Einselection without pointer states

Einselection without pointer states - Decoherence under weak interaction

Christian Gogolin

Universit¨atW¨urzburg

2009-12-16

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 1 / 26 Einselection without pointer states | Introduction New foundation for Statistical Mechanics

X Thermodynamics

X Statistical Mechanics

Second Law ergodicity equal a priory probabilities

Classical Mechanics

[2, 3] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 2 / 26 Einselection without pointer states | Introduction New foundation for Statistical Mechanics

X Thermodynamics

X Statistical Mechanics

Second Law ergodicity ? equal a priory probabilities

Classical Mechanics ?

[2, 3] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 2 / 26 Einselection without pointer states | Introduction New foundation for Statistical Mechanics

X Thermodynamics

X Statistical Mechanics

[2, 3] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 2 / 26 Einselection without pointer states | Introduction New foundation for Statistical Mechanics

X Thermodynamics

X Statistical Mechanics

⇑

Quantum Mechanics X

[2, 3] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 2 / 26 quantum mechanical orbitals discrete energy levels

coherent superpositions hopping

Einselection without pointer states | Introduction Why do electrons hop between energy eigenstates?

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 3 / 26 discrete energy levels

hopping

Einselection without pointer states | Introduction Why do electrons hop between energy eigenstates?

quantum mechanical orbitals

coherent superpositions

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 3 / 26 Einselection without pointer states | Introduction Why do electrons hop between energy eigenstates?

quantum mechanical orbitals discrete energy levels

coherent superpositions hopping

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 3 / 26 Einselection without pointer states | Introduction Table of contents

1 Technical introduction

2 Setup

3 Subsystem equilibration

4 Decoherence under weak interaction

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 4 / 26 Einselection without pointer states | Technical introduction

Technical introduction

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 5 / 26 Include classical randomness

ρ, ψ ∈ M(H) ψ = |ψihψ| X Tr[ρ] = hi|ρ|ii = 1 hAiρ = Tr[A ρ] i † − i H t ρt = Ut ρ0 Ut Ut = e

mixtures: ρ = p ψ1 + (1 − p) ψ2

Einselection without pointer states | Technical introduction Quantum Mechanics on one slide

Pure Quantum Mechanics

|ψi ∈ H A = A†

hψ|ψi = 1 hAiψ = hψ|A|ψi − i H t |ψti = Ut |ψ0i Ut = e

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 6 / 26 X Tr[ρ] = hi|ρ|ii = 1 hAiρ = Tr[A ρ] i † − i H t ρt = Ut ρ0 Ut Ut = e

mixtures: ρ = p ψ1 + (1 − p) ψ2

Einselection without pointer states | Technical introduction Quantum Mechanics on one slide

Pure Quantum Mechanics

|ψi ∈ H A = A†

hψ|ψi = 1 hAiψ = hψ|A|ψi − i H t |ψti = Ut |ψ0i Ut = e

Include classical randomness

ρ, ψ ∈ M(H) ψ = |ψihψ|

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 6 / 26 † − i H t ρt = Ut ρ0 Ut Ut = e

mixtures: ρ = p ψ1 + (1 − p) ψ2

Einselection without pointer states | Technical introduction Quantum Mechanics on one slide

Pure Quantum Mechanics

|ψi ∈ H A = A†

hψ|ψi = 1 hAiψ = hψ|A|ψi − i H t |ψti = Ut |ψ0i Ut = e

Include classical randomness

ρ, ψ ∈ M(H) ψ = |ψihψ| X Tr[ρ] = hi|ρ|ii = 1 hAiρ = Tr[A ρ] i

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 6 / 26 mixtures: ρ = p ψ1 + (1 − p) ψ2

Einselection without pointer states | Technical introduction Quantum Mechanics on one slide

Pure Quantum Mechanics

|ψi ∈ H A = A†

hψ|ψi = 1 hAiψ = hψ|A|ψi − i H t |ψti = Ut |ψ0i Ut = e

Include classical randomness

ρ, ψ ∈ M(H) ψ = |ψihψ| X Tr[ρ] = hi|ρ|ii = 1 hAiρ = Tr[A ρ] i † − i H t ρt = Ut ρ0 Ut Ut = e

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 6 / 26 Einselection without pointer states | Technical introduction Quantum Mechanics on one slide

Pure Quantum Mechanics

|ψi ∈ H A = A†

hψ|ψi = 1 hAiψ = hψ|A|ψi − i H t |ψti = Ut |ψ0i Ut = e

Include classical randomness

ρ, ψ ∈ M(H) ψ = |ψihψ| X Tr[ρ] = hi|ρ|ii = 1 hAiρ = Tr[A ρ] i † − i H t ρt = Ut ρ0 Ut Ut = e

mixtures: ρ = p ψ1 + (1 − p) ψ2

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 6 / 26 1 deff (ψ) = 1 deff ( ) = d d

= max Tr[A ρ] − Tr[A σ] 0≤A≤1

Effective dimension

1 deff (ρ) = Tr(ρ2)

Time average

1 Z T ω = hρtit = lim ρt dt T →∞ T 0

Einselection without pointer states | Technical introduction Technical introduction

Trace distance

1 D(ρ, σ) = kρ − σk 2 1

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 7 / 26 1 deff (ψ) = 1 deff ( ) = d d

Effective dimension

1 deff (ρ) = Tr(ρ2)

Time average

1 Z T ω = hρtit = lim ρt dt T →∞ T 0

Einselection without pointer states | Technical introduction Technical introduction

Trace distance

1 D(ρ, σ) = kρ − σk 2 1 = max Tr[A ρ] − Tr[A σ] 0≤A≤1

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 7 / 26 1 deff (ψ) = 1 deff ( ) = d d

Time average

1 Z T ω = hρtit = lim ρt dt T →∞ T 0

Einselection without pointer states | Technical introduction Technical introduction

Trace distance

1 D(ρ, σ) = kρ − σk 2 1 = max Tr[A ρ] − Tr[A σ] 0≤A≤1

Effective dimension

1 deff (ρ) = Tr(ρ2)

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 7 / 26 1 deff ( ) = d d

Time average

1 Z T ω = hρtit = lim ρt dt T →∞ T 0

Einselection without pointer states | Technical introduction Technical introduction

Trace distance

1 D(ρ, σ) = kρ − σk 2 1 = max Tr[A ρ] − Tr[A σ] 0≤A≤1

Effective dimension

1 deff (ρ) = deff (ψ) = 1 Tr(ρ2)

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 7 / 26 Time average

1 Z T ω = hρtit = lim ρt dt T →∞ T 0

Einselection without pointer states | Technical introduction Technical introduction

Trace distance

1 D(ρ, σ) = kρ − σk 2 1 = max Tr[A ρ] − Tr[A σ] 0≤A≤1

Effective dimension

1 1 deff (ρ) = deff (ψ) = 1 deff ( ) = d Tr(ρ2) d

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 7 / 26 Einselection without pointer states | Technical introduction Technical introduction

Trace distance

1 D(ρ, σ) = kρ − σk 2 1 = max Tr[A ρ] − Tr[A σ] 0≤A≤1

Effective dimension

1 1 deff (ρ) = deff (ψ) = 1 deff ( ) = d Tr(ρ2) d

Time average

1 Z T ω = hρtit = lim ρt dt T →∞ T 0

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 7 / 26 Explicit construction P 1 expand in basis: |ψi = i ci |ii hi|ji = δij 2 choose ci from a normal distribution P 2 3 normalize 1 = i |ci| hψ|ψi = 1

Einselection without pointer states | Technical introduction Choosing random states

Mathematical construction

Haar measure on SU(n) −→ “uniform” distribution

µ(V ) = µ(UV ) U |0i = |ψi Pr{|ψi} = Pr{U |ψi}

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 8 / 26 Einselection without pointer states | Technical introduction Choosing random states

Mathematical construction

Haar measure on SU(n) −→ “uniform” distribution

µ(V ) = µ(UV ) U |0i = |ψi Pr{|ψi} = Pr{U |ψi}

Explicit construction P 1 expand in basis: |ψi = i ci |ii hi|ji = δij 2 choose ci from a normal distribution P 2 3 normalize 1 = i |ci| hψ|ψi = 1

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 8 / 26 Einselection without pointer states | Setup

Setup

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 9 / 26 Einselection without pointer states | Setup Setup

System, HS, H S Bath, HB, H B

H SB

S B ρt = TrB[ψt] ρt = TrS[ψt]

1 S Tr[(AS ⊗ B)ψt] = Tr[AS ρt ]

reduced state → locally observable

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 10 / 26 1 S Tr[(AS ⊗ B)ψt] = Tr[AS ρt ]

reduced state → locally observable

Einselection without pointer states | Setup Setup

System, HS, H S Bath, HB, H B

H SB

S B ρt = TrB[ψt] ρt = TrS[ψt]

dψt dt = i [ψt, H ]

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 10 / 26 H 0 +

H 0 ∝ 1 Tr[H S] = Tr[H B] = Tr[H SB] = 0

Assumption A Hamiltonian has non–degenerate energy gaps iff:

Ek − El = Em − En

=⇒ k = l ∧ m = n or k = m ∧ l = n

Einselection without pointer states | Setup A very weak assumption on the Hamiltonian

H = H S ⊗1 + 1 ⊗ H B + H SB

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 11 / 26 Assumption A Hamiltonian has non–degenerate energy gaps iff:

Ek − El = Em − En

=⇒ k = l ∧ m = n or k = m ∧ l = n

Einselection without pointer states | Setup A very weak assumption on the Hamiltonian

H = H 0 + H S ⊗1 + 1 ⊗ H B + H SB

H 0 ∝ 1 Tr[H S] = Tr[H B] = Tr[H SB] = 0

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 11 / 26 Einselection without pointer states | Setup A very weak assumption on the Hamiltonian

H = H 0 + H S ⊗1 + 1 ⊗ H B + H SB

H 0 ∝ 1 Tr[H S] = Tr[H B] = Tr[H SB] = 0

Assumption A Hamiltonian has non–degenerate energy gaps iff:

Ek − El = Em − En

=⇒ k = l ∧ m = n or k = m ∧ l = n

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 11 / 26 Einselection without pointer states | Subsystem equilibration

Subsystem equilibration and fluctuations around equilibrium

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 12 / 26 =⇒ If d is large then deff (ω) is large.

Einselection without pointer states | Subsystem equilibration Measure concentration in Hilbert space

Theorem 1

For random ψ0 ∈ P1(H) with d = dim(H)

 d √ Pr deff (ω) < ≤ 2 e−c d 4

eff d (ω) ∼ # energy eigenstates in |ψ0i

[6] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 13 / 26 Einselection without pointer states | Subsystem equilibration Measure concentration in Hilbert space

Theorem 1

For random ψ0 ∈ P1(H) with d = dim(H)

 d √ Pr deff (ω) < ≤ 2 e−c d 4

eff d (ω) ∼ # energy eigenstates in |ψ0i

=⇒ If d is large then deff (ω) is large.

[6] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 13 / 26 eff 2 S =⇒ If d (ω)  dS then ρt equilibrates.

Einselection without pointer states | Subsystem equilibration Equilibration

Theorem 2

For every ψ0 ∈ P1(H) s 1 d2 hD(ρS, ωS)i ≤ S t t 2 deff (ω) where

S S S ρt = TrB ψt ω = hρt it ω = hψtit

[6] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 14 / 26 Einselection without pointer states | Subsystem equilibration Equilibration

Theorem 2

For every ψ0 ∈ P1(H) s 1 d2 hD(ρS, ωS)i ≤ S t t 2 deff (ω) where

S S S ρt = TrB ψt ω = hρt it ω = hψtit

eff 2 S =⇒ If d (ω)  dS then ρt equilibrates.

[6] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 14 / 26 S dρt = i TrB[ψt, H ] dt

Theorem 3

For every ψ0 ∈ P1(H)

s 3 dS hvS(t)it ≤ k H S ⊗1 + H SB k∞ deff(ω)

eff 3 S =⇒ If d (ω)  dS then ρt is slow.

Einselection without pointer states | Subsystem equilibration Speed of the fluctuations around equilibrium

S S S D(ρt , ρt+δt) 1 dρt vS(t) = lim = δt→0 δt 2 dt 1

[5] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 15 / 26 Theorem 3

For every ψ0 ∈ P1(H)

s 3 dS hvS(t)it ≤ k H S ⊗1 + H SB k∞ deff(ω)

eff 3 S =⇒ If d (ω)  dS then ρt is slow.

Einselection without pointer states | Subsystem equilibration Speed of the fluctuations around equilibrium

S S S S D(ρt , ρt+δt) 1 dρt dρt vS(t) = lim = = i TrB[ψt, H ] δt→0 δt 2 dt 1 dt

[5] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 15 / 26 eff 3 S =⇒ If d (ω)  dS then ρt is slow.

Einselection without pointer states | Subsystem equilibration Speed of the fluctuations around equilibrium

S S S S D(ρt , ρt+δt) 1 dρt dρt vS(t) = lim = = i TrB[ψt, H ] δt→0 δt 2 dt 1 dt

Theorem 3

For every ψ0 ∈ P1(H)

s 3 dS hvS(t)it ≤ k H S ⊗1 + H SB k∞ deff(ω)

[5] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 15 / 26 Einselection without pointer states | Subsystem equilibration Speed of the fluctuations around equilibrium

S S S S D(ρt , ρt+δt) 1 dρt dρt vS(t) = lim = = i TrB[ψt, H ] δt→0 δt 2 dt 1 dt

Theorem 3

For every ψ0 ∈ P1(H)

s 3 dS hvS(t)it ≤ k H S ⊗1 + H SB k∞ deff(ω)

eff 3 S =⇒ If d (ω)  dS then ρt is slow.

[5] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 15 / 26 Einselection without pointer states | Subsystem equilibration Summary

Typical states of large quantum systems have a high average effective dimension, their subsystems equilibrate and fluctuate slowly around the equilibrium state.

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 16 / 26 Einselection without pointer states | Decoherence under weak interaction

Decoherence under weak interaction

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 17 / 26 effective dynamics:

H S H B

H SB L

S dρt S S = i [ρ , H S] + i L(ρ ) dt t t

Einselection without pointer states | Decoherence under weak interaction Approach 1: Effective dynamics

standard QM:

H S H B

H SB

S dρt = i TrB[ψt, H ] dt

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 18 / 26 Einselection without pointer states | Decoherence under weak interaction Approach 1: Effective dynamics

standard QM: effective dynamics:

H S H B H S H B

H SB H SB L

S S dρt dρt S S = i TrB[ψt, H ] = i [ρ , H S] + i L(ρ ) dt dt t t

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 18 / 26 Einselection Off-diagonal elements in the pointer basis are suppressed:

0 † S 0 S 0 B (p ) (p) B hp|ρt |p i = hp|ρ0 |p i hψ0 |Ut Ut |ψ0 i | {z } ≤1

Einselection without pointer states | Decoherence under weak interaction Approach 2: Decoherence `ala Zurek

Special Hamiltonian with pointer sates |pi: X H = |pihp| ⊗ H (p) p

S B Initial product state ψ0 = ρ0 ⊗ ψ0

[7] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 19 / 26 Einselection without pointer states | Decoherence under weak interaction Approach 2: Decoherence `ala Zurek

Special Hamiltonian with pointer sates |pi: X H = |pihp| ⊗ H (p) p

S B Initial product state ψ0 = ρ0 ⊗ ψ0

Einselection Off-diagonal elements in the pointer basis are suppressed:

0 † S 0 S 0 B (p ) (p) B hp|ρt |p i = hp|ρ0 |p i hψ0 |Ut Ut |ψ0 i | {z } ≤1

[7] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 19 / 26 Can we find a more general mechanism based on standard Quantum Mechanics?

Einselection without pointer states | Decoherence under weak interaction Comparison

Pros and cons unitary evolution general mechanism effective dynamics X X einselection X X

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 20 / 26 Einselection without pointer states | Decoherence under weak interaction Comparison

Pros and cons unitary evolution general mechanism effective dynamics X X einselection X X

Can we find a more general mechanism based on standard Quantum Mechanics?

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 20 / 26 when is the subsystem slow?

S 1 dρt vS(t) = 2 dt 1

= i TrB[ψt, H 0 + H S ⊗1 + 1 ⊗ H B + H SB]

[5] =⇒ = i TrB[ψt, H S ⊗1 + H SB]

Given the interaction is weak

k H SB k∞  k H S k∞, H S H B

H SB

S dρt = i TrB[ψt, H ] dt

Einselection without pointer states | Decoherence under weak interaction Yes we can!

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 21 / 26 = i TrB[ψt, H 0 + H S ⊗1 + 1 ⊗ H B + H SB]

[5] =⇒ = i TrB[ψt, H S ⊗1 + H SB]

when is the subsystem slow?

S 1 dρt vS(t) = 2 dt 1

S dρt = i TrB[ψt, H ] dt

Einselection without pointer states | Decoherence under weak interaction Yes we can!

Given the interaction is weak

k H SB k∞  k H S k∞, H S H B

H SB

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 21 / 26 = i TrB[ψt, H 0 + H S ⊗1 + 1 ⊗ H B + H SB]

[5] =⇒ = i TrB[ψt, H S ⊗1 + H SB]

S dρt = i TrB[ψt, H ] dt

Einselection without pointer states | Decoherence under weak interaction Yes we can!

Given the interaction is weak

k H SB k∞  k H S k∞, H S H B

when is the subsystem slow? H SB

S 1 dρt vS(t) = 2 dt 1

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 21 / 26 = i TrB[ψt, H 0 + H S ⊗1 + 1 ⊗ H B + H SB]

[5] =⇒ = i TrB[ψt, H S ⊗1 + H SB]

Einselection without pointer states | Decoherence under weak interaction Yes we can!

Given the interaction is weak

k H SB k∞  k H S k∞, H S H B

when is the subsystem slow? H SB

S 1 dρt vS(t) = 2 dt 1

S dρt = i TrB[ψt, H ] dt

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 21 / 26 [5] =⇒ = i TrB[ψt, H S ⊗1 + H SB]

Einselection without pointer states | Decoherence under weak interaction Yes we can!

Given the interaction is weak

k H SB k∞  k H S k∞, H S H B

when is the subsystem slow? H SB

S 1 dρt vS(t) = 2 dt 1

S dρt = i TrB[ψt, H ] dt = i TrB[ψt, H 0 + H S ⊗1 + 1 ⊗ H B + H SB]

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 21 / 26 Einselection without pointer states | Decoherence under weak interaction Yes we can!

Given the interaction is weak

k H SB k∞  k H S k∞, H S H B

when is the subsystem slow? H SB

S 1 dρt vS(t) = 2 dt 1

S dρt = i TrB[ψt, H ] dt = i TrB[ψt, H 0 + H S ⊗1 + 1 ⊗ H B + H SB]

[5] =⇒ = i TrB[ψt, H S ⊗1 + H SB]

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 21 / 26 Einselection without pointer states | Decoherence under weak interaction Tow competing forces

S dρt S = i TrB[ψt, H S ⊗1 + H SB] =i[ ρ , H S]+ i Tr B[ψt, H SB] dt t

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 22 / 26 Einselection without pointer states | Decoherence under weak interaction Tow competing forces

S dρt S = i TrB[ψt, H S ⊗1 + H SB] =i[ ρ , H S]+ i Tr B[ψt, H SB] dt t

S ρt

H S

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 22 / 26 Einselection without pointer states | Decoherence under weak interaction Tow competing forces

S dρt S = i TrB[ψt, H S ⊗1 + H SB] =i[ ρ , H S]+ i Tr B[ψt, H SB] dt t

S ρt

H S H SB

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 22 / 26 Einselection without pointer states | Decoherence under weak interaction Tow competing forces

S dρt S = i TrB[ψt, H S ⊗1 + H SB] =i[ ρ , H S]+ i Tr B[ψt, H SB] dt t

S ρt

H S H SB

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 22 / 26 Einselection without pointer states | Decoherence under weak interaction Tow competing forces

S dρt S = i TrB[ψt, H S ⊗1 + H SB] =i[ ρ , H S]+ i Tr B[ψt, H SB] dt t

S ρt

H S H SB

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 22 / 26 S =⇒ If ρt is slow its off-diagonal elements are small.

Einselection without pointer states | Decoherence under weak interaction Decoherence through weak interaction

Theorem 4 S All reduced states ρt satisfy

S S S S dρt max 2 |Ek − El | |ρkl| ≤ 2 k H SB k∞ + k6=l dt 1 where S S S S ρkl = hEk |ρt |El i

[1] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 23 / 26 Einselection without pointer states | Decoherence under weak interaction Decoherence through weak interaction

Theorem 4 S All reduced states ρt satisfy

S S S S dρt max 2 |Ek − El | |ρkl| ≤ 2 k H SB k∞ + k6=l dt 1 where S S S S ρkl = hEk |ρt |El i

S =⇒ If ρt is slow its off-diagonal elements are small.

[1] C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 23 / 26 Einselection without pointer states | Decoherence under weak interaction Conclusions

Pros and cons unitary evolution general mechanism effective dynamics X X einselection X X our mechanism XX

Applications electronic excitations of gases at moderate temperature radioactive decay environment assisted entanglement creation ...

C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 24 / 26 Einselection without pointer states | Decoherence under weak interaction References

[1] C. Gogolin, “Einselection without pointer states”, 0908.2921v2. [2] S. Lloyd, “Black holes, demons and the loss of coherence: How complex systems get information, and what they do with it”,. PhD thesis, Rockefeller University, April, 1991. [3] S. Popescu, A. J. Short, and A. Winter, “Entanglement and the foundations of statistical mechanics”, Nature Physics 2 (2006) no. 11, 754. [4] N. Linden, S. Popescu, A. J. Short, and A. Winter, “Quantum mechanical evolution towards thermal equilibrium”, Physical Review E 79 (2009) no. 6, 061103, 0812.2385v1. [5] N. Linden, S. Popescu, A. J. Short, and A. Winter, “On the speed of fluctuations around thermodynamic equilibrium”, 0907.1267v1. [6] N. Linden, S. Popescu, A. J. Short, and A. Winter, “Quantum mechanical evolution towards thermal equilibrium”, 0812.2385v1. [7] W. H. Zurek, “Environment-induced superselection rules”, Phys. Rev. D (1982) no. 26, 1862–1880. C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 25 / 26 Einselection without pointer states

Thank you for your attention!

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C. Gogolin | Universit¨atW¨urzburg | 2009-12-16 26 / 26