Unifying paradigms ofUnifying paradigms ofquantum refrigeration:quantum refrigeration:

A universal and attainableA universal and attainablebound on coolingbound on cooling

, , , , ,

Fabien Clivaz Ralph Silva Géraldine Haack JonatanBohr Brask Nicolas Brunner Marcus Huber

QTD 2019, 23-28 June 2019, Espoo, FinlandFabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

General IdeaS

S arbitrary (finite dim.)S initially at

room temperature

 

T R

T =R

T R

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

General IdeaS

S arbitrary (finite dim.)S initially at

room temperature

 

T R

T =R

T R

closed

no cooling

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

General IdeaS

S arbitrary (finite dim.)S initially at

room temperature

 

T R

T =R

T R

closed fully open

no coolingLoose track of

Resource

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

General IdeaS

S arbitrary (finite dim.)S initially at

room temperature

 

T R

T =R

T R

closed fully open

no coolingLoose track of

ResourceOpen in a

Controlled way

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

General IdeaS

S arbitrary (finite dim.)S initially at

room temperature

 

T R

T =R

T R

closed fully open

no coolingLoose track of

ResourceOpen in a

Controlled way

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

General IdeaS

S arbitrary (finite dim.)S initially at

room temperature

 

T R

T =R

T R

closed fully open

no coolingLoose track of

ResourceOpen in a

Controlled way

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

General IdeaS

S arbitrary (finite dim.)S initially at

room temperature

 

T R

T =R

T R

closed fully open

no coolingLoose track of

ResourceOpen in a

Controlled way

ΛFabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

2 Paradigms: Coherent & Incoherent

Related Paradigms

Universal Bound

Attainability of Bound

Summary

 

Table of Contents

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

N. dim.Machine

S

Coherent & Incoherent

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

N. dim.Machine

S

Battery

thermal at Λ (ρ ) =coh S Tr [U ρ ⊗M S ρ U ], ρ , ρ :MR †

S M T R

Cohe

rent

Coherent & Incoherent

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

N. dim.Machine

S

Battery Hot Bath

thermal at Λ (ρ ) =coh S Tr [U ρ ⊗M S ρ U ], ρ , ρ :MR †

S M T R

part at part at Λ (ρ ) =inc S Tr [U ρ ⊗M cons S ρ U ], ρ :MR,H

cons†

MR,H T R T H

Cohe

rent

IncoherentCoherent & Incoherent

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

T R

N. dim.Machine

S

Battery Hot Bath

T R

thermal at Λ (ρ ) =coh S Tr [U ρ ⊗M S ρ U ], ρ , ρ :MR †

S M T R

part at part at Λ (ρ ) =inc S Tr [U ρ ⊗M cons S ρ U ], ρ :MR,H

cons†

MR,H T R T H

Cohe

rent

Incoherent

allow repetitions: Λ (ρ ) :nS = Λ(⋯ Λ(Λ(ρ ))) →S Λ (ρ )∞

S

Coherent & Incoherent

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

Finite machineCoherent

Incoherent

Λ coh

Λ inc

Related Paradigms

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

Infinite machineCPTP map

Thermal operations

Finite machineCoherent

Incoherent

Λ coh

Λ inc

Related Paradigms

1,2,3

Brandao, Horodecki, Oppenheim, Renes,  Spekkens, PRL 111 (2013)1

Gour, Müller, Narasimhachar, Spekkens, Yunger Halpern, Phys. Rep. 583 (2015)2

Gallego, Eisert, Wilming, NJP.18 (2016)3

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

Infinite machineCPTP map

Thermal operations

Finite machineCoherent

Incoherent

Λ coh

Λ inc

Related Paradigms

Heat bath algorithmic cooling Quantum Otto engines

4,5,6

7

1,2,3

Brandao, Horodecki, Oppenheim, Renes,  Spekkens, PRL 111 (2013)1

Schulman, Vazirani, Proc. 31’st ACMSTOC, 322-329 (1999)4

Rodriguez-Briones, Martin-Martinez, Kempf, Laflamme, PRL.119 (2017)5

Skrzypczyk, Brunner, Linden, Popescu, JPA. 44 (2011)8

Niedenzu, Gelbwaser-Klimovsky, Kof-man, Kurizki, NJP. 18 (2016).7

, , 6 Alhambra Lostaglio Perry, arXiv:1807.07974 (2018)

Gour, Müller, Narasimhachar, Spekkens, Yunger Halpern, Phys. Rep. 583 (2015)2

Gallego, Eisert, Wilming, NJP.18 (2016)3

Coherent

Autonomous cooling 8Incoherent

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

Definition:

Universal Bound

colder than iff ρ 1 ρ 2 ρ ≻1 ρ 2

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

Definition:

Universal Bound

colder than iff ρ 1 ρ 2 ρ ≻1 ρ 2

ground state population - purity - average energy - entropy

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

Definition:

the ground state population of isupper bounded by:

Λ (ρ )∞S

p =0∗

(e )∑n=0d −1S − E

T R

1max n

1

Theorem:

Universal Bound

colder than iff ρ 1 ρ 2 ρ ≻1 ρ 2

ground state population - purity - average energy - entropy

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

Definition:

the ground state population of isupper bounded by:

Λ (ρ )∞S

p =0∗

(e )∑n=0d −1S − E

T R

1max n

1

Theorem:

Universal Bound

colder than iff ρ 1 ρ 2 ρ ≻1 ρ 2

ground state population - purity - average energy - entropy

E max

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

Definition:

the ground state population of isupper bounded by:

Λ (ρ )∞S

p =0∗

(e )∑n=0d −1S − E

T R

1max n

1

Theorem:

Proof (sketch): Find a such that if ρ S∗ ρ ≺S ρ ⇒S

∗ Λ (ρ ) ≺coh S ρ S∗

⇒ Λ (ρ ) ≺coh∞

S ρ S∗

Universal Bound

colder than iff ρ 1 ρ 2 ρ ≻1 ρ 2

ground state population - purity - average energy - entropy

E max

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

Definition:

the ground state population of isupper bounded by:

Λ (ρ )∞S

p =0∗

(e )∑n=0d −1S − E

T R

1max n

1

Theorem:

Proof (sketch): Find a such that if

Hot bath not more powerful than a battery

ρ S∗ ρ ≺S ρ ⇒S

∗ Λ (ρ ) ≺coh S ρ S∗

⇒ Λ (ρ ) ≺coh∞

S ρ S∗

⇒ Λ (ρ ) ≺inc∞

S Λ (ρ )coh∞

S

Universal Bound

colder than iff ρ 1 ρ 2 ρ ≻1 ρ 2

ground state population - purity - average energy - entropy

E max

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

the bound is reachable within thecoherent paradigm

Theorem:

Attainability

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

the bound is reachable within thecoherent paradigm

Theorem:

Example: E S

S

T =∗ T E max

E SR

Attainability

E max

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

the bound is reachable within thecoherent paradigm

Theorem:

For S qubit, can incoherently cool to ifadd one qubit (of gap ) to machine

T ∗

E −max E S

Theorem:

Example: E S

S

T =∗ T E max

E SR

Attainability

E max

E max

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

For S qubit, can autonomously cool to ifadd one qubit (of gap ) to machine

T ∗

E −max E S

Theorem:

Attainability

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

For S qubit, can autonomously cool to ifadd one qubit (of gap ) to machine

T ∗

E −max E S

Theorem:

S

Attainability

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

H int

For S qubit, can autonomously cool to ifadd one qubit (of gap ) to machine

T ∗

E −max E S

Theorem:

S

Attainability

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

H int

For S qubit, can autonomously cool to ifadd one qubit (of gap ) to machine

T ∗

E −max E S

Theorem:

ST H

T R

Attainability

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

H H

H R

H int

For S qubit, can autonomously cool to ifadd one qubit (of gap ) to machine

T ∗

E −max E S

Theorem:

ST H

T R

Attainability

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

H H

H R

H int

For S qubit, can autonomously cool to ifadd one qubit (of gap ) to machine

T ∗

E −max E S

Theorem:

ST H

T R

Attainability

E max

Fabien Clivaz et al., Fabien Clivaz et al.,

arXiv:1903.04970arXiv:1710.11624 (to appear in PRE)

Derived Universal boundCoherent: bound reachableIncoherent (+ qubit): bound reachable

Summary

T =∗ T E max

E SR

Derived Universal boundCoherent: bound reachableIncoherent (+ qubit): bound reachable

  , , , , , Unifying paradigms of quantum refrigeration:A universal and attainable bound on cooling

Unifying paradigms of quantum refrigeration:fundamental limits of cooling and associated work costs

F. Clivaz R. Silva G. Haack J. Bohr Brask N. Brunner M. Huber

arXiv:1903.04970

arXiv:1710.11624 (to appear in PRE)

For more details:

Summary

T =∗ T E max

E SR