Robert Myers Lecture 5

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAA

“It from Qubit”: New Collision of Ideas

“It from Qubit”: New Collision of Ideas

Gravity

Quantum

“Holography”

Information

Theory

Quantum

“It from Qubit”: New Collision of Ideas

Gravity

Quantum

Confluence

“Holography”

Information

Theory

Quantum

“It from Qubit”: New Collision of Ideas

Gravity

Quantum

“Holography”

Information

Theory

Quantum

General Relativity is the geometric arena for physics

on very large scales: planets, stars, galaxies, cosmology

Einstein:

Gravity? It’s all about geometry!

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

General Relativity + Quantum Theory = Quantum Gravity?

+ ℏ

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

• 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?

• 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?

“It from Qubit”: New Collision of Ideas

Gravity

Quantum

“Holography”

Information

Theory

Quantum

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

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 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

• 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

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)

Holographic Entanglement Entropy: (Ryu & Takayanagi `06)

A

AdS boundary

AdS bulk spacetime

boundary conformal field theory

A –

Bekenstein- Hawking formula

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

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

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

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, ……)

●

(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

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, ….)

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

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

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

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

• 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)

• 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)

• 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)

• 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)

• 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)

• 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)

• 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)

“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

• 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)

• 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)

• 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 –

• 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:

• 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

• 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

• 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

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

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!

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

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

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

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

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

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

Susskind: Entanglement is not enough! • “to understand the rich geometric structures that exist behind

the horizon and which are predicted by general relativity.”

V

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’

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’

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

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

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

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!

Complexity?

• recall 𝑆𝐸𝐸 only probes the eigenvalues of the density matrix

SEE = ¡Tr [½A log ½A] =¡X

¸i log¸i

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

Complexity:

• computational complexity: how difficult is it to implement a task? eg, how difficult is it to prepare a particular state?

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

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

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

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

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:

• 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

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

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? ???

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!

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

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

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)

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:

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:

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

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

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

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

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

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

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

+ ??????

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

+ ??????

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

Complexity of Formation: jTFDi = Z¡1=2

X

®

e¡E®=(2T )jE®iLjE®iR

Shira Chapman, Hugo Marrochio & RCM

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

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

• 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)

• 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)

• 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)

• 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)

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)

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

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)

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)

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)

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

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

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

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

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

• 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

• 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

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)

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)

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)

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

Holographic Complexity: • UV divergences appear as local integrals of geometric invariants

C(§) ' c0 V(§)=±d¡1 + ¢ ¢ ¢

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

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 + ¢ ¢ ¢

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

“It from Qubit”: New Collision of Ideas

Gravity

Quantum

“Holography”

Information

Theory

Quantum

“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