A Berry–Esseen Theorem for Quantum Lattice Systems and the Equivalence of Statistical Mechanical Ensembles M. Cramer Ulm University with F.G.S.L. Brandão University College London M. Guta University of Nottingham
The Berry–Esseen Theorem a N X X = X i i =1 Berry-Esseen: sup x | F ( x ) − G ( x ) | ≤ C √ N Central limit theorem: Z x 1 d y e − ( y − µ )2 [ X ≤ x ] = F ( x ) G ( x ) = 2 σ 2 − − − − → √ 2 πσ 2 N →∞ −∞ µ = h X i , σ 2 = h ( X � µ ) 2 i
Quantum Central Limit Theorems a X X x k | k ih k | X = X i = i ∈ Λ k X i bounded and -local X i k k i sufficiently clustering: k % Λ = { 1, . . . , n } × d Central limit theorem (quantum): Z x 1 d y e − ( y − µ )2 X h k | % | k i = F ( x ) G ( x ) = 2 σ 2 − − − − → √ 2 πσ 2 N →∞ −∞ x k ≤ x Goderis, Vets (1989); Hartmann, Mahler, Hess (2004) relation to density of states for , : X = H % = 2 N F ( E ) − F ( E − ∆ E ) ∝ |{ k : E − ∆ E < E k ≤ E }| µ = h X i , σ 2 = h ( X � µ ) 2 i
Quantum Berry–Esseen Theorem a X X x k | k ih k | A X = X i B L = i ∈ Λ k X i bounded and -local X i k k i sufficiently clustering: k % Λ = { 1, . . . , n } × d | h AB i � h A ih B i | N z e − L/ ξ k A kk B k µ = h X i , σ 2 = h ( X � µ ) 2 i
Quantum Berry–Esseen Theorem a X X x k | k ih k | A X = X i B L = i ∈ Λ k X i bounded and -local X i k k i % sufficiently clustering: k Λ = { 1, . . . , n } × d | h AB i � h A ih B i | N z e − L/ ξ k A kk B k x | F ( x ) − G ( x ) | ≤ C ln 2 d ( N ) sup √ N (max { k , ξ } ( z +1)) 2 d n o 1 1 C = C d max max { k , ξ } ( z +1) ln( N ) , σ 2 /N √ σ / N µ = h X i , σ 2 = h ( X � µ ) 2 i
Quantum Berry–Esseen Theorem: Proof Idea a main ingredient (also for (quantum) central limit): Z T d t | φ ( t ) − e − σ 2 t 2 / 2+i µ | x | F ( x ) − G ( x ) | ≤ c 1 sup T + | t | 0 Esseen (1945)
Quantum Berry–Esseen Theorem: Proof Idea a main ingredient (also for (quantum) central limit): Z T d t | φ ( t ) − e − σ 2 t 2 / 2+i µ | x | F ( x ) − G ( x ) | ≤ c 1 sup T + | t | 0 Esseen (1945) bound | φ ( t ) − e − σ 2 t 2 / 2+i µ | • characteristic function φ ( t ) = h e i Xt i • : X = H •pure state: Loschmidt echo • : Fourier transform of d.o.s % = 2 N
Quantum Berry–Esseen Theorem: Proof Idea a bound | φ ( t ) − e − σ 2 t 2 / 2+i µ | set up differential equation for and bound its derivative φ
Quantum Berry–Esseen Theorem: Proof Idea a bound | φ ( t ) − e − σ 2 t 2 / 2+i µ | set up differential equation for and bound its derivative φ Tikhomirov (1980), Sunklodas (1984)
Quantum Berry–Esseen Theorem: Proof Idea a bound | φ ( t ) − e − σ 2 t 2 / 2+i µ | set up differential equation for and bound its derivative φ
Quantum Berry–Esseen Theorem: Application a X X E k | k ih k | H = H i = A L B i ∈ Λ k H i k i canonical state % T = e − H/T k Λ = { 1, . . . , n } × d Z u ( T ) = tr( H % T ) with energy density ( ) = µ N N c ( T ) = ∂ u ( T ) σ 2 specific heat capacity ( ) = ∂ T NT 2 finite correlation length | h AB i�h A ih B i | ≤ N z e � L/ ξ k A kk B k : Araki (1969) d = 1 : Kliesch, Gogolin, Kastoryano, Riera, Eisert (2014) d > 1 , T > T c
Quantum Berry–Esseen Theorem: Application a X X E k | k ih k | H = H i = i ∈ Λ k H i √ k i n o k : | E k − eN | ≤ δ M e , δ = N k Λ = { 1, . . . , n } × d state ρ on subspace spanned by those | k i q C ln 2 d ( N ) c ( T ) T 2 δ if and √ | e − u ( T ) | ≤ c ( T ) T 2 ≤ 1 ≤ √ N N then ⇣p ⌘ ln 2 d ( N ) S ( ⇢ k % T ) log( | M e , δ | ) � S ( ⇢ ) + c ( T ) C + 4 special case: microcanonical state 1 P k ∈ M e , δ | k ih k | | M e , δ | for which S ( ρ ) = log( | M e , δ | )
Equivalence of Ensembles a | h AB i�h A ih B i | : ≤ N z e � L/ ξ % = % T k A kk B k for which (and which ) is l ρ C l small? k ⇢ C � % C k 1 l Λ = { 1, . . . , n } × d • Question goes back to Boltzmann and Gibbs • Thermodynamic limit •Thermodynamical functions: Lebowitz, Lieb (1969); Lima (1971/72); Touchette (2009) •States: Mueller, Adlam, Masanes, Wiebe (2013) • see also: Popescu, Short, Winter (2005); Riera, Gogolin, Eisert (2011)
Equivalence of Ensembles a | h AB i�h A ih B i | : ≤ N z e � L/ ξ % = % T k A kk B k for which (and which ) is l ρ C l small? k ⇢ C � % C k 1 l Λ = { 1, . . . , n } × d Here: • Finite size, explicit bounds in system size • More general states than microcanonical • Equivalence of microcanonical states • Not necessarily translational invariant
Equivalence of Ensembles a | h AB i�h A ih B i | : ≤ N z e � L/ ξ % = % T k A kk B k for which (and which ) is l ρ ≤ 7 √ ✏ C l ? k ⇢ C � % C k 1 l Λ = { 1, . . . , n } × d non-t.i.: same holds for the expectation over all cubic regions of edge [ k ⇢ C � % C k 1 � a ] 7 √ ✏ length by Markov’s inequality l a
Equivalence of Ensembles a | h AB i�h A ih B i | : ≤ N z e � L/ ξ % = % T k A kk B k for which (and which ) is l ρ ≤ 7 √ ✏ C l ? k ⇢ C � % C k 1 l Λ = { 1, . . . , n } × d For microcanonical states √ 1 n o X | k ih k | where k : | E k − eN | ≤ δ ρ = M e , δ = N | M e, δ | k ∈ M e, δ q C ln 2 d ( N ) c ( T ) T 2 with and δ √ | e − u ( T ) | ≤ c ( T ) T 2 ≤ 1 ≤ √ N N z +5+ √ 1 ⇣ ⌘ c ( T ) C d +1 + l d ≤ ln 2 d ( N ) + l +1+ ⇠ d and such that ✏ N l ✏ ln(2) 4 d ⇠ d ⇠ δ = 0 : Eigenstate Thermalization
Equivalence of Ensembles a | h AB i�h A ih B i | : ≤ N z e � L/ ξ % = % T k A kk B k for which (and which ) is l ρ C l small? k ⇢ C � % C k 1 l Λ = { 1, . . . , n } × d For pure states drawn from ρ subspace : span {| k i } k ∈ M e, δ k ⇢ C � (m.c.) C k 1 p ✏ + | Me, � | ✏ 2 ld p ⇥ ⇤ � 1 � 2e − 18 ⇡ 3 | M e, � | Popescu, Short, Winter (2005)
Equivalence of Ensembles a | h AB i�h A ih B i | : ≤ N z e � L/ ξ % = % T k A kk B k for which (and which ) is l ρ C l small? k ⇢ C � % C k 1 l Λ = { 1, . . . , n } × d For pure states drawn from ρ subspace span {| k i } k ∈ M e, δ : k ⇢ C � (m.c.) C k 1 p ✏ + | Me, � | ✏ 2 ld p ⇥ ⇤ � 1 � 2e − 18 ⇡ 3 | M e, � | Popescu, Short, Winter (2005) C √ ✓ ✓ ◆◆� c ( T ) ln 2 d ( N ) ≥ 1 − 2 exp 18 ⇡ 3 exp s ( % ) − =: p ✏ N − √ N QBE as before with prob. at least p : M e, δ , e, δ , l C p k ⇢ C � % C k 1 8 p ✏ + 2 l d exp ✓ ✓ ◆ ◆ c ( T ) ln 2 d ( N ) � N s ( % ) � / 2 √ N cp. Riera, Gogolin, Eisert (2011); Mueller, Adlam, Masanes, Wiebe (2013)
Equivalence of Ensembles a | h AB i�h A ih B i | : ≤ N z e � L/ ξ % = % T k A kk B k for which (and which ) is l ρ ≤ 7 √ ✏ C l ? k ⇢ C � % C k 1 l Λ = { 1, . . . , n } × d For those which fulfil ✓ ✏ N 1 ◆ S ( ⇢ k % T ) + 3 + l + 1 + ⇠ d d +1 + l d + ln( N z +1 ) 4 d ⇠ d ✏ ⇠ • quantum substate theorem Jain, Radhakrishnan, Sen (2009); Jain, Nayak (2011) • Lemma Datta, Renner (2009); Brandão, Plenio (2010); Brandão, Horodecki (2012) • Pinsker’s inequality k ⇢ � % k 2 1 ln(4) S ( ⇢ k % ) • Super-additivity P M j =1 S ( ⇢ C j k % C j ) S ( ⇢ C 1 ··· C M k % C 1 ··· C M ) • S ( ⇢ k % ) S max ( ⇢ k % ) Datta (2009)
Equivalence of Ensembles a | h AB i�h A ih B i | : ≤ N z e � L/ ξ % = % T k A kk B k for which (and which ) is l ρ ≤ 7 √ ✏ C l ? k ⇢ C � % C k 1 l Λ = { 1, . . . , n } × d For those which fulfil ✓ ✏ N 1 ◆ S ( ⇢ k % T ) + 3 + l + 1 + ⇠ d d +1 + l d + ln( N z +1 ) 4 d ⇠ d ✏ ⇠ i.e., as TS ( ⇢ k % T ) = F T ( ⇢ ) � F T ( % T ) , it holds for states with small free energy F T ( ρ ) = tr[ H ρ ] − TS ( ρ ) ρ cp. Th. 2 of Mueller, Adlam, Masanes, Wiebe (2013)
Equivalence of Ensembles a | h AB i�h A ih B i | : ≤ N z e � L/ ξ % = % T k A kk B k States are locally thermal C l ρ ( is small) if l k ⇢ C � % C k 1 Λ = { 1, . . . , n } × d as before and l 1 • ⇣ ⌘ d +1 ✏ N F T ( ⇢ ) ≤ F T ( % ) + T ✏ • a 4 d ⇠ d or • on the subspace corresponding to (as before) M e, δ ρ and 1 ⇣ ⌘ d +1 ✏ N S ( ⇢ ) ≥ log( | M e, � | ) − ✏ 4 d ⇠ d (in fact, “almost all” states in this subspace)
Local Thermalization After Quantum Quench a ρ ( t ) = | ψ ( t ) ih ψ ( t ) | | ψ ( t ) i = e − i Ht | ψ 0 i C l X E k | k ih k | H = l Λ = { 1, . . . , n } × d k non-degen. energy gaps Z T 1 X | h ψ 0 | k i | 2 | k ih k | ω = lim d t ρ ( t ) = T T →∞ 0 k Z T 1 ≤ 2 l d p lim d t k ρ ( t ) � ω C k 1 tr[ ω 2 ] T T →∞ 0 Linden, Popescu, Short, Winter (2008); Reimann (2008)
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