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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
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