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

• n work with

F.G.S.L. Brandão

Microsoft Research and University College London

• M. Guta

University of Nottingham

X =

N

X

i=1

Xi Central Limit Theorem:

The Rate of Convergence in the Central Limit Theorem

= F(x)

N→∞

− − − → G(x) = 1 √ 2πσ2 Z x

−∞

dy e− (y−µ)2

2σ2

[X ≤ x]

µ = hXi, σ2 = ⌦ (X µ)2↵

Xi : “weakly correlated”

X =

N

X

i=1

Xi

= F(x)

N→∞

− − − → G(x) = 1 √ 2πσ2 Z x

−∞

dy e− (y−µ)2

2σ2

Berry—Esseen:

[X ≤ x] sup

x |F(x) − G(x)| ≤ C

√ N

µ = hXi, σ2 = ⌦ (X µ)2↵

Xi : Central Limit Theorem: “weakly correlated”

The Rate of Convergence in the Central Limit Theorem

Λ = {1, . . . , n}×d, N = nd

= F(x)

N→∞

− − − → G(x) = 1 √ 2πσ2 Z x

−∞

dy e− (y−µ)2

2σ2

Berry—Esseen:

sup

x |F(x) − G(x)| ≤

local Quantum Central Limit Theorem:

The Rate of Convergence in the Quantum Central Limit Theorem

ˆ X = X

i∈Λ

ˆ Xi = X

k

xk|kihk| : |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ ˆ % ˆ Xi X

xk≤x

hk|ˆ %|ki L ˆ A ˆ B

µ = h ˆ Xi, σ2 = ⌦ ( ˆ X µ)2↵

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

C log2d(N) √ N

Cramer, Brandão, Guta, in prep. (2015)

Λ = {1, . . . , n}×d, N = nd

= F(x)

N→∞

− − − → G(x) = 1 √ 2πσ2 Z x

−∞

dy e− (y−µ)2

2σ2

Berry—Esseen:

sup

x |F(x) − G(x)| ≤

local Quantum Central Limit Theorem:

The Rate of Convergence in the Quantum Central Limit Theorem

ˆ X = X

i∈Λ

ˆ Xi = X

k

xk|kihk| : |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ ˆ % ˆ Xi X

xk≤x

hk|ˆ %|ki L ˆ A ˆ B

µ = h ˆ Xi, σ2 = ⌦ ( ˆ X µ)2↵

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

C log2d(N) √ N

Cramer, Brandão, Guta, in prep. (2015)

relation to density of states: ˆ % ∝ ∝ F(E) − F(E − ∆E)

• k : E − ∆E < Ek ≤ E

The Rate of Convergence in the Quantum Central Limit Theorem: Proof Idea

main ingredient (also for (quantum) central limit): sup

x |F(x) − G(x)| ≤ 1

T + Z T dt |φ(t) − e−t2/2| |t|

Esseen (1945)

characteristic function :

The Rate of Convergence in the Quantum Central Limit Theorem: Proof Idea

main ingredient (also for (quantum) central limit): sup

x |F(x) − G(x)| ≤ 1

T + Z T dt |φ(t) − e−t2/2| |t|

Esseen (1945)

• φ(t) − e−t2/2
• need to bound

: φ(t) = ⌦ ei ˆ

Xt↵

The Rate of Convergence in the Quantum Central Limit Theorem: Proof Idea

main ingredient (also for (quantum) central limit): sup

x |F(x) − G(x)| ≤ 1

T + Z T dt |φ(t) − e−t2/2| |t|

Esseen (1945)

• φ(t) − e−t2/2
• ˆ

X = ˆ H need to bound :

characteristic function :

pure state: Loschmidt echo,
 return probability : Fourier transform of d.o.s ˆ % = 2N φ(t) = ⌦ ei ˆ

Xt↵

The Rate of Convergence in the Quantum Central Limit Theorem: Proof Idea

main ingredient (also for (quantum) central limit): sup

x |F(x) − G(x)| ≤ 1

T + Z T dt |φ(t) − e−t2/2| |t|

Esseen (1945)

• φ(t) − e−t2/2
• ˆ

X = ˆ H need to bound :

characteristic function :

pure state: Loschmidt echo,
 return probability : Fourier transform of d.o.s ˆ % = 2N φ(t) = ⌦ ei ˆ

Xt↵

dynamical “phase transitions”

The Rate of Convergence in the Quantum Central Limit Theorem: Proof Idea

• φ(t) − e−t2/2
• need to bound

set up differential equation for char. function and bound its derivative

• cf. Tikhomirov (1980), Sunklodas (1984)

A

B

L ˆ H = X

i∈Λ

ˆ Hi = X

k

Ek|kihk|

Λ = {1, . . . , n}×d, N = nd

: |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ ˆ Hi local

The Rate of Convergence in the Quantum Central Limit Theorem: Application

ˆ %T = e− ˆ

H/T /Z

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

canonical state ˆ %T

specific heat capacity ( ) A B L ˆ H = X

i∈Λ

ˆ Hi = X

k

Ek|kihk|

Λ = {1, . . . , n}×d, N = nd

: |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ ˆ Hi local

The Rate of Convergence in the Quantum Central Limit Theorem: Application

ˆ %T = e− ˆ

H/T /Z

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

canonical state ˆ %T u(T) = tr[ ˆ

Hˆ %T ] N

= µ

N

c(T) =

∂ ∂T u(T)

=

σ2 NT 2

with energy density ( )

A

B

L ˆ H = X

i∈Λ

ˆ Hi = X

k

Ek|kihk|

Λ = {1, . . . , n}×d, N = nd

: |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ ˆ Hi local

The Rate of Convergence in the Quantum Central Limit Theorem: Application

ˆ %T = e− ˆ

H/T /Z

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

canonical state ˆ %T : state on microcanonical subspace ˆ % Mδ =

• |ki : |Ek Nu(T)|  δ

p N ,

log2d(N) √ N

. δ . 1 S(ˆ %kˆ %T ) . log(|Mδ|) S(ˆ %) + log2d(N)

quantum Berry—Esseen

ˆ %T = e− ˆ

H/T /Z

Why Do Systems Thermalize?

Why Do Systems Thermalize?

lack of knowledge, ignorance

Jaynes’ principle

ˆ %T = e− ˆ

H/T /Z

ˆ %C = tr\C[ˆ %]

Why Do Systems Thermalize? – Kinematics and Dynamics

part of a large (closed) system C

ˆ %C = tr\C[ˆ %] part of a large (closed) system ≈ e− ˆ

HC/T /Z

C

Why Do Systems Thermalize? – Kinematics and Dynamics

≈ tr\C ⇥ e− ˆ

H/T /Z

⇤ part of a large (closed) system ˆ %C = tr\C[ˆ %] C

Why Do Systems Thermalize? – Kinematics and Dynamics

part of a large (closed) system ˆ %C ≈ in contact with heat bath C ˆ %C( ) ⊗ ˆ %B tr\C ⇥ e− ˆ

H/T /Z

⇤

Why Do Systems Thermalize? – Kinematics and Dynamics

part of a large (closed) system ˆ %C ≈ in contact with heat bath, unitary evolution C e−it ˆ

H

ˆ %C( ) ⊗ ˆ %B

• eit ˆ

H

tr\C ⇥ e− ˆ

H/T /Z

⇤

Why Do Systems Thermalize? – Kinematics and Dynamics

part of a large (closed) system ˆ %C ≈ in contact with heat bath, unitary evolution C tr\C ⇥ e−it ˆ

H

ˆ %C( ) ⊗ ˆ %B

• eit ˆ

H⇤

tr\C ⇥ e− ˆ

H/T /Z

⇤ ˆ %C(t)

=

Why Do Systems Thermalize? – Kinematics and Dynamics

tr\C ⇥ e−it ˆ

H

ˆ %C( ) ⊗ ˆ %B

• eit ˆ

H⇤

t→∞

− − → tr\C ⇥ e− ˆ

H/T /Z

⇤ part of a large (closed) system ˆ %C ≈ in contact with heat bath, unitary evolution C tr\C ⇥ e− ˆ

H/T /Z

⇤ ˆ %C(t)

=

Why Do Systems Thermalize? – Kinematics and Dynamics

tr\C ⇥ e−it ˆ

H

ˆ %C( ) ⊗ ˆ %B

• eit ˆ

H⇤

t→∞

− − → tr\C ⇥ e− ˆ

H/T /Z

⇤ part of a large (closed) system ˆ %C ≈ quantum quench C tr\C ⇥ e− ˆ

H/T /Z

⇤ ˆ %C(t)

=

Why Do Systems Thermalize? – Kinematics and Dynamics

part of a large (closed) system ˆ %C ≈ tr\C ⇥ e− ˆ

H/T /Z

⇤

t→∞

− − → tr\C ⇥ e− ˆ

H/T /Z

⇤ ˆ %C(t) quantum quench

Time-dependence of correlation functions following a quantum quench


Calabrese, Cardy, Phys. Rev. Lett. (2006); arXiv:cond-mat/0601225

Relaxation in a Completely Integrable Many-Body Quantum System


Rigol, Dunjko, Yurovsky, Olshanii, Phys. Rev. Lett. (2007); arXiv:cond-mat/0604476

Effect of suddenly turning on interactions in the Luttinger model


Cazalilla,Phys. Rev. Lett. (2006); arXiv:cond-mat/0606236

Quenching, Relaxation, and a Central Limit Theorem for Quantum Lattice Systems


Cramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arXiv:cond-mat/0703314

Thermalization and its mechanism for generic isolated quantum systems


Rigol, Dunjko, Olshanii, Nature (2008); arXiv:0708.1324

Foundation of Statistical Mechanics under Experimentally Realistic Conditions


Reimann, Phys. Rev. Lett. (2008); arXiv:0810.3092

Quantum mechanical evolution towards thermal equilibrium


Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arXiv:0812.2385

Canonical Typicality


Goldstein, Lebowitz, Tumulka, Zanghi, Phys. Rev. Lett. (2006); arXiv:cond-mat/0511091

Entanglement and the foundations of statistical mechanics


Popescu, Short, Winter, Nature Physics (2006); arXiv:quant-ph/0511225

Thermalization in Nature and on a Quantum Computer


Riera, Gogolin, Eisert, Phys. Rev. Lett. (2012); arXiv:1102.2389

Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems


Mueller, Adlam, Masanes,Wiebe, arXiv:1312.7420

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

Why Do Systems Thermalize? – Kinematics and Dynamics

part of a large (closed) system ˆ %C ≈ tr\C ⇥ e− ˆ

H/T /Z

⇤

t→∞

− − → tr\C ⇥ e− ˆ

H/T /Z

⇤ ˆ %C(t) quantum quench

Time-dependence of correlation functions following a quantum quench


Calabrese, Cardy, Phys. Rev. Lett. (2006); arXiv:cond-mat/0601225

Relaxation in a Completely Integrable Many-Body Quantum System


Rigol, Dunjko, Yurovsky, Olshanii, Phys. Rev. Lett. (2007); arXiv:cond-mat/0604476

Effect of suddenly turning on interactions in the Luttinger model


Cazalilla,Phys. Rev. Lett. (2006); arXiv:cond-mat/0606236

Quenching, Relaxation, and a Central Limit Theorem for Quantum Lattice Systems


Cramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arXiv:cond-mat/0703314

Thermalization and its mechanism for generic isolated quantum systems


Rigol, Dunjko, Olshanii, Nature (2008); arXiv:0708.1324

Foundation of Statistical Mechanics under Experimentally Realistic Conditions


Reimann, Phys. Rev. Lett. (2008); arXiv:0810.3092

Quantum mechanical evolution towards thermal equilibrium


Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arXiv:0812.2385

Canonical Typicality


Goldstein, Lebowitz, Tumulka, Zanghi, Phys. Rev. Lett. (2006); arXiv:cond-mat/0511091

Entanglement and the foundations of statistical mechanics


Popescu, Short, Winter, Nature Physics (2006); arXiv:quant-ph/0511225

Thermalization in Nature and on a Quantum Computer


Riera, Gogolin, Eisert, Phys. Rev. Lett. (2012); arXiv:1102.2389

Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems


Mueller, Adlam, Masanes,Wiebe, arXiv:1312.7420

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

Why Do Systems Thermalize? – Kinematics and Dynamics

part of a large (closed) system ˆ %C ≈ tr\C ⇥ e− ˆ

H/T /Z

⇤

t→∞

− − → tr\C ⇥ e− ˆ

H/T /Z

⇤ ˆ %C(t) quantum quench

Time-dependence of correlation functions following a quantum quench


Calabrese, Cardy, Phys. Rev. Lett. (2006); arXiv:cond-mat/0601225

Relaxation in a Completely Integrable Many-Body Quantum System


Rigol, Dunjko, Yurovsky, Olshanii, Phys. Rev. Lett. (2007); arXiv:cond-mat/0604476

Effect of suddenly turning on interactions in the Luttinger model


Cazalilla,Phys. Rev. Lett. (2006); arXiv:cond-mat/0606236

Quenching, Relaxation, and a Central Limit Theorem for Quantum Lattice Systems


Cramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arXiv:cond-mat/0703314

Thermalization and its mechanism for generic isolated quantum systems


Rigol, Dunjko, Olshanii, Nature (2008); arXiv:0708.1324

Foundation of Statistical Mechanics under Experimentally Realistic Conditions


Reimann, Phys. Rev. Lett. (2008); arXiv:0810.3092

Quantum mechanical evolution towards thermal equilibrium


Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arXiv:0812.2385

Canonical Typicality


Goldstein, Lebowitz, Tumulka, Zanghi, Phys. Rev. Lett. (2006); arXiv:cond-mat/0511091

Entanglement and the foundations of statistical mechanics


Popescu, Short, Winter, Nature Physics (2006); arXiv:quant-ph/0511225

Thermalization in Nature and on a Quantum Computer


Riera, Gogolin, Eisert, Phys. Rev. Lett. (2012); arXiv:1102.2389

Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems


Mueller, Adlam, Masanes,Wiebe, arXiv:1312.7420

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

• gen. can. pricniple

for random with high probability …thermal? |ψi 2 HR ⇢ HC ⌦ HB ˆ %C ≈ tr\C[

R/dR]

Why Do Systems Thermalize? – Kinematics and Dynamics

part of a large (closed) system ˆ %C ≈ tr\C ⇥ e− ˆ

H/T /Z

⇤

t→∞

− − → tr\C ⇥ e− ˆ

H/T /Z

⇤ ˆ %C(t) quantum quench

Time-dependence of correlation functions following a quantum quench


Calabrese, Cardy, Phys. Rev. Lett. (2006); arXiv:cond-mat/0601225

Relaxation in a Completely Integrable Many-Body Quantum System


Rigol, Dunjko, Yurovsky, Olshanii, Phys. Rev. Lett. (2007); arXiv:cond-mat/0604476

Effect of suddenly turning on interactions in the Luttinger model


Cazalilla,Phys. Rev. Lett. (2006); arXiv:cond-mat/0606236

Quenching, Relaxation, and a Central Limit Theorem for Quantum Lattice Systems


Cramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arXiv:cond-mat/0703314

Thermalization and its mechanism for generic isolated quantum systems


Rigol, Dunjko, Olshanii, Nature (2008); arXiv:0708.1324

Foundation of Statistical Mechanics under Experimentally Realistic Conditions


Reimann, Phys. Rev. Lett. (2008); arXiv:0810.3092

Quantum mechanical evolution towards thermal equilibrium


Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arXiv:0812.2385

Canonical Typicality


Goldstein, Lebowitz, Tumulka, Zanghi, Phys. Rev. Lett. (2006); arXiv:cond-mat/0511091

Entanglement and the foundations of statistical mechanics


Popescu, Short, Winter, Nature Physics (2006); arXiv:quant-ph/0511225

Thermalization in Nature and on a Quantum Computer


Riera, Gogolin, Eisert, Phys. Rev. Lett. (2012); arXiv:1102.2389

Thermalization and Canonical Typicality in Translation-Invariant Quantum Lattice Systems


Mueller, Adlam, Masanes,Wiebe, arXiv:1312.7420

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

• gen. can. pricniple

for random with high probability …thermal? |ψi 2 HR ⇢ HC ⌦ HB ˆ %C ≈ tr\C[

R/dR]

Integrable

no thermalization instead: generalized Gibbs ensemble

non-integrable

“equilibrium state”, close to it for most times …thermal? time scale?

Why Do Systems Thermalize? – Kinematics and Dynamics

Quench: Quasi-Free Bosons

ˆ %( ) ∈ HC ⊗ HB sufficiently clustering (not necessarily Gaussian) ˆ H = P

ij

ˆ b†

i Aijˆ

bj + ˆ biBijˆ bj + h.c.

• local, t.i.

C

n n

N = nd

A Quantum Central Limit Theorem for Non-Equilibrium Systems: Exact Local Relaxation of Correlated States


Cramer, Eisert, New J. Phys. (2010)

ˆ %( ) ∈ HC ⊗ HB sufficiently clustering (not necessarily Gaussian) t ∈ [t1(✏, N), t2(✏, N)] kˆ %C(t) ˆ G(t)ktr  ✏ ˆ H = P

ij

ˆ b†

i Aijˆ

bj + ˆ biBijˆ bj + h.c.

• local, t.i.

for all : Gaussian with same second moments as ˆ G(t) ˆ %C(t) C

n n

N = nd

A Quantum Central Limit Theorem for Non-Equilibrium Systems: Exact Local Relaxation of Correlated States


Cramer, Eisert, New J. Phys. (2010)

Quench: Quasi-Free Bosons

ˆ %( ) ∈ HC ⊗ HB sufficiently clustering (not necessarily Gaussian) t ∈ [t1(✏, N), t2(✏, N)] kˆ %C(t) ˆ G(t)ktr  ✏ ˆ H = P

ij

ˆ b†

i Aijˆ

bj + ˆ biBijˆ bj + h.c.

• local, t.i.

for all : Gaussian with same second moments as ˆ G(t) ˆ %C(t) maximum entropy state

equilibration, non-thermal: Tegmark, Yeh (1994)

C

n n

N = nd

A Quantum Central Limit Theorem for Non-Equilibrium Systems: Exact Local Relaxation of Correlated States


Cramer, Eisert, New J. Phys. (2010)

Quench: Quasi-Free Bosons

Quench: Quasi-Free Bosons, Proof Idea

ˆ

%C(t)(β) = trC[ˆ

%C(t) ˆ D(β)] ˆ D(β) = Y

i∈C

eβiˆ

b†

i −β∗ i ˆ

bi

characteristic function (FT of Wigner function, Bochner’s theorem) C

n n

N = nd

A Quantum Central Limit Theorem for Non-Equilibrium Systems: Exact Local Relaxation of Correlated States


Cramer, Eisert, New J. Phys. (2010)

A Quantum Central Limit Theorem for Non-Equilibrium Systems: Exact Local Relaxation of Correlated States


Cramer, Eisert, New J. Phys. (2010)

ˆ

%C(t)(β) = trC[ˆ

%C(t) ˆ D(β)] ˆ D(β) = Y

i∈C

eβiˆ

b†

i −β∗ i ˆ

bi

αi = X

j∈C

βj

• eitA

ij

= tr ⇥ ˆ %( ) ˆ D(α(t, β)) ⇤

C

n n

N = nd

Quench: Quasi-Free Bosons, Proof Idea

characteristic function (FT of Wigner function, Bochner’s theorem)

l

l N = nd

C for which states , (and which ) is

l

ˆ % ˆ τ

kˆ %C ˆ ⌧Cktr  ✏ ?

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

Local Closeness – A Lemma

l

l N = nd

C for which states , (and which ) is

l

ˆ % ˆ τ

kˆ %C ˆ ⌧Cktr  ✏ ?

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

Local Closeness – A Lemma

non-t.i.: [ ]

l

l N = nd

C A

B

L for which states , (and which ) is

l

ˆ % ˆ τ : |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ kˆ %C ˆ ⌧Cktr  ✏ ?

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

Local Closeness – A Lemma

l

l N = nd

C ˆ τ kˆ %C ˆ ⌧Cktr  ✏

: |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ

A

B

L

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

for those with

S(ˆ %kˆ ⌧) ✏2

+ ld . (✏2N)

1 d+1

ln(N)

for which states , (and which ) is l ˆ %

Local Closeness – A Lemma

?

l

l N = nd

C ˆ τ kˆ %C ˆ ⌧Cktr  ✏

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

: |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ

A

B

L

Quantum Substate Theorem Lemma Pinsker’s inequality Super-additivity

S2p✏(ˆ %kˆ ⌧)  S2p✏

max(ˆ

%kˆ ⌧)  S(ˆ

%kˆ ⌧)+1 ✏

+ log(

1 1✏)

for those with

S(ˆ %kˆ ⌧) ✏2

+ ld . (✏2N)

1 d+1

ln(N)

for which states , (and which ) is l ˆ %

Jain, Radhakrishnan, Sen (2009); Jain, Nayak (2011)

Local Closeness – A Lemma

?

l

l N = nd

C ˆ τ kˆ %C ˆ ⌧Cktr  ✏

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

: |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ

A

B

L

Quantum Substate Theorem Lemma Pinsker’s inequality Super-additivity

Smax(ˆ %kˆ ⇡)  κ = 2λkˆ τ ˆ πktr S

√ 8κ max (ˆ

%kˆ ⌧)  + log(

1 1−κ)

S(ˆ %kˆ ⌧) ✏2

+ ld . (✏2N)

1 d+1

ln(N)

for those with for which states , (and which ) is l ˆ %

Jain, Radhakrishnan, Sen (2009); Jain, Nayak (2011) Datta, Renner (2009); Brandão, Plenio (2010); Brandão, Horodecki (2012)

Local Closeness – A Lemma

?

l

l N = nd

C ˆ τ kˆ %C ˆ ⌧Cktr  ✏

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

: |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ

A

B

L

Quantum Substate Theorem Lemma Pinsker’s inequality Super-additivity

kˆ τC1···CM ˆ τC1 ⌦ · · · ⌦ ˆ τC1k  PM

j=2 cov( ˆ

A1 · · · ˆ Aj−1, ˆ Aj)

S(ˆ %kˆ ⌧) ✏2

+ ld . (✏2N)

1 d+1

ln(N)

for those with for which states , (and which ) is l ˆ %

Jain, Radhakrishnan, Sen (2009); Jain, Nayak (2011) Datta, Renner (2009); Brandão, Plenio (2010); Brandão, Horodecki (2012)

Local Closeness – A Lemma

?

l

l N = nd

C ˆ τ kˆ %C ˆ ⌧Cktr  ✏

Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems


Brandão, Cramer, arxiv:1502.03263

: |h ˆ

A ˆ Bih ˆ Aih ˆ Bi| k ˆ Akk ˆ Bk

≤ NzeL/ξ

A

B

L

S(ˆ %kˆ ⌧) ✏2

+ ld . (✏2N)

1 d+1

ln(N)

Quantum Substate Theorem Lemma Pinsker’s inequality Super-additivity

PM

j=1 S(ˆ

%Cjkˆ ⌧Cj)  S(ˆ %kˆ ⌧C1 ⌦ · · · ⌦ ˆ ⌧CM) kˆ % ˆ ⌧ktr  ln(4)S(ˆ %kˆ ⌧)

Jain, Radhakrishnan, Sen (2009); Jain, Nayak (2011) Datta, Renner (2009); Brandão, Plenio (2010); Brandão, Horodecki (2012)

for those with for which states , (and which ) is l ˆ %

Local Closeness – A Lemma

?

l

l

C

Local Equivalence of Micro- and Macrocanonical Ensembles

kˆ %C ˆ ⌧Cktr  ✏ ? for which states , (and which ) is

l

ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ

l

l

C

Local Equivalence of Micro- and Macrocanonical Ensembles

kˆ %C ˆ ⌧Cktr  ✏ ? for which states , (and which ) is

l

ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ which states are locally thermal? ˆ %

l

l

C

Local Equivalence of Micro- and Macrocanonical Ensembles

kˆ %C ˆ ⌧Cktr  ✏ ? for which states , (and which ) is

l

ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ which states are locally thermal? ˆ % ˆ % =

Mδ

|Mδ|

for microcanonical states this question goes back to Boltzmann and Gibbs

Thermodynamical functions


[Lebowitz, Lieb (1969); Lima (1971/72); Touchette (2009)]

States [Mueller, Adlam, Masanes, Wiebe (2013)]

Popescu, Short, Winter (2005); Riera, Gogolin, Eisert (2011)

thermodynamical limit, t.i.

previous work:

l

l

C

Local Equivalence of Micro- and Macrocanonical Ensembles

kˆ %C ˆ ⌧Cktr  ✏ ? for which states , (and which ) is

l

ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ which states are locally thermal? ˆ % ˆ % =

Mδ

|Mδ|

for microcanonical states this question goes back to Boltzmann and Gibbs here:

Finite size, explicit bounds Not necessarily translational invariant More general than microcanonical

l

l

C

Local Equivalence of Micro- and Macrocanonical Ensembles

kˆ %C ˆ ⌧Cktr  ✏ ? for which states , (and which ) is

l

ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ which states are locally thermal? ˆ % ˆ % =

Mδ

|Mδ|

microcanonical states Mδ =

• |ki : |Ek Nu(T)|  δ

p N ,

log2d(N) √ N

. δ . 1 with and such that ld . (✏2N)

1 d+1

ln(N)

l

δ = 0 : Eigenstate Thermalization

l

l

C

Local Equivalence of Micro- and Macrocanonical Ensembles

kˆ %C ˆ ⌧Cktr  ✏ ? for which states , (and which ) is

l

ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ which states are locally thermal? ˆ % ˆ % =

Mδ

|Mδ|

microcanonical states Mδ =

• |ki : |Ek Nu(T)|  δ

p N ,

log2d(N) √ N

. δ . 1 with and such that ld . (✏2N)

1 d+1

ln(N)

l

l

l

C

Canonical Typicality

kˆ %C ˆ ⌧Cktr  ✏ ? for which states , (and which ) is

l

ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ which states are locally thermal? ˆ % pure states ˆ % drawn from the subspace spanned by : Mδ ⇥ kˆ %c (m.c.)Cktr  p✏ + 2ld/ p |M| ⇤ 1 2e−|M|✏

Popescu, Short, Winter (2005)

l

l

C kˆ %C ˆ ⌧Cktr  ✏ ? for which states , (and which ) is

l

ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ which states are locally thermal? ˆ % pure states ˆ % drawn from the subspace spanned by : Mδ ⇥ kˆ %c (m.c.)Cktr  p✏ + 2ld/ p |M| ⇤ 1 2e−|M|✏

Popescu, Short, Winter (2005)

QBE

≥ 1 − 2 exp ⇥ −✏ exp

• S(ˆ

⌧) − log2d(N) √ N ⇤ =: p

Canonical Typicality

l

l

C kˆ %C ˆ ⌧Cktr  ✏ ? for which states , (and which ) is

l

ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ which states are locally thermal? ˆ % pure states ˆ % drawn from the subspace spanned by : Mδ as before with probability at least kˆ %C ˆ ⌧Cktr  ✏ + 2ld exp ⇥ (S(ˆ ⌧) log2d(N) p N ⇤ ˆ τ, Mδ, δ, l p

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

Canonical Typicality

l

l

C

Sufficient Conditions for Local Thermalization: Summary

kˆ %C ˆ ⌧Cktr  ✏ ? for which states is ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ which states are locally thermal? ˆ % with small free energy in microcanonical subspace
 with large entropy as before then those

FT (ˆ %) = tr[ ˆ Hˆ %] − TS(ˆ %)

FT (ˆ %) . FT (ˆ ⌧) + T ✏2(✏2N)

1 d+1

ln(N)

ˆ τ, l

l

l

C

Sufficient Conditions for Local Thermalization: Summary

kˆ %C ˆ ⌧Cktr  ✏ ? for which states is ˆ % = e− ˆ

H/T /Z

canonical state ˆ τ which states are locally thermal? ˆ % with small free energy in microcanonical subspace
 with large entropy as before then those FT (ˆ %) . FT (ˆ ⌧) + T ✏2(✏2N)

1 d+1

ln(N)

S(ˆ %) ≥ log(|M|) − ✏2(✏2N)

1 d+1

ln(N)

ˆ τ, Mδ, δ, l (in fact, “almost all” states in this subspace)

• r

C

Local Thermalization after Quantum Quench

ˆ %(t) = e−it ˆ

H ˆ

%0eit ˆ

H

ˆ ! = lim

T →∞

1 T Z T dt ˆ %(t) ˆ H = P

k Ek|kihk|

C

non-degen. energy gaps

Local Thermalization after Quantum Quench

ˆ %(t) = e−it ˆ

H ˆ

%0eit ˆ

H

ˆ ! = lim

T →∞

1 T Z T dt ˆ %(t) ˆ H = P

k Ek|kihk|

= P

khk|ˆ

%0|ki|kihk|

Linden, Popescu, Short, Winter (2008)

lim

T →∞

1 T Z T dt kˆ %C(t) ˆ !Cktr  2|C|p tr[ˆ !2]

C

non-degen. energy gaps

Local Thermalization after Quantum Quench

ˆ %(t) = e−it ˆ

H ˆ

%0eit ˆ

H

ˆ ! = lim

T →∞

1 T Z T dt ˆ %(t) ˆ H = P

k Ek|kihk|

= P

khk|ˆ

%0|ki|kihk|

Linden, Popescu, Short, Winter (2008)

lim

T →∞

1 T Z T dt kˆ %C(t) ˆ !Cktr  2|C|p tr[ˆ !2] fraction of times for which is at 1 − 2|C|p tr[ˆ !2]/✏ least kˆ %C(t) ˆ !Cktr  ✏

C

non-degen. energy gaps

Local Thermalization after Quantum Quench

ˆ %(t) = e−it ˆ

H ˆ

%0eit ˆ

H

ˆ ! = lim

T →∞

1 T Z T dt ˆ %(t) ˆ H = P

k Ek|kihk|

= P

khk|ˆ

%0|ki|kihk|

Linden, Popescu, Short, Winter (2008)

lim

T →∞

1 T Z T dt kˆ %C(t) ˆ !Cktr  2|C|p tr[ˆ !2] fraction of times for which is at 1 − 2|C|p tr[ˆ !2]/✏ least kˆ %C(t) ˆ !Cktr  ✏

Geometry irrelevant Even “global” observables Also “local” quenches

C

non-degen. energy gaps

Local Thermalization after Quantum Quench

ˆ %(t) = e−it ˆ

H ˆ

%0eit ˆ

H

ˆ ! = lim

T →∞

1 T Z T dt ˆ %(t) ˆ H = P

k Ek|kihk|

= P

khk|ˆ

%0|ki|kihk|

Linden, Popescu, Short, Winter (2008)

lim

T →∞

1 T Z T dt kˆ %C(t) ˆ !Cktr  2|C|p tr[ˆ !2] fraction of times for which is at 1 − 2|C|p tr[ˆ !2]/✏ least kˆ %C(t) ˆ !Cktr  ✏

Purity? Thermal? Time scale?

local Hamiltonian, sufficiently weakly correlated initial state:

Purity

QBE

C

Local Thermalization after Quantum Quench: Summary

tr[ˆ ω2] . ln2d(N)

√ N

local Hamiltonian, sufficiently weakly correlated initial state:

Purity

QBE

C

Local Thermalization after Quantum Quench: Summary

tr[ˆ ω2] . ln2d(N)

√ N

Thermalization

integrable: no thermalization (instead generalized Gibbs ensemble)

local Hamiltonian, sufficiently weakly correlated initial state:

Purity

QBE

C

Local Thermalization after Quantum Quench: Summary

tr[ˆ ω2] . ln2d(N)

√ N

QBE

*the subsystem spends most of the times in close to the maximally mixed state [0, N

1 5d − 1 2 ]

Thermalization

integrable: no thermalization (instead generalized Gibbs ensemble) most Hamiltonians that are unitarily equivalent to a local Hamiltonian lead to fast thermalization*

Cramer, Thermalization under randomized local Hamiltonians (2012)

local Hamiltonian, sufficiently weakly correlated initial state:

Purity

QBE

C

Local Thermalization after Quantum Quench: Summary

tr[ˆ ω2] . ln2d(N)

√ N

Thermalization

integrable: no thermalization (instead generalized Gibbs ensemble)

QBE

*the subsystem spends most of the times in close to the maximally mixed state [0, N

1 5d − 1 2 ]

• transl. inv., thermodynamic limit: entropic condition
• n initial state implies thermalization

Mueller, Adlam, Masanes, Wiebe, Thermalization and canonical typicality in translation-invariant quantum lattice systems (2013)

most Hamiltonians that are unitarily equivalent to a local Hamiltonian lead to fast thermalization*

Cramer, Thermalization under randomized local Hamiltonians (2012)

QBE

non-t.i., finite size