg,
M K
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 83: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/83.jpg)
+ ??????
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities: total derivatives, extra boundary terms, . . . .
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 84: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/84.jpg)
+ ??????
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities: total derivatives, extra boundary terms, . . . .
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
ambiguities in circuit complexity?
![Page 85: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/85.jpg)
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 86: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/86.jpg)
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 87: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/87.jpg)
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 88: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/88.jpg)
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
Aside: make choices without referring to particular metric/coordinates, which allow for meaningful comparison of different states/geometries
![Page 89: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/89.jpg)
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
2) a = 𝐚𝟎 + 휀 log |𝒌1 ∙ 𝒏2|
log |𝒌1 ∙ 𝒌2/2|
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 90: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/90.jpg)
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
2) a = 𝐚𝟎 + 휀 log |𝒌1 ∙ 𝒏2|
log |𝒌1 ∙ 𝒌2/2|
addivity of grav. action: 𝒂𝟎 = 0
=
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 91: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/91.jpg)
3) constant rescaling: 𝑘𝜇 → 𝜶 𝑘𝜇
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
2) a = 𝐚𝟎 + 휀 log |𝒌1 ∙ 𝒏2|
log |𝒌1 ∙ 𝒌2/2|
addivity of grav. action: 𝒂𝟎 = 0
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 92: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/92.jpg)
fix normalization asymptotically:
𝒌 ∙ 𝒕 = ±1
3) constant rescaling: 𝑘𝜇 → 𝜶 𝑘𝜇
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
2) a = 𝐚𝟎 + 휀 log |𝒌1 ∙ 𝒏2|
log |𝒌1 ∙ 𝒌2/2|
addivity of grav. action: 𝒂𝟎 = 0
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
![Page 93: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/93.jpg)
fix normalization asymptotically:
𝒌 ∙ 𝒕 = ±1
3) constant rescaling: 𝑘𝜇 → 𝜶 𝑘𝜇
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
2) a = 𝐚𝟎 + 휀 log |𝒌1 ∙ 𝒏2|
log |𝒌1 ∙ 𝒌2/2|
addivity of grav. action: 𝒂𝟎 = 0
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
dCAdt
¯̄¯t!1
=2M
¼
![Page 94: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/94.jpg)
fix normalization asymptotically:
𝒌 ∙ 𝒕 = ±1
3) constant rescaling: 𝑘𝜇 → 𝜶 𝑘𝜇
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
2) a = 𝐚𝟎 + 휀 log |𝒌1 ∙ 𝒏2|
log |𝒌1 ∙ 𝒌2/2|
addivity of grav. action: 𝒂𝟎 = 0
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
dCAdt
¯̄¯t!1
=2M
¼
![Page 95: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/95.jpg)
fix normalization asymptotically:
𝒌 ∙ 𝒕 = ±1
3) constant rescaling: 𝑘𝜇 → 𝜶 𝑘𝜇
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
2) a = 𝐚𝟎 + 휀 log |𝒌1 ∙ 𝒏2|
log |𝒌1 ∙ 𝒌2/2|
addivity of grav. action: 𝒂𝟎 = 0
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
dCAdt
¯̄¯t!1
=2M
¼
![Page 96: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/96.jpg)
fix normalization asymptotically:
𝒌 ∙ 𝒕 = ±1
3) constant rescaling: 𝑘𝜇 → 𝜶 𝑘𝜇
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
2) a = 𝐚𝟎 + 휀 log |𝒌1 ∙ 𝒏2|
log |𝒌1 ∙ 𝒌2/2|
addivity of grav. action: 𝒂𝟎 = 0
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
dCAdt
¯̄¯t!1
=2M
¼
![Page 97: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/97.jpg)
fix normalization asymptotically:
𝒌 ∙ 𝒕 = ±1
3) constant rescaling: 𝑘𝜇 → 𝜶 𝑘𝜇
1) 𝑘𝜈 𝛻𝜈 𝑘𝜇 = 𝜿 𝑘𝜇
I =1
16¼GN
Z
Mdd+1x
p¡g
µR+
d(d¡ 1)
L2
¶
+1
8¼GN
Z
Bddx
pjhjK +
1
8¼GN
Z
§
dd¡1xp¾ ´
¡ 1
8¼GN
Z
B0d¸dd¡1µ
p° · +
1
8¼GN
Z
§0dd¡1x
p¾ a
Complexity = Action: • gravitational action:
• ambiguities:
affinely parameterize: 𝜿 = 0
2) a = 𝐚𝟎 + 휀 log |𝒌1 ∙ 𝒏2|
log |𝒌1 ∙ 𝒌2/2|
addivity of grav. action: 𝒂𝟎 = 0
Luis Lehner, RCM, Eric Poisson & Rafael Sorkin
dCAdt
¯̄¯t!1
=2M
¼
Aside: make choices without referring to particular metric/coordinates, which allow for meaningful comparison of different states/geometries
![Page 98: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/98.jpg)
dCAdt
¯̄¯t!1
=2M
¼
Holographic Complexity:
Complexity = Volume Complexity = Action
Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao
dCVdt
¯̄¯t!1
=8¼
d¡ 1M (planar)
![Page 99: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/99.jpg)
Complexity of Formation: jTFDi = Z¡1=2
X
®
e¡E®=(2T )jE®iLjE®iR
Shira Chapman, Hugo Marrochio & RCM
![Page 100: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/100.jpg)
Complexity of Formation:
• additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state?
jTFDi = Z¡1=2X
®
e¡E®=(2T )jE®iLjE®iR
¢C = C(jTFDi)¡C(j0i j0i)
Shira Chapman, Hugo Marrochio & RCM
![Page 101: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/101.jpg)
Complexity of Formation:
• additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state?
jTFDi = Z¡1=2X
®
e¡E®=(2T )jE®iLjE®iR
¢C = C(jTFDi)¡C(j0i j0i)
¢CA = ¡ 2£
Shira Chapman, Hugo Marrochio & RCM
![Page 102: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/102.jpg)
• additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state?
¢C = C(jTFDi)¡C(j0i j0i)
¢CA =d¡ 2
d¼cot
µ¼
d
¶S + ¢ ¢ ¢
thermal/ent. entropy
curvature corrections
Complexity of Formation: Shira Chapman, Hugo Marrochio & RCM
(d= boundary dimension)
![Page 103: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/103.jpg)
• additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state?
¢C = C(jTFDi)¡C(j0i j0i)
¢CA =d¡ 2
d¼cot
µ¼
d
¶S + ¢ ¢ ¢
thermal/ent. entropy
curvature corrections
Complexity of Formation: Shira Chapman, Hugo Marrochio & RCM
(d= boundary dimension)
![Page 104: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/104.jpg)
• additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state?
¢C = C(jTFDi)¡C(j0i j0i)
¢CA =d¡ 2
d¼cot
µ¼
d
¶S + ¢ ¢ ¢
thermal/ent. entropy
curvature corrections
Complexity of Formation:
d¡ 2
d¼cot
µ¼
d
¶•
d
d¡ 2
¼2
Shira Chapman, Hugo Marrochio & RCM
(d= boundary dimension)
![Page 105: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/105.jpg)
• additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state?
¢C = C(jTFDi)¡C(j0i j0i)
¢CA =d¡ 2
d¼cot
µ¼
d
¶S + ¢ ¢ ¢
thermal/ent. entropy
curvature corrections
Complexity of Formation:
¢CV = 4p¼(d¡ 2) ¡(1 + 1
d)
(d¡ 1)¡(12+ 1
d)S + ¢ ¢ ¢
Shira Chapman, Hugo Marrochio & RCM
(d= boundary dimension)
![Page 106: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/106.jpg)
dCAdt
¯̄¯t!1
=2M
¼
Holographic Complexity:
Complexity = Volume Complexity = Action
Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao
dCVdt
¯̄¯t!1
=8¼
d¡ 1M (planar)
![Page 107: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/107.jpg)
Compare?
Rrate ¡Rform =log 2
2¼2+O(1=d)
Rform =¢CA¢CV
=d¡ 1
4¼3=2
¡¡1¡ 1
d
¢
¡¡12¡ 1
d
¢ ' d
4¼2
Rrate =dCA=dtdCV =dt
=d¡ 1
4¼2' d
4¼2
![Page 108: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/108.jpg)
Compare?
Rrate ¡Rform =log 2
2¼2+O(1=d)
d
RrateRform
Rform =¢CA¢CV
=d¡ 1
4¼3=2
¡¡1¡ 1
d
¢
¡¡12¡ 1
d
¢ ' d
4¼2
Rrate =dCA=dtdCV =dt
=d¡ 1
4¼2' d
4¼2
(d= boundary dimension)
![Page 109: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/109.jpg)
Compare?
Rrate ¡Rform =log 2
2¼2+O(1=d)
d
RrateRform
Rform =¢CA¢CV
=d¡ 1
4¼3=2
¡¡1¡ 1
d
¢
¡¡12¡ 1
d
¢ ' d
4¼2
Rrate =dCA=dtdCV =dt
=d¡ 1
4¼2' d
4¼2
points to consistency of C=V and C=A dualities up to differences in microscopic rules, eg, gate set
(d= boundary dimension)
![Page 110: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/110.jpg)
Complexity of Formation for d=2:
• additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state?
¢C = C(jTFDi)¡C(j0i j0i)
¢CA =d¡ 2
d¼cot
µ¼
d
¶S + ¢ ¢ ¢
¢CV = 4p¼(d¡ 2) ¡(1 + 1
d)
(d¡ 1)¡(12+ 1
d)S + ¢ ¢ ¢
(d= boundary dimension)
' 0!d= 2
' 0!d= 2
leading term vanishes
• actually holographic calculations apply for 𝑑 ≥ 3 (but still correct)
![Page 111: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/111.jpg)
Complexity of Formation for d=2:
• additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state?
¢C = C(jTFDi)¡C(j0i j0i)
(d= boundary dimension)
• reconsider holographic calculations for 3D BTZ black hole:
¢CA = ¡c
3¢CV = +
8¼
3c
• perhaps related to BTZ black hole being locally AdS geometry
𝒄 = central charge of boundary CFT
![Page 112: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/112.jpg)
Complexity of Formation for d=2:
• additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state?
¢C = C(jTFDi)¡C(j0i j0i)
(d= boundary dimension)
• reconsider holographic calculations for 3D BTZ black hole:
¢CA = ¡c
3¢CV = +
8¼
3c
• perhaps related to BTZ black hole being locally AdS geometry
(NS vac)
(R vac) = 0 = 0
𝒄 = central charge of boundary CFT
![Page 113: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/113.jpg)
Complexity of Formation from MERA?
Cc¡g(vac) ' CTV Td¡1
= k0 S
? ?
? ? ¢C = C(TFD)¡ 2C(vac) = (kT ¡ 2k0)S ¸ 0
Cc¡g(TFD) ' CTV Td¡1
= kT S
![Page 114: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/114.jpg)
CA(§) =IWDW
¼ ¹h
Holographic Complexity:
Complexity = Volume Complexity = Action
CV(§) = max§=@B
·V(B)GN `
¸
Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao
Dean Carmi, RCM & Pratik Rath
![Page 115: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/115.jpg)
CA(§) =IWDW
¼ ¹h
Holographic Complexity:
Complexity = Volume Complexity = Action
CV(§) = max§=@B
·V(B)GN `
¸
Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao
Dean Carmi, RCM & Pratik Rath
![Page 116: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/116.jpg)
• UV divergences naturally associated with establishing correlations or entanglement down to arbitrarily small length scales
Holographic Complexity:
Complexity = Volume Complexity = Action
Dean Carmi, RCM & Pratik Rath
![Page 117: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/117.jpg)
• UV divergences naturally associated with establishing correlations or entanglement down to arbitrarily small length scales
Holographic Complexity:
Complexity = Volume Complexity = Action
• regulate volume/action with the introduction of UV regulator surface at large radius (𝑟𝑚𝑎𝑥 = 𝐿2/𝛿), as usual
Dean Carmi, RCM & Pratik Rath
![Page 118: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/118.jpg)
CV (§) =Ld¡1
(d¡ 1)GN
Z
§
dd¡1¾ph
·1
±d¡1
Holographic Complexity:
• UV divergences naturally associated with establishing correlations down to arbitrarily small length scales
• regulate volume/action with the introduction of UV regulator surface at large radius (𝑟𝑚𝑎𝑥 = 𝐿2/𝛿), as usual
¡ (d¡ 1)
2(d¡ 2)(d¡ 3)±d¡3
µRaa ¡
1
2R¡ (d¡ 2)2
(d¡ 1)2K2
¶+ ¢ ¢ ¢
¸
• UV divergences appear as local integrals of geometric invariants (as with holographic entanglement entropy)
![Page 119: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/119.jpg)
CV (§) =Ld¡1
(d¡ 1)GN
Z
§
dd¡1¾ph
·1
±d¡1
Holographic Complexity:
• UV divergences naturally associated with establishing correlations down to arbitrarily small length scales
• regulate volume/action with the introduction of UV regulator surface at large radius (𝑟𝑚𝑎𝑥 = 𝐿2/𝛿), as usual
¡ (d¡ 1)
2(d¡ 2)(d¡ 3)±d¡3
µRaa ¡
1
2R¡ (d¡ 2)2
(d¡ 1)2K2
¶+ ¢ ¢ ¢
¸CV (§) =
8¼d+22 ¡(d=2)
¡(d+ 2)CT
Z
§
dd¡1¾ph
·1
±d¡1
• UV divergences appear as local integrals of geometric invariants (as with holographic entanglement entropy)
![Page 120: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/120.jpg)
CA(§) =1
±d¡1
Z
§
dd¡1¾ph
·v1(R;K) + log
µL
®±
¶v2(R;K)
¸
CV (§) =1
±d¡1
Z
§
dd¡1¾ph v0(R;K)
Holographic Complexity:
• UV divergences naturally associated with establishing correlations down to arbitrarily small length scales
• regulate volume/action with the introduction of UV regulator surface at large radius (𝑟𝑚𝑎𝑥 = 𝐿2/𝛿), as usual
with vk(R;K) =
bd¡12cX
n=0
X
i
c[k]
i;n(d) ±2n [R;K]2ni
• UV divergences appear as local integrals of geometric invariants (as with holographic entanglement entropy)
![Page 121: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/121.jpg)
CA(§) =1
±d¡1
Z
§
dd¡1¾ph
·v1(R;K) + log
µL
®±
¶v2(R;K)
¸
CV (§) =1
±d¡1
Z
§
dd¡1¾ph v0(R;K)
Holographic Complexity:
• UV divergences naturally associated with establishing correlations down to arbitrarily small length scales
• regulate volume/action with the introduction of UV regulator surface at large radius (𝑟𝑚𝑎𝑥 = 𝐿2/𝛿), as usual
with vk(R;K) =
bd¡12cX
n=0
X
i
c[k]
i;n(d) ±2n [R;K]2ni
• UV divergences appear as local integrals of geometric invariants (as with holographic entanglement entropy)
AdS scale
normalization
from asymptotic joint
![Page 122: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/122.jpg)
Holographic Complexity: • UV divergences appear as local integrals of geometric invariants
C(§) ' c0 V(§)=±d¡1 + ¢ ¢ ¢
![Page 123: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/123.jpg)
Holographic Complexity: • UV divergences appear as local integrals of geometric invariants
C(§) ' c0 V(§)=±d¡1 + ¢ ¢ ¢
¢C ' 2c0 (p¢`2 ¡¢t2 ¡¢`)Vtrans=±d¡1 + ¢ ¢ ¢ < 0
![Page 124: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/124.jpg)
Holographic Complexity: • UV divergences appear as local integrals of geometric invariants
C(§) ' c0 V(§)=±d¡1 + ¢ ¢ ¢
¢C ' 2c0 (p¢`2 ¡¢t2 ¡¢`)Vtrans=±d¡1 + ¢ ¢ ¢ < 0
C(§3) ' c1 Vtrans=±d¡2 + ¢ ¢ ¢
![Page 125: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/125.jpg)
Questions? • What is “holographic complexity”?
QFT/path integral description of “complexity” in boundary CFT? what is boundary dual of these gravitational observables?
• ambiguities? ambiguities? ambiguities?
• is there a privileged role for (states on) null Cauchy surfaces? provide distinguished reference states?
• is there a “renormalized holographic complexity”?
• why is complexity of formation positive?
what’s it good for?; (EE vs mutual information versions of F)
• subregion complexity?
connections between ambiguities in gravity and boundary?
• more boundary terms: higher codim. intersections; “complex” joint contributions; boundary “counterterms”
• contribution of spacetime singularity? CA
![Page 126: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/126.jpg)
“It from Qubit”: New Collision of Ideas
Gravity
Quantum
“Holography”
Information
Theory
Quantum
![Page 127: Robert Myers - [ SBF ] - Sociedade Brasileira de Físicaevjaspc/xix/images/myers-lecture5...• different subsystems are correlated through global state of full system Quantum Entanglement](https://reader031.fdocuments.net/reader031/viewer/2022030505/5ab301f27f8b9a284c8df8b8/html5/thumbnails/127.jpg)
“It from Qubit”: New Collision of Ideas
Gravity
Quantum
“Holography”
Information
Theory
Quantum
http://www.perimeterinstitute.ca/it-qubit-summer-school /it-qubit-summer-school-resources