Quantum Entanglement

Robert Myers Lecture 5

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AA AAA A A A A A “It from Qubit”: New Collision of Ideas “It from Qubit”: New Collision of Ideas

“Holography” Quantum Gravity

Quantum Information Theory Confluence “It from Qubit”: New Collision of Ideas

“Holography” Quantum Gravity

Quantum Information Theory “It from Qubit”: New Collision of Ideas

“Holography” Quantum Gravity

Quantum Information Theory 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!

SpacetimeGeneral moves Relativity from issimply the geometric stage for physicalarena for phenomena, physics to beingon very both large the scales: stage planets,and an active stars, playergalaxies, in thecosmology dynamics

General Relativity + Quantum Theory = Quantum Gravity?

+ ℏ General Relativity + Quantum Theory = Quantum Gravity?

+ ℏ

• quantum fluctuations become manifest at small scales e.g., magnetic moment of the electron, ¹ e = g e ¹h = 4 m e , with 푔 ≈ 2 but modified by quantum fluctuations

gtheory = 2:0023193043070    theory experiment  1010 experiment 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? 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 “It from Qubit”: New Collision of Ideas

“Holography” Quantum Gravity

Quantum Information Theory Quantum Entanglement • different subsystems are correlated through global state of full system Einstein-Podolsky-Rosen Paradox: • polarizations of pair of photons connected, no matter how far apart they travel “spukhafte Fernwirkung” = spooky action at a distance Quantum Entanglement • different subsystems are correlated through global state of full system Einstein-Podolsky-Rosen Paradox: • polarizations of pair of photons connected, no matter how far apart they travel “spukhafte Fernwirkung” = spooky action at a distance

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, . . . . Quantum Entanglement • different subsystems are correlated through global state of full system Einstein-Podolsky-Rosen Paradox: • polarizations of pair of photons connected, no matter how far apart they travel “spukhafte Fernwirkung” = spooky action at a distance

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, . . . . Quantum Fields & Quantum Gravity Entanglement Entropy in QFT • general diagnostic to give a quantitative measure of entanglement using entropy to detect correlations between two subsystems • 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 calculate von Neumann entropy:

– A

A Holography: AdS/CFT correspondence Boundary: quantum field theory Bulk: gravity with negative Λ without intrinsic scales in d+1 dimensions in d dimensions “holography” anti-de Sitter conformal space field theory

radius energy (Maldacena ‘97) (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein- AdS bulk Hawking spacetime formula (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein- AdS bulk Hawking spacetime formula

• conjecture many detailed consistency tests (Ryu, Takayanagi, Hubeny, Rangamani, Headrick, Hung, Smolkin, RM, Faulkner, . . . ) (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein- AdS bulk Hawking spacetime formula

• conjecture many detailed consistency tests (Ryu, Takayanagi, Hubeny, Rangamani, Headrick, Hung, Smolkin, RM, Faulkner, . . . ) • 2013 proof (for static geometries) (Maldacena & Lewkowycz) • 2016 proof (for general geometries) (Dong, Lewkowycz & Rangamani) (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein- AdS bulk Hawking spacetime formula

• conjecture many detailed consistency tests (Ryu, Takayanagi, Hubeny, Rangamani, Headrick, Hung, Smolkin, RM, Faulkner, . . . ) • 2013 proof (for static geometries) (Maldacena & Lewkowycz) • 2016 proof (for general geometries) (Dong, Lewkowycz & Rangamani) • holographic EE: fruitful forum for bulk-boundary dialogue (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein- AdS bulk Hawking spacetime formula

• holographic EE teaches us lessons about QFTs, eg, diagnostic in RG flows and c-theorms, eg, F-theorem (Sinha & RM, ……) gravity/holography + EE RG flows in (2+1)-dimensions

B C(R) = R@RS(R) S(R) ¡ A R

● (F)UV ●

●

F-theorem: (F) (F) UV ¸ IR (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein- AdS bulk Hawking spacetime formula

• holographic EE teaches us lessons about QFTs, eg, diagnostic in RG flows and c-theorms, eg, F-theorem (Sinha & RM, ……) geometric properties of entanglement entropy in QFT’s (Mezei, Perlmutter, Lewkowycz, Bueno, RM, Witczak-Krempa, ….) diagnostic for quantum quenches/phase transitions (Lopez, Johnson, Balasubramanian, Bernamonti, Craps, Galli, ….) (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein- AdS bulk Hawking spacetime formula

• holographic EE teaches us lessons about (quantum) gravity, eg, BH formula applies beyond black holes/horizons connectivity of spacetime requires entanglement (van Raamsdonk) (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein- AdS bulk Hawking spacetime formula

• holographic EE teaches us lessons about (quantum) gravity, eg, BH formula applies beyond black holes/horizons connectivity of spacetime requires entanglement (van Raamsdonk) Spacetime Geometry = Entanglement Spacetime Geometry = Entanglement

Bekenstein-Hawking formula: spacetime geometry encodes SBH black hole entropy is entanglement entropy (Sorkin, ….) use BH formula for holographic entanglement entropy (Ryu & Takayanagi; ….) connectivity of spacetime requires entanglement (van Raamsdonk)

spacetime entanglement conjecture (Bianchi & RM)

AdS spacetime as a tensor network (MERA) (Swingle, Vidal, ….)

“ER = EPR” conjecture (Maldacena & Susskind)

hole-ographic spacetime (Balasubramanian, Chowdhury, Czech, de Boer & Heller; RM, Rao & Sugishita; Czech, Dong & Sully; ….)

B Σ A Spacetime Geometry = Entanglement

spacetime entanglement conjecture (Bianchi & RM)

AdS spacetime as a tensor network (MERA) (Swingle, Vidal, ….)

“ER = EPR” conjecture (Maldacena & Susskind)

hole-ographic spacetime (Balasubramanian, Chowdhury, Czech, de Boer & Heller; RM, Rao & Sugishita; Czech, Dong & Sully; ….)

B spacetime provides bothΣ the stageA for physical phenomena and the agent which manifests gravitational dynamics Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)

• entanglement entropy: S(½A) = tr(½A log ½A) ¡ • make a small perturbation of state: ½~ = ½A + ±½ Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)

• entanglement entropy: S(½A) = tr(½A log ½A) ¡ • make a small perturbation of state: ½~ = ½A + ±½

• modular (or entanglement) Hamiltonian: ½A = exp( HA) ¡ Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)

• entanglement entropy: S(½A) = tr(½A log ½A) ¡ • make a small perturbation of state: ½~ = ½A + ±½

• modular (or entanglement) Hamiltonian: ½A = exp( HA) (Blanco, Casini¡ , Hung & RM)

±SA = ± HA h i “1st law” of entanglement entropy Gravitational Dynamics from Entanglement: (Lashkari, McDermott & Van Raamsdonk; Swingle & Van Raamsdonk; Faulkner, Guica, Hartman, RM & Van Raamsdonk)

• entanglement entropy: S(½A) = tr(½A log ½A) ¡ • make a small perturbation of state: ½~ = ½A + ±½

• modular (or entanglement) Hamiltonian: ½A = exp( HA) (Blanco, Casini¡ , Hung & RM)

±SA = ± HA h i “1st law” of entanglement entropy

st • this is the 1 law for thermal state: ½A = exp( H=T) ¡ “1st law” of entanglement entropy:

• generally is “nonlocal mess” and flow is not geometric

d 1 ¹º d 1 d 1 ¹º;½¾ HA = d ¡ x ° (x) T¹º + d ¡ x d ¡ y ° (x; y) T¹ºT½¾ + 1 2 ¢ ¢ ¢ Z Z Z hence usefulness of first law is very limited, in general “1st law” of entanglement entropy:

• generally is “nonlocal mess” and flow is not geometric

d 1 ¹º d 1 d 1 ¹º;½¾ HA = d ¡ x ° (x) T¹º + d ¡ x d ¡ y ° (x; y) T¹ºT½¾ + 1 2 ¢ ¢ ¢ Z Z Z hence usefulness of first law is very limited, in general

• famous exception: Rindler wedge • any QFT in Minkowski vacuum; choose Σ = (x = 0, t = 0)

HA = 2¼K boost generator d 2 Σ = 2¼ d ¡ y dx [ x Ttt ] + c0 ● B ZA(x>0) A • by causality, and H A describe physics throughout domain of dependence ; eg, generate boost flows (Bisognano & Wichmann; Unruh) “1st law” of entanglement entropy:

• generally is “nonlocal mess” and flow is not geometric

d 1 ¹º d 1 d 1 ¹º;½¾ HA = d ¡ x ° (x) T¹º + d ¡ x d ¡ y ° (x; y) T¹ºT½¾ + 1 2 ¢ ¢ ¢ Z Z Z hence usefulness of first law is very limited, in general

• famous exception: Rindler wedge • any QFT in Minkowski vacuum; choose Σ = (x = 0, t = 0)

• another exception: CFT in vacuum of d-dim. flat space and entangling surface which is Sd-2 with radius R 2 2 d 1 R ~y H = 2¼ d ¡ y ¡ j j T (~y) + c0 A 2R tt ZA (Casini, Huerta & RM; Hislop & Longo)

– A A “1st law” of entanglement entropy:

– A A

• construct with conformal transformation: ¹ º K = 2¼ d§ T¹º K ZA “1st law” of entanglement entropy:

• HA has simply form for CFT and spherical entangling surface: 2 2 d 1 R ~y H = 2¼ d ¡ y ¡ j j T (~y) + c0 A 2R tt ZA • holographic realization: “1st law” of entanglement entropy:

• HA has simply form for CFT and spherical entangling surface: 2 2 d 1 R ~y H = 2¼ d ¡ y ¡ j j T (~y) + c0 A 2R tt ZA • holographic realization:

• apply 1st law for spheres of all sizes, positions and in all frames:

st bulk geometry satisfies 1 law of SEE linearized Einstein eq’s “1st law” of entanglement entropy:

• HA has simply form for CFT and spherical entangling surface: 2 2 d 1 R ~y H = 2¼ d ¡ y ¡ j j T (~y) + c0 A 2R tt ZA • holographic realization:

spacetime provides both the stage for physical phenomena and the agent which manifests gravitational dynamics

• apply 1st law for spheres of all sizes, positions and in all frames:

st bulk geometry satisfies 1 law of SEE linearized Einstein eq’s (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein- AdS bulk Hawking spacetime formula

• holographic EE teaches us lessons about (quantum) gravity, eg, BH formula applies beyond black holes/horizons connectivity of spacetime requires entanglement (van Raamsdonk) Spacetime Geometry = Entanglement (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein- AdS bulk Hawking Susskindspacetime: Entanglement is not enough!formula

• holographic EE teaches us lessons about (quantum) gravity, eg, BH formula applies beyond black holes/horizons connectivity of spacetime requires entanglement (van Raamsdonk) Spacetime Geometry = Entanglement (Ryu & Takayanagi `06) Holographic Entanglement Entropy:

AdS boundary – A A boundary conformal field theory

Bekenstein-

AdS bulk V Hawking Susskindspacetime: Entanglement is not enough!formula

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

r2 ¹ dr2 ds2 = + 1 dt2 + + r2d•2 2 d 2 r2 ¹ d 1 ¡ L ¡ r ¡ 2 + 1 d 2 ¡ µ ¶ L ¡ r ¡

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

singularity

asymptotic asymptotic AdS bndry horizons AdS bndry

singularity r2 ¹ dr2 ds2 = + 1 dt2 + + r2d•2 2 d 2 r2 ¹ d 1 ¡ L ¡ r ¡ 2 + 1 d 2 ¡ µ ¶ L ¡ r ¡

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

E®=(2T ) TFD e¡ E® L E® R j i ' ® j i j i X • pure state: 푆퐸퐸 = 0

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

E®=(2T ) TFD e¡ E® L E® R j i ' ® j i j i X • pure state: 푆퐸퐸 = 0

½R = TrL TFD TFD j ih j E®=T e¡ E® R E® R ' ® j i h j X 퐴퐻 • mixed state: 푆퐸퐸 = 4퐺

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

E®=(2T ) TFD e¡ E® L E® R j i ' ® j i j i X • pure state: 푆퐸퐸 = 0

½R = TrL TFD TFD j ih j E®=T e¡ E® R E® R ' ® j i h j X 퐴퐻 • mixed state: 푆퐸퐸 = 4퐺

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

A’ A

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

A’ A

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

A’ A

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

A’ A

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

A’ A

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

A’ A

Holographic Complexity! Complexity?

• recall 푆퐸퐸 only probes the eigenvalues of the density matrix

SEE = Tr [½A log ½A] = ¸i log ¸i ¡ ¡ X Complexity?

• recall 푆퐸퐸 only probes the eigenvalues of the density matrix

SEE = Tr [½A log ½A] = ¸i log ¸i ¡ ¡ • would like a new probe “sensitive to phases”X

E®=(2T ) iE®(tL+tR) TFD e¡ ¡ E® L E® R j i ' ® j i j i X 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:

Ã = U Ã0 j i j i Complexity:

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

• quantum circuit model:

Ã = U Ã0 j i j i simple reference state eg, 00000 0 j ¢ ¢ ¢ i Complexity:

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

• quantum circuit model:

Ã = U Ã0 j i j i unitary operator simple reference state built from set of eg, 00000 0 simple gates j ¢ ¢ ¢ i Complexity:

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

• quantum circuit model:

Ã = U Ã0 j i j i unitary operator simple reference state built from set of eg, 00000 0 simple gates j ¢ ¢ ¢ i Complexity:

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

• quantum circuit model:

Ã = U Ã0 j i j i unitary operator simple reference state built from set of eg, 00000 0 simple gates j ¢ ¢ ¢ i 2 tolerance: Ã Ã Target · " j j i ¡ j i j Complexity:

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

• quantum circuit model:

Ã = U Ã0 j i j i unitary operator simple reference state built from set of eg, 00000 0 simple gates j ¢ ¢ ¢ i 2 tolerance: Ã Ã Target · " j j i ¡ j i j • 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:

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

• quantum circuit model:

Susskind: Entanglement isV not enough! • “to understand the rich geometric structures that exist behind the horizon and which are predicted by general relativity.”

Holographic Complexity! A Tale of Two Dualities: Holographic Complexity

Complexity = Volume Complexity = Action

( ) IWDW V(§) = max V B A(§) = C §=@ GN ` ¼ ¹h B · ¸ C

Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao A Tale of Two Dualities: Holographic Complexity

Complexity = Volume Complexity = Action

( ) IWDW V(§) = max V B A(§) = C §=@ GN ` ¼ ¹h B · ¸ C

Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao A Tale of Two Dualities: Holographic Complexity

Complexity = Volume Complexity = Action

d V 8¼ IWDW C = M (planar) A(§) = dt t d 1 C ¼ ¹h ¯ !1 ¡ ¯ (d = boundary dimension) (universal; Lloyd bound) Team¯ Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action:

Complexity = Action

null boundaries null boundaries

Wheeler-DeWitt patch: domain of dependence of Cauchy surface ending on boundary time slice Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action:

Complexity = Action

null boundaries null boundaries

evaluate the action? what action?? Wheeler-DeWitt patch: domain of dependence of Cauchy surface ending on boundary time slice Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

1 d+1 d(d 1) I = d xp g R + ¡2 16¼GN ¡ L ZM µ ¶ 1 d 1 d 1 + d x h K + d ¡ xp¾ ´ 8¼GN j j 8¼GN § ZB p Z 1 d 1 1 d 1 d¸ d ¡ µp° · + d ¡ xp¾ a ¡ 8¼GN 8¼GN § ZB0 Z 0 Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

1 d+1 d(d 1) I = d xp g R + ¡2 16¼GN ¡ L ZM µ ¶ 1 d 1 d 1 + d x h K + d ¡ xp¾ ´ 8¼GN j j 8¼GN § ZB p Z 1 d 1 1 d 1 d¸ d ¡ µp° · + d ¡ xp¾ a ¡ 8¼GN 8¼GN § ZB0 Z 0 • well-defined action principle requires boundary terms

M Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

1 d+1 d(d 1) I = d xp g R + ¡2 16¼GN ¡ L ZM µ ¶ 1 d 1 d 1 + d x h K Gibbons+ -Hawkingd -¡Yorkxp ¾ ´ 8¼GN j j 8¼GN § ZB p Z 1 d 1 1 d 1 d¸ d ¡ µp° · + d ¡ xp¾ a e¡g, 8¼GN 8¼GN § ZB0 Z 0 • well-defined action principle requires boundary terms

ab K = h anb M r Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

1 d+1 d(d 1) I = d xp g R + ¡2 16¼GN ¡ L ZM µ ¶ Hayward 1 d 1 d 1 + d x h K + d ¡ xp¾ ´ 8¼GN j j 8¼GN § ZB p Z 1 d 1 1 d 1 d¸ d ¡ µp° · + d ¡ xp¾ a ¡ 8¼GN 8¼GN § ZB0 Z 0 • well-defined action principle requires boundary terms K • •´ ab K = h anb M r Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

1 d+1 d(d 1) I = d xp g R + ¡2 16¼GN ¡ L ZM µ ¶ 1 d 1 d 1 + d x h K + d ¡ xp¾ ´ 8¼GN j j 8¼GN § ZB p Z 1 d 1 1 d 1 d¸ d ¡ µp° · + d ¡ xp¾ a e¡g, 8¼GN 8¼GN § ZB0 Z 0 • well-defined action principle requires boundary terms K • • •´ ab K = h anb r M • · •a Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

1 d+1 d(d 1) I = d xp g R + ¡2 16¼GN ¡ L ZM µ ¶ 1 d 1 d 1 + d x h K + d ¡ xp¾ ´ 8¼GN j j 8¼GN § ZB p Z 1 d 1 1 d 1 d¸ d ¡ µp° · + d ¡ xp¾ a ¡ 8¼GN 8¼GN § ZB0 Z 0 + ?????? • ambiguities: total derivatives, extra boundary terms, . . . . Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

ambiguities in circuit complexity? Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

1 d+1 d(d 1) I = d xp g R + ¡2 16¼GN ¡ L ZM µ ¶ 1 d 1 d 1 + d x h K + d ¡ xp¾ ´ 8¼GN j j 8¼GN § ZB p Z 1 d 1 1 d 1 d¸ d ¡ µp° · + d ¡ xp¾ a ¡ 8¼GN 8¼GN § ZB0 Z 0 • ambiguities: 휈 휇 휇 1) 푘 훻휈 푘 = 휿 푘 Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

1 d+1 d(d 1) I = d xp g R + ¡2 16¼GN ¡ L ZM µ ¶ 1 d 1 d 1 + d x h K + d ¡ xp¾ ´ 8¼GN j j 8¼GN § ZB p Z 1 d 1 1 d 1 d¸ d ¡ µp° · + d ¡ xp¾ a ¡ 8¼GN 8¼GN § ZB0 Z 0 • ambiguities: 휈 휇 휇 1) 푘 훻휈 푘 = 휿 푘 affinely parameterize: 휿 = 0 Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

1 d+1 d(d 1) I = d xp g R + ¡2 16¼GN ¡ L ZM µ ¶ 1 d 1 d 1 + d x h K + d ¡ xp¾ ´ 8¼GN j j 8¼GN § ZB p Z 1 d 1 1 d 1 d¸ d ¡ µp° · + d ¡ xp¾ a ¡ 8¼GN 8¼GN § ZB0 Z 0 • ambiguities: 휈 휇 휇 1) 푘 훻휈 푘 = 휿 푘 affinely parameterize: 휿 = 0

Aside: make choices without referring to particular metric/coordinates, which allow for meaningful comparison of different states/geometries Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

log |풌1 ∙ 풏2| 2) a = 퐚ퟎ + 휀 log |풌1 ∙ 풌2/2| Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

log |풌1 ∙ 풏2| addivity of grav. action: 2) a = 퐚ퟎ + 휀 log |풌1 ∙ 풌2/2| 풂ퟎ = 0

= Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

log |풌1 ∙ 풏2| addivity of grav. action: 2) a = 퐚ퟎ + 휀 log |풌1 ∙ 풌2/2| 풂ퟎ = 0

3) constant rescaling: 푘휇 → 휶 푘휇 Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

log |풌1 ∙ 풏2| addivity of grav. action: 2) a = 퐚ퟎ + 휀 log |풌1 ∙ 풌2/2| 풂ퟎ = 0 fix normalization 3) constant rescaling: 푘휇 → 휶 푘휇 asymptotically: 풌 ∙ 풕 = ±1 Luis Lehner, RCM, Eric Poisson & Rafael Sorkin Complexity = Action: • gravitational action:

1 d+1 d(d 1) I = d xp g R + ¡2 16¼GN ¡ L ZM µ ¶ 1 d Ad 2M 1 d 1 + d x h K + d ¡ xp¾ ´ 8¼GN C j j = 8¼GN § dZBt tp ¼ Z 1 !d 11 1 d 1 d¯¸ d ¡ µp° · + d ¡ xp¾ a ¡ 8¼GN 8¼GN 0 ¯ §0 ZB ¯ Z • ambiguities: 휈 휇 휇 1) 푘 훻휈 푘 = 휿 푘 affinely parameterize: 휿 = 0

log |풌1 ∙ 풏2| addivity of grav. action: Aside2) : amake= 퐚ퟎ choices+ 휀 without referring to particular metric/coordinates, log |풌1 ∙ 풌2/2| 풂 = 0 which allow for meaningful comparison of different states/geometriesퟎ fix normalization 3) constant rescaling: 푘휇 → 휶 푘휇 asymptotically: 풌 ∙ 풕 = ±1 Holographic Complexity:

Complexity = Volume Complexity = Action

d V 8¼ d A 2M C = M (planar) C = dt t d 1 dt t ¼ ¯ !1 ¡ ¯ !1 ¯ ¯ ¯Team Lenny, including Brown, Roberts, Swingle, ¯Stanford, Susskind & Zhao Complexity of Formation: Shira Chapman, Hugo Marrochio & RCM

1=2 E®=(2T ) TFD = Z¡ e¡ E® L E® R j i ® j i j i X Complexity of Formation: Shira Chapman, Hugo Marrochio & RCM

1=2 E®=(2T ) TFD = Z¡ e¡ E® L E® R j i ® j i j i • additional complexity involvedX in forming thermofield double state compared to preparing two copies of vacuum state? ¢ = ( TFD ) ( 0 0 ) C C j i ¡ C j i • j i Complexity of Formation: Shira Chapman, Hugo Marrochio & RCM

¢ A = 2 C ¡ £ Complexity of Formation: Shira Chapman, Hugo Marrochio & RCM • additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state? ¢ = ( TFD ) ( 0 0 ) C C j i ¡ C j i • j i d 2 ¼ ¢ A = ¡ cot S + C d ¼ d ¢ ¢ ¢ µ ¶ curvature corrections thermal/ent. entropy

(d = boundary dimension) Complexity of Formation: Shira Chapman, Hugo Marrochio & RCM • additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state? ¢ = ( TFD ) ( 0 0 ) C C j i ¡ C j i • j i d 2 ¼ ¢ A = ¡ cot S + C d ¼ d ¢ ¢ ¢ µ ¶ curvature corrections thermal/ent. entropy

d 2 ¼ ¡ cot • d ¼ d µ ¶ d 2 ¡ ¼2 (d = boundary dimension) d Complexity of Formation: Shira Chapman, Hugo Marrochio & RCM • additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state? ¢ = ( TFD ) ( 0 0 ) C C j i ¡ C j i • j i d 2 ¼ ¢ A = ¡ cot S + C d ¼ d ¢ ¢ ¢ µ ¶ curvature corrections thermal/ent. entropy

1 (d 2) ¡(1 + d ) ¢ V = 4p¼ ¡ S + C (d 1) ¡( 1 + 1 ) ¢ ¢ ¢ ¡ 2 d

(d = boundary dimension) Holographic Complexity:

Complexity = Volume Complexity = Action

d V 8¼ d A 2M C = M (planar) C = dt t d 1 dt t ¼ ¯ !1 ¡ ¯ !1 ¯ ¯ ¯Team Lenny, including Brown, Roberts, Swingle, ¯Stanford, Susskind & Zhao Compare? ¢ d 1 ¡ 1 1 d R = A = d form C ¡3=2 1 ¡ 1 2 ¢ V 4¼ ¡ ' 4¼ C ¡2 ¡ d¢ d A=dt d 1 ¡ ¢ d Rrate = C = ¡2 2 d V =dt 4¼ ' 4¼ C log 2 Rrate Rform = + (1=d) ¡ 2¼2 O Compare? ¢ d 1 ¡ 1 1 d R = A = d form C ¡3=2 1 ¡ 1 2 ¢ V 4¼ ¡ ' 4¼ C ¡2 ¡ d¢ d A=dt d 1 ¡ ¢ d Rrate = C = ¡2 2 d V =dt 4¼ ' 4¼ C log 2 Rrate Rform = + (1=d) ¡ 2¼2 O

Rrate Rform

(d = boundary dimension) d Compare? ¢ d 1 ¡ 1 1 d R = A = d form C ¡3=2 1 ¡ 1 2 ¢ V 4¼ ¡ ' 4¼ C ¡2 ¡ d¢ d A=dt d 1 ¡ ¢ d Rrate = C = ¡2 2 d V =dt 4¼ ' 4¼ C log 2 Rrate Rform = + (1=d) ¡ 2¼2 O points to consistency of C=V and C=A dualities up to differences in microscopic rules, eg, gate set Rrate Rform

(d = boundary dimension) d Complexity of Formation for d=2: • additional complexity involved in forming thermofield double state compared to preparing two copies of vacuum state? ¢ = ( TFD ) ( 0 0 ) C C j i ¡ C j i • j i d 2 ¼ ¢ A = ¡ cot S + C d ¼ d ¢ ¢ ¢ lµeading¶ term vanishes 1 (d 2) ¡(1 + d ) ¢ V = 4p¼ ¡ S + 0! C (d 1) ¡( 1 + 1 ) ¢ ¢ ¢ ' ¡ 2 d d = 2

• actually holographic calculations apply for 푑 ≥ 3 (but still correct)

(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? ¢ = ( TFD ) ( 0 0 ) C C j i ¡ C j i • j i • reconsider holographic calculations for 3D BTZ black hole: c 8¼ ¢ A = ¢ V = + c C ¡3 C 3

풄 = central charge of boundary CFT

• perhaps related to BTZ black hole being locally AdS geometry

= 0 (R vac) 풄 = central charge of boundary CFT

• perhaps related to BTZ black hole being locally AdS geometry

(d = boundary dimension) Complexity of Formation from MERA?

d 1 c g(TFD) CT T ¡ C ¡ ' V = kT S

d 1 c g(vac) CT T ¡ C ¡ ' V = k0 S

? ? ¢ = (TFD) 2 (vac) = (kT 2 k0) S 0 C C ¡ C ¡ ?¸ ? Dean Carmi, RCM & Pratik Rath Holographic Complexity:

Complexity = Volume Complexity = Action

( ) IWDW V(§) = max V B A(§) = C §=@ GN ` ¼ ¹h B · ¸ C

Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao Dean Carmi, RCM & Pratik Rath Holographic Complexity:

Complexity = Volume Complexity = Action

( ) IWDW V(§) = max V B A(§) = C §=@ GN ` ¼ ¹h B · ¸ C

Team Lenny, including Brown, Roberts, Swingle, Stanford, Susskind & Zhao Dean Carmi, RCM & Pratik Rath Holographic Complexity:

Complexity = Volume Complexity = Action

• UV divergences naturally associated with establishing correlations or entanglement down to arbitrarily small length scales Dean Carmi, RCM & Pratik Rath Holographic Complexity:

Complexity = Volume Complexity = Action

• UV divergences naturally associated with establishing correlations or entanglement down to arbitrarily small length scales • regulate volume/action with the introduction of UV regulator surface 2 at large radius (푟푚푎푥 = 퐿 /훿), as usual 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 2 at large radius (푟푚푎푥 = 퐿 /훿), as usual

Ld 1 1 ¡ d 1 p V (§) = d ¡ ¾ h d 1 C (d 1)GN ± ¡ Z§ · ¡ (d 1) 1 (d 2)2 ¡ a ¡ K2 + ¡2(d 2)(d 3)±d 3 Ra ¡ 2 R ¡ (d 1)2 ¢ ¢ ¢ ¡ ¡ ¡ µ ¡ ¶ ¸

• UV divergences appear as local integrals of geometric invariants (as with holographic entanglement entropy) 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 2 at large radius (푟푚푎푥 = 퐿 /훿), as usual

d+d2 1 8¼L2 ¡(d=2) 1 1 (§) = ¡ dd 1¾pdh1 p V (§) = CT ¡ d ¡ ¾ d h1 d 1 C (d¡(d1)+G2N) ± ± ¡ Z§ Z§ · ¡ · ¡ (d 1) 1 (d 2)2 ¡ a ¡ K2 + ¡2(d 2)(d 3)±d 3 Ra ¡ 2 R ¡ (d 1)2 ¢ ¢ ¢ ¡ ¡ ¡ µ ¡ ¶ ¸

1 d 1 V (§) = d ¡ ¾ ph v0( ; K) C ±d 1 R ¡ Z§

1 d 1 L A(§) = d ¡ ¾ ph v1( ; K) + log v2( ; K) C ±d 1 R ® ± R ¡ Z§ · µ ¶ ¸ d 1 ¡ b 2 c [k] 2n 2n with vk( ; K) = c (d) ± [ ; K] R i;n R i n=0 i X X • UV divergences appear as local integrals of geometric invariants (as with holographic entanglement entropy) 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 2 at large radius (푟푚푎푥 = 퐿 /훿), as usual 1 d 1 p V (§) = d 1 d ¡ ¾ h v0( ; K) AdS scale C ± ¡ § R Z normalization 1 d 1 L A(§) = d ¡ ¾ ph v1( ; K) + log v2( ; K) C ±d 1 R ® ± R ¡ Z§ · µ ¶ ¸ d 1 from asymptotic joint ¡ b 2 c [k] 2n 2n with vk( ; K) = c (d) ± [ ; K] R i;n R i n=0 i X X • UV divergences appear as local integrals of geometric invariants (as with holographic entanglement entropy) Holographic Complexity: • UV divergences appear as local integrals of geometric invariants d 1 (§) c0 (§)=± ¡ + C ' V ¢ ¢ ¢ Holographic Complexity: • UV divergences appear as local integrals of geometric invariants d 1 (§) c0 (§)=± ¡ + C ' V ¢ ¢ ¢

2 2 d 1 ¢ 2c0 ( ¢` ¢t ¢`) trans=± ¡ + < 0 C ' ¡ ¡ V ¢ ¢ ¢ p Holographic Complexity: • UV divergences appear as local integrals of geometric invariants d 1 (§) c0 (§)=± ¡ + C ' V ¢ ¢ ¢

2 2 d 1 ¢ 2c0 ( ¢` ¢t ¢`) trans=± ¡ + < 0 C ' ¡ ¡ V ¢ ¢ ¢ p d 2 (§3) c1 trans=± ¡ + C ' V ¢ ¢ ¢ Questions? • What is “holographic complexity”?  QFT/path integral description of “complexity” in boundary CFT?  what is boundary dual of these gravitational observables? • is there a privileged role for (states on) null Cauchy surfaces?  provide distinguished reference states? • is there a “renormalized holographic complexity”?  what’s it good for?; (EE vs mutual information versions of F) • ambiguities? ambiguities? ambiguities?  connections between ambiguities in gravity and boundary? • more boundary terms: higher codim. intersections; “complex” joint contributions; boundary “counterterms” • why is complexity of formation positive?

• A contribution of spacetime singularity? • subregion complexity? C “It from Qubit”: New Collision of Ideas

“Holography” Quantum Gravity

Quantum Information Theory “It from Qubit”: New Collision of Ideas

“Holography” http://www.perimeterinstitute.ca/it-qubitQuantum-summer-school /it-qubit-summerGravity-school-resources

Quantum Information Theory