On the Rate of Convergence in the Quantum Central Limit Theorem M. - PowerPoint PPT Presentation

On the Rate of Convergence in the Quantum Central Limit Theorem M. Cramer Ulm University on work with F.G.S.L. Brandão Microsoft Research and University College London M. Guta University of Nottingham

The Rate of Convergence in the Central Limit Theorem N X X = X i i =1 X i : “weakly correlated” Central Limit Theorem: Z x 1 d y e − ( y − µ )2 [ X ≤ x ] = F ( x ) → G ( x ) = N →∞ 2 σ 2 √ − − − 2 πσ 2 −∞ µ = h X i , σ 2 = ( X � µ ) 2 ↵ ⌦

The Rate of Convergence in the Central Limit Theorem N X X = X i i =1 “weakly correlated” X i : Central Limit Theorem: Z x 1 d y e − ( y − µ )2 [ X ≤ x ] = F ( x ) → G ( x ) = N →∞ 2 σ 2 √ − − − 2 πσ 2 −∞ x | F ( x ) − G ( x ) | ≤ C Berry—Esseen: sup √ N µ = h X i , σ 2 = ( X � µ ) 2 ↵ ⌦

The Rate of Convergence in the Quantum Central Limit Theorem X X ˆ ˆ local x k | k ih k | ˆ ˆ X = X i = L A B i ∈ Λ k : | h ˆ A ˆ B i�h ˆ A ih ˆ ˆ B i | X i ≤ N z e � L/ ξ ˆ % k ˆ A kk ˆ B k Λ = { 1 , . . . , n } × d , N = n d Quantum Central Limit Theorem: Z x 1 d y e − ( y − µ )2 X h k | ˆ % | k i = F ( x ) → G ( x ) = N →∞ 2 σ 2 √ − − − 2 πσ 2 −∞ x k ≤ x Goderis, Vets (1989); Hartmann, Mahler, Hess (2004) C log 2 d ( N ) Berry—Esseen: sup x | F ( x ) − G ( x ) | ≤ √ N Cramer, Brandão, Guta, in prep. (2015) σ 2 = µ = h ˆ ( ˆ X � µ ) 2 ↵ ⌦ X i ,

The Rate of Convergence in the Quantum Central Limit Theorem X X ˆ ˆ local x k | k ih k | ˆ ˆ X = X i = L A B i ∈ Λ k : | h ˆ A ˆ B i�h ˆ A ih ˆ ˆ B i | X i ≤ N z e � L/ ξ ˆ % k ˆ A kk ˆ B k Λ = { 1 , . . . , n } × d , N = n d Quantum Central Limit Theorem: Z x 1 d y e − ( y − µ )2 X relation to density of states: ˆ h k | ˆ % | k i = F ( x ) → G ( x ) = N →∞ 2 σ 2 √ − − − % ∝ 2 πσ 2 −∞ � � �� x k ≤ x k : E − ∆ E < E k ≤ E Goderis, Vets (1989); Hartmann, Mahler, Hess (2004) � ∝ F ( E ) − F ( E − ∆ E ) C log 2 d ( N ) Berry—Esseen: sup x | F ( x ) − G ( x ) | ≤ √ N Cramer, Brandão, Guta, in prep. (2015) σ 2 = µ = h ˆ ( ˆ X � µ ) 2 ↵ ⌦ X i ,

The Rate of Convergence in the Quantum Central Limit Theorem: Proof Idea main ingredient (also for (quantum) central limit): Z T d t | φ ( t ) − e − t 2 / 2 | x | F ( x ) − G ( x ) | ≤ 1 sup T + | t | 0 Esseen (1945)

The Rate of Convergence in the Quantum Central Limit Theorem: Proof Idea main ingredient (also for (quantum) central limit): Z T d t | φ ( t ) − e − t 2 / 2 | x | F ( x ) − G ( x ) | ≤ 1 sup T + | t | 0 Esseen (1945) � φ ( t ) − e − t 2 / 2 � need to bound � � e i ˆ : characteristic function Xt ↵ ⌦ φ ( t ) = :

The Rate of Convergence in the Quantum Central Limit Theorem: Proof Idea main ingredient (also for (quantum) central limit): Z T d t | φ ( t ) − e − t 2 / 2 | x | F ( x ) − G ( x ) | ≤ 1 sup T + | t | 0 Esseen (1945) � φ ( t ) − e − t 2 / 2 � need to bound � � e i ˆ : characteristic function Xt ↵ ⌦ φ ( t ) = : X = ˆ ˆ H pure state: Loschmidt echo, 
 return probability : Fourier transform of d.o.s % = 2 N ˆ

The Rate of Convergence in the Quantum Central Limit Theorem: Proof Idea main ingredient (also for (quantum) central limit): Z T d t | φ ( t ) − e − t 2 / 2 | x | F ( x ) − G ( x ) | ≤ 1 sup T + | t | 0 Esseen (1945) � φ ( t ) − e − t 2 / 2 � need to bound � � e i ˆ : characteristic function Xt ↵ ⌦ φ ( t ) = : X = ˆ ˆ H pure state: Loschmidt echo, 
 dynamical “phase transitions” return probability : Fourier transform of d.o.s % = 2 N ˆ

The Rate of Convergence in the Quantum Central Limit Theorem: Proof Idea � φ ( t ) − e − t 2 / 2 � need to bound � � set up differential equation for char. function and bound its derivative cf. Tikhomirov (1980), Sunklodas (1984)

The Rate of Convergence in the Quantum Central Limit Theorem: Application X X ˆ ˆ local E k | k ih k | H = H i = L A B i ∈ Λ k : | h ˆ A ˆ B i�h ˆ A ih ˆ B i | ≤ N z e � L/ ξ ˆ ˆ % T H i k ˆ A kk ˆ B k Araki (1969) d = 1 : d > 1 , T > T c : Kliesch, Gogolin, Kastoryano, Riera, Eisert (2014) Λ = { 1 , . . . , n } × d , N = n d H/T /Z canonical state ˆ = e − ˆ % T

The Rate of Convergence in the Quantum Central Limit Theorem: Application X X ˆ ˆ local E k | k ih k | H = H i = L A B i ∈ Λ k : | h ˆ A ˆ B i�h ˆ A ih ˆ B i | ≤ N z e � L/ ξ ˆ ˆ % T H i k ˆ A kk ˆ B k Araki (1969) d = 1 : d > 1 , T > T c : Kliesch, Gogolin, Kastoryano, Riera, Eisert (2014) Λ = { 1 , . . . , n } × d , N = n d H/T /Z canonical state ˆ = e − ˆ % T u ( T ) = tr[ ˆ H ˆ % T ] with energy density ( ) = µ N N specific heat capacity ( ) σ 2 ∂ c ( T ) = ∂ T u ( T ) = NT 2

The Rate of Convergence in the Quantum Central Limit Theorem: Application X X ˆ ˆ local E k | k ih k | H = H i = L A B i ∈ Λ k : | h ˆ A ˆ B i�h ˆ A ih ˆ B i | ≤ N z e � L/ ξ ˆ ˆ % T H i k ˆ A kk ˆ B k Araki (1969) d = 1 : d > 1 , T > T c : Kliesch, Gogolin, Kastoryano, Riera, Eisert (2014) Λ = { 1 , . . . , n } × d , N = n d H/T /Z canonical state ˆ = e − ˆ % T : state on microcanonical subspace ˆ % p log 2 d ( N ) � M δ = | k i : | E k � Nu ( T ) |  δ . δ . 1 N , √ N quantum Berry—Esseen % ) + log 2 d ( N ) S (ˆ % k ˆ % T ) . log( | M δ | ) � S (ˆ

Why Do Systems Thermalize? H/T /Z % T = e − ˆ ˆ

Why Do Systems Thermalize? lack of knowledge, ignorance Jaynes’ principle H/T /Z % T = e − ˆ ˆ

Why Do Systems Thermalize? – Kinematics and Dynamics part of a large (closed) system % C = tr \ C [ˆ ˆ % ] C

Why Do Systems Thermalize? – Kinematics and Dynamics part of a large (closed) system % C = tr \ C [ˆ ˆ % ] H C /T /Z ≈ e − ˆ C

Why Do Systems Thermalize? – Kinematics and Dynamics part of a large (closed) system ˆ % C = tr \ C [ˆ % ] H/T /Z e − ˆ ⇥ ⇤ ≈ tr \ C C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ tr \ C % C ≈ in contact with heat bath % C ( ˆ 0 ) ⊗ ˆ % B C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ tr \ C % C ≈ in contact with heat bath, unitary evolution e − i t ˆ e i t ˆ H � H � % C ( ˆ 0 ) ⊗ ˆ % B C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ tr \ C % C ≈ in contact with heat bath, unitary evolution e − i t ˆ e i t ˆ H � H ⇤ ⇥ � tr \ C % C ( ˆ 0 ) ⊗ ˆ % C ( t ) ˆ % B = C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ tr \ C % C ≈ in contact with heat bath, unitary evolution e − i t ˆ e i t ˆ H � H ⇤ ⇥ � tr \ C % C ( ˆ 0 ) ⊗ ˆ % C ( t ) ˆ % B = H/T /Z e − ˆ ⇥ ⇤ t →∞ → tr \ C − − C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ tr \ C % C ≈ quantum quench e − i t ˆ e i t ˆ H � H ⇤ ⇥ � tr \ C % C ( ˆ 0 ) ⊗ ˆ % C ( t ) ˆ % B = H/T /Z e − ˆ ⇥ ⇤ t →∞ → tr \ C − − C

Why Do Systems Thermalize? – Kinematics and Dynamics H/T /Z e − ˆ part of a large (closed) system ˆ ⇥ ⇤ % C ≈ tr \ C Canonical Typicality 
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