Thermodynamics beyond free
energy relations or
“Quantum”
Quantum Thermodynamics
Matteo Lostaglio
Imperial College London
David Jennings
Kamil
Korzekwa
Terry Rudolph
This is wind, he is actually slim
What makes quantum
thermodynamics really quantum?
Can we develop a general
framework to understand
coherence?
Single-shot
Try to maximize
average work
𝑊
Single-shot
𝑊
Truly work-like work extraction
Johan Aberg
Nature Communications 4, 1925 (2013)
Single-shot
𝑊
Single-shot thermodynamics
Single-shot
𝑊
Single-shot thermodynamics
Classical sector
“Classical” Quantum Thermodynamics
system
𝜌
bath
𝛾𝐵 ∝ 𝑒 −𝛽𝐻𝐵
catalyst
𝑐
“Classical” Quantum Thermodynamics
system
𝜌
bath
𝛾𝐵 ∝ 𝑒 −𝛽𝐻𝐵
𝑈, 𝐻 = 0
overall energy conservation
catalyst
𝑐
“Classical” Quantum Thermodynamics
system
𝜌
bath
𝛾𝐵 ∝ 𝑒 −𝛽𝐻𝐵
catalyst
𝑐
𝑈, 𝐻 = 0
overall energy conservation
𝜎
𝑐
“Classical” Quantum Thermodynamics
𝜌, 𝐻𝑆 = 0
𝑒 −𝛽𝐻𝐵
“Classical” Quantum Thermodynamics
𝜌, 𝐻𝑆 = 0
Theory becomes ``classical’’
: 𝐵
𝑒 −𝛽𝐻
1. Diagonalise everything in energy eigenbasis.
2. Hence, only probabilities of occupying
different energies matters.
3. Thermal operations are particular classical
stochastic processes.
“Classical” Quantum Thermodynamics
𝜌, 𝐻𝑆 = 0
Theory becomes ``classical’’
: 𝐵
𝑒 −𝛽𝐻
1. Diagonalise everything in energy eigenbasis.
2. Hence, only probabilities of occupying
different energies matters.
3. Thermal operations are particular classical
stochastic processes.
c.f. two-measurement
protocol in fluctuation
theorems (?)
“Classical” Quantum Thermodynamics
If 𝜌, 𝐻𝑆 = 0,
𝜌→𝜎
𝑒 −𝛽𝐻𝐵
𝐹𝛼 𝜌 ≥ 𝐹𝛼 𝜎
The second laws of quantum thermodynamics
Fernando G.S.L. Brandao, Michał Horodecki, Nelly Huei Ying Ng, Jonathan
Oppenheim, Stephanie Wehner
PNAS 112, 3275 (2015)
∀𝛼
“Classical” Quantum Thermodynamics
If 𝜌, 𝐻𝑆 = 0
𝜌→𝜎
𝑒 −𝛽𝐻𝐵
𝐹𝛼 𝜌 ≥ 𝐹𝛼 𝜎
∀𝛼
where
𝐹𝛼 𝜌 = −𝑘𝑇 log 𝑍 + 𝑘𝑇𝑆𝛼 (𝜌||𝛾)
𝛼/|𝛼|
𝑆𝛼 (𝜌| 𝛾 =
log 𝑇𝑟𝜌𝛼 𝛾 1−𝛼
𝛼−1
The second laws of quantum thermodynamics
Fernando G.S.L. Brandao, Michał Horodecki, Nelly Huei Ying Ng, Jonathan
Oppenheim, Stephanie Wehner
PNAS 112, 3275 (2015)
𝑒 −𝛽𝐻𝑆
𝛾=
𝑍𝑆
“Quantum” Quantum Thermodynamics
𝑒 −𝛽𝐻𝐵
𝜎 = 𝜀𝑇 𝜌 = 𝑇𝑟𝐵 [𝑈 −𝛽𝐻
𝜌⊗
𝑈ϯ]
𝐵
𝑍𝐵
𝑒
𝑈, 𝐻 = 0
Now 𝜌, 𝐻𝑆 ≠ 0
(superpositions)
“Quantum” Quantum Thermodynamics
|0>
?
?
|1>
𝑒 −𝛽𝐻𝑆
𝛾=
𝑍𝑆
“Quantum” Quantum Thermodynamics
|0>
𝑒 −𝛽𝐻𝐵
𝑒 −𝛽𝐻𝑆
𝛾=
𝑍𝑆
|1>
“Quantum” Quantum Thermodynamics
|0>
𝑒 −𝛽𝐻𝐵
``classical’’
thermalisation
dephasing
|1>
𝑒 −𝛽𝐻𝑆
𝛾=
𝑍𝑆
“Quantum” Quantum Thermodynamics
|0>
𝑒 −𝛽𝐻𝐵
𝑒 −𝛽𝐻𝑆
𝛾=
𝑍𝑆
No processing
of coherence
|1>
Free energy is not enough
If first law 𝑈, 𝐻 = 0, this is impossible:
𝑒 −𝛽𝐻𝐵
𝜎
Despite any condition on F being satisfied.
What new ideas can we use to incorporate
coherence in the description?
Description of quantum coherence in thermodynamic processes requires constraints beyond
free energy
Matteo Lostaglio, David Jennings, Terry Rudolph
Nature Communications 6, 6383 (2015)
Quantum coherence, time-translation symmetry and thermodynamics
Matteo Lostaglio, Kamil Korzekwa, David Jennings, Terry Rudolph
Phys. Rev. X 5, 021001 (2015)
“Quantum” Quantum Thermodynamics
Simple but powerful symmetry consideration:
Non-equilibrium
thermodynamics
𝜀𝑇𝑒 −𝛽𝐻
⊂ 𝐵𝜀𝑆
Time-translation
symmetry
“Quantum” Quantum Thermodynamics
Simple but powerful symmetry consideration:
𝜀𝑇𝑒 −𝛽𝐻
⊂ 𝐵𝜀𝑆
𝜀𝑆 𝑒 −𝑖𝐻𝑆𝑡 𝜌𝑒 𝑖𝐻𝑆𝑡 = 𝑒 −𝑖𝐻𝑆𝑡 𝜀𝑆 𝜌 𝑒 𝑖𝐻𝑆𝑡
time-translation symmetric maps
Extending Noether's theorem by quantifying the asymmetry of quantum states
Nature Communications 5, 3821 (2014)
Modes of asymmetry: the application of harmonic analysis to symmetric quantum dynamics and
quantum reference frames
Phys. Rev. A 90, 062110 (2014)
Iman Marvian, Robert W. Spekkens
“Quantum” Quantum Thermodynamics
Simple but powerful symmetry consideration:
𝜀𝑇𝑒 −𝛽𝐻
⊂ 𝐵𝜀𝑆
𝜀𝑆 𝑒 −𝑖𝐻𝑆𝑡 𝜌𝑒 𝑖𝐻𝑆𝑡 = 𝑒 −𝑖𝐻𝑆𝑡 𝜀𝑆 𝜌 𝑒 𝑖𝐻𝑆𝑡
time-translation symmetric maps
Symmetry that describes how energy and coherence
are NOT conserved in the system.
Extending Noether's theorem by quantifying the asymmetry of quantum states
Nature Communications 5, 3821 (2014)
Modes of asymmetry: the application of harmonic analysis to symmetric quantum dynamics and
quantum reference frames
Phys. Rev. A 90, 062110 (2014)
Iman Marvian, Robert W. Spekkens
“Quantum” Quantum Thermodynamics
Simple but powerful symmetry consideration:
𝜀𝑇𝑒 −𝛽𝐻
⊂ 𝐵𝜀𝑆
𝜀𝑆 𝑒 −𝑖𝐻𝑆𝑡 𝜌𝑒 𝑖𝐻𝑆𝑡 = 𝑒 −𝑖𝐻𝑆𝑡 𝜀𝑆 𝜌 𝑒 𝑖𝐻𝑆𝑡
time-translation symmetric maps
Symmetry that describes how energy and coherence
are NOT conserved in the system.
1. Even if closed, 𝐻𝑆 , 𝐻𝑆 2 , … , 𝐻𝑆 𝑘 not enough for mixed states
2. Open dynamics
“Quantum” Quantum Thermodynamics
Simple but powerful symmetry consideration:
𝜀𝑇𝑒 −𝛽𝐻
⊂ 𝐵𝜀𝑆
𝜀𝑆 𝑒 −𝑖𝐻𝑆𝑡 𝜌𝑒 𝑖𝐻𝑆𝑡 = 𝑒 −𝑖𝐻𝑆𝑡 𝜀𝑆 𝜌 𝑒 𝑖𝐻𝑆𝑡
time-translation symmetric maps
Symmetry that describes how energy and coherence
are NOT conserved in the system.
1. Even if closed, 𝐻𝑆 , 𝐻𝑆 2 , … , 𝐻𝑆 𝑘 not enough for mixed states
2. Open dynamics
THERMODYNAMICS!
“Quantum” Quantum Thermodynamics
Some consequences
𝐴𝛼 𝜌 = 𝑆𝛼 (𝜌||𝐷 𝜌 )
𝐵 𝑍 + 𝑘𝑇𝑆 (𝜌||𝛾)
c.f. 𝐹𝛼 𝜌 = −𝑘𝑇
𝑒 −𝛽𝐻log
𝛼
Under any thermal operation:
∆𝐴𝛼 𝜌 ≤ 0
``second laws’’ for quantum coherence
“Quantum” Quantum Thermodynamics
Some consequences
𝐴𝛼 𝜌 = 𝑆𝛼 (𝜌||𝐷 𝜌 )
𝐵 𝑍 + 𝑘𝑇𝑆 (𝜌||𝛾)
c.f. 𝐹𝛼 𝜌 = −𝑘𝑇
𝑒 −𝛽𝐻log
𝛼
Under any thermal operation:
∆𝐴𝛼 𝜌 ≤ 0
``second laws’’ for quantum coherence
Macroscopically irrelevant:
lim
𝑁→∞
𝐴𝛼 𝜌⊗𝑁
𝑁
=0
∆𝑨𝜶 ≤ 𝟎
∆𝑭𝜶 ≤ 𝟎
“Quantum” Quantum Thermodynamics
Some consequences
𝑒 −𝛽𝐻𝐵
Standard quantum free energy:
𝑆 𝜌
𝐹 𝜌 = 𝐻 −
=𝐹 𝐷 𝜌
𝛽
both decrease
1
+ 𝐴(𝜌)
𝛽
Work locked
“Quantum” Quantum Thermodynamics
Some consequences
𝑒 −𝛽𝐻𝐵
Standard quantum free energy:
𝑆 𝜌
𝐹 𝜌 = 𝐻 −
=𝐹 𝐷 𝜌
𝛽
both decrease
1
+ 𝐴(𝜌)
𝛽
Work locked
Activation:
𝐹 𝐷 𝜌⊗2
≥ 2𝐹 𝐷 𝜌
c.f. passive states…
(strict if 𝜌, 𝐻𝑆 ≠ 0 )
“Quantum” Quantum Thermodynamics
Concrete bounds? If 𝜌 → 𝜎
𝜌𝑘𝑙 𝑒 −𝛽
|𝜎𝑛𝑚 | ≤
𝑘,𝑙
𝐸𝑘 <𝐸𝑛
𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚
𝐸𝑛 −𝐸𝑘
𝑒 −𝛽𝐻𝐵
+
|𝜌𝑘𝑙 |
𝑘,𝑙
𝐸𝑘 ≥𝐸𝑛
𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚
“Quantum” Quantum Thermodynamics
Concrete bounds? If 𝜌 → 𝜎
𝜌𝑘𝑙 𝑒 −𝛽
|𝜎𝑛𝑚 | ≤
𝑘,𝑙
𝐸𝑘 <𝐸𝑛
𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚
𝜌
𝜌56
𝜌23
𝐸𝑛 −𝐸𝑘
𝑒 −𝛽𝐻𝐵
+
|𝜌𝑘𝑙 |
𝑘,𝑙
𝐸𝑘 ≥𝐸𝑛
𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚
𝜎
𝜎56
“Quantum” Quantum Thermodynamics
Concrete bounds? If 𝜌 → 𝜎
𝜌𝑘𝑙 𝑒 −𝛽
|𝜎𝑛𝑚 | ≤
𝑘,𝑙
𝐸𝑘 <𝐸𝑛
𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚
𝜌
𝐸𝑛 −𝐸𝑘
𝑒 −𝛽𝐻𝐵
+
|𝜌𝑘𝑙 |
𝑘,𝑙
𝐸𝑘 ≥𝐸𝑛
𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚
𝜎
𝜌56
𝜎23
“Quantum” Quantum Thermodynamics
Concrete bounds? If 𝜌 → 𝜎
𝜌𝑘𝑙 𝑒 −𝛽
|𝜎𝑛𝑚 | ≤
𝑘,𝑙
𝐸𝑘 <𝐸𝑛
𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚
𝑒 −𝛽𝐻𝐵
+
|𝜌𝑘𝑙 |
𝑘,𝑙
𝐸𝑘 ≥𝐸𝑛
𝐸𝑘 −𝐸𝑙 =𝐸𝑛 −𝐸𝑚
𝜎
𝜌
𝑒 −𝛽∆𝐸
𝜌23
𝐸𝑛 −𝐸𝑘
𝜎56
“Quantum” Quantum Thermodynamics
Concrete bounds? If 𝜌 → 𝜎
𝑒 −𝛽𝐻𝐵
Kamil: Uououo!! This is
explained better in my
poster!
𝑒 −𝛽∆𝐸
𝜌23
𝜎56
Conclusions
1.
2.
3.
4.
Coherence in thermodynamics can be
understood through symmetry principles
Coherence + Thermo > Free energy
consideration
Bias towards energetic considerations?
More to be done!
Thank you for your attention!
More details:
Description of quantum coherence in thermodynamic processes requires constraints beyond
free energy
Matteo Lostaglio, David Jennings, Terry Rudolph
Nature Communications 6, 6383 (2015)
Quantum coherence, time-translation symmetry and thermodynamics
Matteo Lostaglio, Kamil Korzekwa, David Jennings, Terry Rudolph
Phys. Rev. X 5, 021001 (2015)
Work extraction from coherence
Kamil Korzekwa, Matteo Lostaglio , Jonathan Oppenheim, David Jennings
Soon on ArXiv!
“Classical” Quantum Thermodynamics
𝜌𝛼 𝛾 1−𝛼
𝛼/|𝛼|
𝑆𝛼 (𝜌| 𝛾 =
log 𝑇𝑟
1−𝛼
−𝛽𝐻𝐵 1−𝛼
𝑒
𝛼−1
(𝜌 2𝛼 𝛾𝜌 2𝛼 )𝛼
0≤𝛼≤1
𝛼>1
Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative
entropies
Milan Mosonyi, Tomohiro Ogawa
Communications in Mathematical Physics: Volume 334, Issue 3 (2015), Page 1617-1648