Non-equilibrium fluctuations and athermality as quantum resources

169 Pages·2017·2.95 MB·English

Non-equilibrium fluctuations and
athermality as quantum resources
By
Alvaro Martin Alhambra
A thesis submitted to
University College London
for the degree of
Doctor of Philosophy
Department of Physics and Astronomy
University College London
October 31, 2017
I, Alvaro Martin Alhambra, confirm that the work presented in this thesis is my own. Where
information has been derived from other sources, I confirm that this has been indicated in the
thesis.
Signed ............................................ Date ................
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Non-equilibrium fluctuations and athermality as quantum resources
Alvaro Martin Alhambra
Doctor of Philosophy of Physics
University College London
Prof. Jonathan Oppenheim, Supervisor
Abstract
Within quantum information, a very productive way to look at physical phenomena is under
the light of the framework of resource theories. Through it, one can focus on questions such as
in what sense a particular physical feature can be thought of as a resource for certain tasks, and
how can it be quantified. A particularly prominent example of this is the theory of quantum
entanglement.
In recent years, a research program has emerged in which researchers are trying to build a
resource theory relevant for thermodynamical and non-equilibrium phenomena at the quantum
and nanoscales. The overarching aim is to find how the perspective given by quantum informa-
tion tools and techniques help us understand new and existing results of the field. In this thesis
we advance these efforts by introducing different kinds of non-equilibrium fluctuations into the
theory: fluctuationsofworkandfluctuationsofstates. Inthisway, wemanagetomergeanum-
ber of ideas from the field of stochastic thermodynamics into the resource-theoretic framework.
We further explore the structure of this resource theory by introducing results in which the
finitenessoftheheatbath’ssizeisconsidered,derivingcorrectionstowell-knownexpressionsre-
garding work extraction. On top of this, we also consider the usefulness within non-equilibrium
thermodynamics of other tools coming from quantum information theory, such as the relative
entropy and the idea of a recovery map. The connection between the recovery map and the
property of quantum detailed balance (and time reversal more generally) is highlighted, and
a bound on the entropy production of Davies maps is derived inspired by recent results on
quantum information.
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List of Publications and Preprints
The work presented in this thesis contains material from the following publications:
A´lvaro M Alhambra, Jonathan Oppenheim, and Christopher Perry. Fluctuating States: What
is the Probability of a Thermodynamical Transition? Physical Review X, 6(4):041016, 2016.
A´lvaro M Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry. Fluctu-
ating Work: From Quantum Thermodynamical Identities to a Second Law Equality Physical
Review X, 6(4):041017, 2016.
A´lvaro M Alhambra and Mischa P Woods. Dynamical maps, quantum detailed balance and the
Petz recovery map. Physical Review A, 96(2):022118, 2017.
Jonathan G Richens, A´lvaro M Alhambra, and Lluis Masanes. Finite-bath corrections to
the second law of thermodynamics. arXiv preprint arXiv:1702.03357, 2017.
Other preprints by the author are:
A´lvaro M Alhambra, Stephanie Wehner, Mark M Wilde, and Mischa P Woods. Work and
reversibility in quantum thermodynamics. arXiv preprint arXiv:1506.08145, 2015.
Matteo Lostaglio, A´lvaro M Alhambra, and Christopher Perry Elementary Thermal Opera-
tions. arXiv preprint arXiv:1607.00394v2, 2016.
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Acknowledgments
Throughout my PhD I have been very lucky to work with and learn from a fantastic group of
people. First of all, I would like to thank my supervisor Jonathan Oppenheim for giving me
the opportunity to come to UCL to do this work, and for being always generous with his time
and his constant stream of ideas.
A very special thanks also goes to Lluis Masanes, who has been a very important source
of support and enlightenment, and is possibly one of the best physicists I know. Many of the
results here would not have happened without my colleague Chris Perry, with whom I have
shared truly endless discussions and boards full of Lorenz curves; working with him has made
this PhD a lot easier. The same goes for Carlo Sparaciari, another dear colleague with whom
I have been lucky to have in the group, in discussions, and in many trips to conferences.
Another big thanks goes to Mischa Woods, with whom I had the pleasure to establish quite
fruitful collaborations. I would also like to thank Jon Richens for all the chats in the office, and
for being a very good colleague.
ManyotherpeoplehavehelpedmegetwhereIamthroughdiscussionsandcollaborations. In
particular: Matteo Lostaglio, Kamil Korzekwa, Felix Leditzky, Toby Cubitt, Joshua Lockhart,
Carlos Gonzalez, Nicole Yunger-Halpern and Joan Camps. I would also like to thank Mark M.
Wilde and Stephanie Wehner for allowing me to collaborate in one of their projects.
I am very grateful for the enormous amount of wonderful people I have crossed paths with
during my time in London. At a personal level these three years have been a really formative
and enjoyable period, and it is all because of them.
This thesis would not have happened without the constant and unconditional support of
my mum and my sister, for whom I don’t have enough words.
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Contents
Introduction 10
0.1 The idea of a (quantum) resource theory . . . . . . . . . . . . . . . . . . . . . . . 13
0.2 What do we expect to learn from this? . . . . . . . . . . . . . . . . . . . . . . . 16
0.3 Where is this thesis placed exactly?. . . . . . . . . . . . . . . . . . . . . . . . . . 18
Guide to the contents of this thesis 20
0.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
0.5 Structure and contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1 Athermality as a quantum resource 23
1.1 The ingredients of the resource theory . . . . . . . . . . . . . . . . . . . . . . . . 24
1.1.1 Thermal states: Understanding the heat bath . . . . . . . . . . . . . . . . 24
1.1.2 Interactions and conservation of energy . . . . . . . . . . . . . . . . . . . 27
1.1.3 Work storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.2 Towards thermomajorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.2.1 Total energy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.2.2 Fully degenerate energy spectrum . . . . . . . . . . . . . . . . . . . . . . 42
1.3 Deterministic work extraction/expenditure. . . . . . . . . . . . . . . . . . . . . . 43
1.3.1 Work extraction in total energy subspaces . . . . . . . . . . . . . . . . . . 43
1.3.2 Work of formation in total energy subspaces . . . . . . . . . . . . . . . . . 45
1.4 The infinite heat bath limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.4.1 Sufficiency of the Renyi divergences . . . . . . . . . . . . . . . . . . . . . 51
1.4.2 Work of transition in the infinite bath limit . . . . . . . . . . . . . . . . . 52
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1.5 Changing the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.6 What about quantum coherence? . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.7 Limitations of thermal operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2 Fluctuations of states 62
2.1 Probability of transition under noisy operations . . . . . . . . . . . . . . . . . . . 63
2.1.1 Non-deterministic transitions . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.1.2 Nonuniformity of transition under noisy operations . . . . . . . . . . . . . 70
2.2 Probability of transition under thermal operations . . . . . . . . . . . . . . . . . 77
2.2.1 Non-deterministic transformations . . . . . . . . . . . . . . . . . . . . . . 78
2.2.2 Bounds on the probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.2.3 Measuring whether the transition occurred under thermal operations . . . 84
2.3 Heralded probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.3.1 Heralded probability with catalysts . . . . . . . . . . . . . . . . . . . . . . 88
2.3.2 Heralded probability for arbitrary quantum states . . . . . . . . . . . . . 88
2.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3 Work fluctuations 92
3.1 Fluctuation theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2 Fluctuating work in thermal operations . . . . . . . . . . . . . . . . . . . . . . . 98
3.2.1 Thermal operations with fluctuating work: definition and properties . . . 99
3.2.2 Arbitrary unitaries and Tasaki-Crooks . . . . . . . . . . . . . . . . . . . . 102
3.3 Quantum identities with an explicit battery . . . . . . . . . . . . . . . . . . . . . 104
3.4 Quasi-classical identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.4.1 Quasiclassical Crooks theorem . . . . . . . . . . . . . . . . . . . . . . . . 114
3.4.2 Connection with fluctuations of states . . . . . . . . . . . . . . . . . . . . 115
3.5 Deterministic work: Landauer erasure revisited . . . . . . . . . . . . . . . . . . . 117
3.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
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4 Convergence to equilibrium and recovery of quantum information 120
4.1 Recovery of quantum information . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.2 Entropy production. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.3 Davies maps and entropy production . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.3.1 Microscopic origin of Davies maps . . . . . . . . . . . . . . . . . . . . . . 127
4.3.2 Davies maps are thermal operations . . . . . . . . . . . . . . . . . . . . . 128
4.4 A bound on the entropy production . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.4.1 Quantum detailed balance and Petz recovery map . . . . . . . . . . . . . 134
4.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
A Technical proofs 144
A.1 Proofs of Lemma 7 on unitaries with translation invariance . . . . . . . . . . . . 144
A.2 Equivalence of definitions of quantum detailed balance . . . . . . . . . . . . . . . 146
A.3 Spohn’s inequality: rate of entropy production . . . . . . . . . . . . . . . . . . . 148
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List of Figures
1.1 Sketch of the bath and a finite-sized region with finite heat capacity. . . . . . . . 26
1.2 Majorization diagram for a given total energy . . . . . . . . . . . . . . . . . . . . 39
1.3 Thermomajorization diagram for a given total energy . . . . . . . . . . . . . . . 40
1.4 Tradeoff between deterministic work and probability of error. . . . . . . . . . . . 48
1.5 Thermomajorization diagrams and allowed transitions . . . . . . . . . . . . . . . 49
1.6 Deterministic work in the thermomajorization diagrams . . . . . . . . . . . . . . 53
2.1 Examples of Lorenz curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.2 Graphical proof of the lower bound of Eq. (2.30). . . . . . . . . . . . . . . . . . . 75
2.3 Graphical proof of the upper bound of Eq. (2.30). . . . . . . . . . . . . . . . . . 76
2.4 Diagrams for the proof of Theorem 9 . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.5 Diagrams for the proof of Theorem 10 . . . . . . . . . . . . . . . . . . . . . . . . 82
2.6 Illustration of the horizontal and vertical monotones, and of definition of p∗. . . . 83
3.1 Illustration of thermal operations with a battery . . . . . . . . . . . . . . . . . . 100
4.1 Example of bound on entropy production. . . . . . . . . . . . . . . . . . . . . . . 138
4.2 Example of tightness of bound on entropy production . . . . . . . . . . . . . . . 141
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I have been asking myself whether natural forms - a tree, a cloud, a river, a stone,
a flower - can be looked at and perceived as messages. Messages - it goes without
saying - which can never be verbalized, and are not particularly addressed to us. Is it
possible to ‘read’ natural appearances as texts? For me there is nothing mystical in
this exercise. It is a gestural exercise, whose aim is to respond to different rhythms
and forms of energy, which I like to imagine as texts from a language that has not
been given to us to read. Yet as I trace the text I physically identify with the thing
I’m drawing and with the limitless, unknown mother tongue in which it is written.
- John Berger in “How To Resist A State Of Forgetfulness”
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