A BerryEsseen Theorem for Quantum Lattice Systems and the - - PowerPoint PPT Presentation

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

a

The Berry–Esseen Theorem

[X ≤ x]

Central limit theorem:

− − − − →

N→∞

= F(x) G(x) = 1 √ 2πσ2 Z x

−∞

dy e− (y−µ)2

2σ2

X =

N

X

i=1

Xi

µ = hXi, σ2 = h(X µ)2i

Berry-Esseen: supx |F(x) − G(x)| ≤

C √ N

X

xk≤x

hk|%|ki

Xi

i

Λ = {1, . . . , n}×d

Central limit theorem (quantum):

a

Quantum Central Limit Theorems

Goderis, Vets (1989); Hartmann, Mahler, Hess (2004)

Xi

%

− − − − →

N→∞

= F(x) G(x) = 1 √ 2πσ2 Z x

−∞

dy e− (y−µ)2

2σ2

X = X

i∈Λ

Xi

µ = hXi, σ2 = h(X µ)2i

bounded and -local k sufficiently clustering: relation to density of states for

F(E) − F(E − ∆E) ∝ |{k : E − ∆E < Ek ≤ E}|

X = H % = 2N , :

= X

k

xk|kihk|

k

k

Xi

i

X = X

i∈Λ

Xi

µ = hXi, σ2 = h(X µ)2i

a

Quantum Berry–Esseen Theorem

Xi

% bounded and -local k

Λ = {1, . . . , n}×d

A

B L |hABi hAihBi| kAkkBk  N ze−L/ξ

sufficiently clustering:

k

= X

k

xk|kihk|

k

Xi

i

Λ = {1, . . . , n}×d

A

B L X = X

i∈Λ

Xi

µ = hXi, σ2 = h(X µ)2i

a

Quantum Berry–Esseen Theorem

Xi

% sufficiently clustering: bounded and -local k

|hABi hAihBi| kAkkBk  N ze−L/ξ

C = Cd

(max{k,ξ}(z+1))2d σ/ √ N

max n

1 max{k,ξ}(z+1) ln(N), 1 σ2/N

• sup

x |F(x) − G(x)| ≤ C ln2d(N)

√ N

k

= X

k

xk|kihk|

k

a

Quantum Berry–Esseen Theorem: Proof Idea

Esseen (1945)

sup

x |F(x) − G(x)| ≤ c1

T + Z T dt |φ(t) − e−σ2t2/2+iµ| |t| main ingredient (also for (quantum) central limit):

• pure state: Loschmidt echo
• : Fourier transform of d.o.s

a

Quantum Berry–Esseen Theorem: Proof Idea

Esseen (1945)

• characteristic function
• :

φ(t) = heiXti

X = H

% = 2N sup

x |F(x) − G(x)| ≤ c1

T + Z T dt |φ(t) − e−σ2t2/2+iµ| |t| main ingredient (also for (quantum) central limit): bound |φ(t) − e−σ2t2/2+iµ|

a

Quantum Berry–Esseen Theorem: Proof Idea

bound |φ(t) − e−σ2t2/2+iµ| set up differential equation for and bound its derivative φ

a

Quantum Berry–Esseen Theorem: Proof Idea

bound |φ(t) − e−σ2t2/2+iµ| φ

Tikhomirov (1980), Sunklodas (1984)

set up differential equation for and bound its derivative

a

Quantum Berry–Esseen Theorem: Proof Idea

bound |φ(t) − e−σ2t2/2+iµ| φ set up differential equation for and bound its derivative

Λ = {1, . . . , n}×d

i Hi

( )

a

Quantum Berry–Esseen Theorem: Application

canonical state %T = e−H/T

Z

with energy density specific heat capacity u(T) = tr(H%T )

N

= µ

N

c(T) = ∂u(T )

∂T

=

σ2 NT 2

( )

|hABihAihBi| kAkkBk

≤ N zeL/ξ

finite correlation length

H = X

i∈Λ

Hi = X

k

Ek|kihk| A B L

: Kliesch, Gogolin, Kastoryano, Riera, Eisert (2014) d > 1, T > Tc : Araki (1969) d = 1

k k

a

Quantum Berry–Esseen Theorem: Application

H = X

i∈Λ

Hi = X

k

Ek|kihk|

state ρ on subspace spanned by those

Me,δ = n k : |Ek − eN| ≤ δ √ N

• |ki

if and |e − u(T)| ≤ q

c(T )T 2 N C ln2d(N) √ N

≤

δ

√

c(T )T 2 ≤ 1

S(⇢k%T )  log(|Me,δ|) S(⇢) + ⇣p c(T)C + 4 ⌘ ln2d(N)

then special case: microcanonical state

1 |Me,δ|

P

k∈Me,δ |kihk|

for which S(ρ) = log(|Me,δ|)

Λ = {1, . . . , n}×d

i Hi

k k

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

k⇢C %Ck1 for which (and which ) is l ρ small?

• 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)

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

k⇢C %Ck1 for which (and which ) is l ρ small?

• Finite size, explicit bounds in system size
• More general states than microcanonical
• Equivalence of microcanonical states
• Not necessarily translational invariant

Here:

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

k⇢C %Ck1 for which (and which ) is l ρ ≤ 7√✏ ?

non-t.i.: same holds for the expectation over all cubic regions of edge length by Markov’s inequality [k⇢C %Ck1 a]  7√✏

a

l

δ = 0 : Eigenstate Thermalization

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

k⇢C %Ck1 for which (and which ) is l ρ ≤ 7√✏ ? For microcanonical states

and Me,δ = n k : |Ek − eN| ≤ δ √ N

• with

ρ = 1 |Me,δ| X

k∈Me,δ

|kihk| where |e − u(T)| ≤ q

c(T )T 2 N C ln2d(N) √ N

≤

δ

√

c(T )T 2 ≤ 1

l and such that

z+5+√ c(T )C ✏ ln(2)

ln2d(N) + l+1+⇠d

⇠

+ ld ≤ ⇣

✏N 4d⇠d

⌘

1 d+1

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

for which (and which ) is l ρ For pure states drawn from subspace

Popescu, Short, Winter (2005)

span{|ki}k∈Me,δ k⇢C %Ck1 small? : ⇥ k⇢C (m.c.)Ck1  p✏ +

2ld

p

|Me,|

⇤ 1 2e−

|Me,|✏ 18⇡3

ρ

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

for which (and which ) is l ρ

Popescu, Short, Winter (2005)

Me,δ, e, δ, l k⇢C %Ck1 small? as before ⇥ k⇢C (m.c.)Ck1  p✏ +

2ld

p

|Me,|

⇤ 1 2e−

|Me,|✏ 18⇡3

with prob. at least p:

• cp. Riera, Gogolin, Eisert (2011); Mueller, Adlam, Masanes, Wiebe (2013)

QBE

k⇢C %Ck1  8p✏ + 2ld exp ✓ N ✓ s(%)

Cp c(T ) ln2d(N) √ N

◆ /2 ◆ ≥ 1 − 2 exp  −

✏ 18⇡3 exp

✓ N ✓ s(%) −

C√ c(T ) ln2d(N) √ N

◆◆ =: p

For pure states drawn from subspace span{|ki}k∈Me,δ: ρ

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l S(⇢k%T ) + 3 ✏ + l + 1 + ⇠d ⇠ + ld + ln(N z+1)  ✓ ✏N 4d⇠d ◆

1 d+1

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

k⇢C %Ck1 for which (and which ) is l ρ ≤ 7√✏ ? For those which fulfil

• quantum substate theorem
• Lemma
• Pinsker’s inequality
• Super-additivity
• Jain, Radhakrishnan, Sen (2009); Jain, Nayak (2011)

Datta, Renner (2009); Brandão, Plenio (2010); Brandão, Horodecki (2012) Datta (2009)

PM

j=1 S(⇢Cjk%Cj)  S(⇢C1···CM k%C1···CM )

S(⇢k%)  Smax(⇢k%) k⇢ %k2

1  ln(4)S(⇢k%)

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

k⇢C %Ck1 for which (and which ) is l ρ ≤ 7√✏ ? i.e., as TS(⇢k%T ) = FT (⇢) FT (%T ), it holds for FT (ρ) = tr[Hρ] − TS(ρ) states with small free energy ρ

• cp. Th. 2 of Mueller, Adlam, Masanes, Wiebe (2013)

S(⇢k%T ) + 3 ✏ + l + 1 + ⇠d ⇠ + ld + ln(N z+1)  ✓ ✏N 4d⇠d ◆

1 d+1

For those which fulfil

l

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

States are locally thermal ( is small) if ρ k⇢C %Ck1 as before and

• on the subspace corresponding to (as before)


and 
 S(⇢) ≥ log(|Me,|) − ✏ ⇣

✏N 4d⇠d

⌘

1 d+1

Me,δ ρ (in fact, “almost all” states in this subspace)

• a
• FT (⇢) ≤ FT (%) + T✏

⇣

✏N 4d⇠d

⌘

1 d+1

• r

Λ = {1, . . . , n}×d

a

Local Thermalization After Quantum Quench

C

l l

|ψ(t)i = e−iHt|ψ0i ρ(t) = |ψ(t)ihψ(t)| H = X

k

Ek|kihk| ω = lim

T →∞

1 T Z T dt ρ(t) = X

k

|hψ0|ki|2|kihk|

Linden, Popescu, Short, Winter (2008); Reimann (2008)

≤ 2ldp tr[ω2] lim

T →∞

1 T Z T dt kρ(t) ωCk1 non-degen. energy gaps

Λ = {1, . . . , n}×d

a

Local Thermalization After Quantum Quench

C

l l

ρ(t) = |ψ(t)ihψ(t)| H = X

k

Ek|kihk| ω = lim

T →∞

1 T Z T dt ρ(t) = X

k

|hψ0|ki|2|kihk|

Linden, Popescu, Short, Winter (2008); Reimann (2008)

≤ 2ldp tr[ω2] lim

T →∞

1 T Z T dt kρ(t) ωCk1 bounded and -local k non-degen. energy gaps, ≤ 2ldC1/2

|ψ0i

lnd(N) N 1/4

|ψ0ias in QBE

|ψ(t)i = e−iHt|ψ0i

Λ = {1, . . . , n}×d

a

Local Thermalization After Quantum Quench

C

l l

non-degen. energy gaps, H = X

k

Ek|kihk|

Linden, Popescu, Short, Winter (2008); Reimann (2008)

bounded and -local k

≤ 2ld s 2C|ψ0i ln2d(N) √ N lim

T →∞

1 T Z T dt k⇢(t) %Ck1 +7√✏

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

l

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

States are locally thermal ( is small) if ρ k⇢C %Ck1 as before and

• on the subspace corresponding to (as before)


and 
 S(⇢) ≥ log(|Me,|) − ✏ ⇣

✏N 4d⇠d

⌘

1 d+1

Me,δ ρ

• a
• FT (⇢) ≤ FT (%) + T✏

⇣

✏N 4d⇠d

⌘

1 d+1

• r

l

Λ = {1, . . . , n}×d

a

Equivalence of Ensembles

C

l l

|hABihAihBi| kAkkBk

≤ N zeL/ξ : % = %T

States are locally thermal ( is small) if ρ k⇢C %Ck1 as before and

• on the subspace corresponding to (as before)


and 
 S(⇢) ≥ log(|Me,|) − ✏ ⇣

✏N 4d⇠d

⌘

1 d+1

Me,δ ρ (in fact, “almost all” states in this subspace)

• a
• FT (⇢) ≤ FT (%) + T✏

⇣

✏N 4d⇠d

⌘

1 d+1

• r