博 士 学 位 論 文

The generalized second law based on
information geometry and its extension to the𝑞-geometry© 情報幾何学に基づく一般化された第二法則の ª­®
再考と𝑞-幾何学への拡張«¬

令和3年 3月
茨城大学大学院 理工学研究科
複雑系システム科学専攻
中村 文美

ABSTRACT

Abstract
Thermodynamics and statistical mechanics, as the name implies, are closely related
to statistics and information theory. In the past, Maxwell’s thought experiment of his
demon and Landauer’s principle showed the direct relationship between information
manipulation and heat flow. With these foundations, energy and information wereconsidered to be equal.
While many studies including thermodynamics of information are still being
conducted, we have attempted to apply information geometry to thermodynamics.
Information geometry provides a geometric understanding of optimization problems
for the space of probabilities. It is recently attracting a lot of attention in thefield of information theory such as artificial intelligence, machine learning, andneuroscience. Due to its potential to provide geometric understanding of statisticalmechanics, it has increasingly become a focus of study.
In this paper, we apply information geometry to the generalized second law of
the maximum work formulation, which is the most fundamental law of thermodynamics. In this application, we introduce a new concept of dimensional divergence
scaled by a parameter such as temperature. We show that the geometric structure of
the generalized second law is obtained by considering it as an optimization problem
for this scaled divergence. Our main result is that scaling the divergence by physicaldimensional quantities significantly changes the geometric structure of the spaceof probabilities. Scaled divergence breaks the invariance of Fisher’s information
matrix given for the original dimensionless divergence. It has, however, a dually flat
structure that fits the maximum work formulation. We also extend this application to
the systems obeying power laws in correlations. In this case, the equilibrium distribution of the system becomes a power distribution such as a Tsallis distribution. The
generalized second law of the maximum 𝑞-work formulation for the Tsallis statistics

1

ABSTRACTis also obtained by considering it as an optimization problem for a dimensional
scaled 𝑞-divergence.
The second law of thermodynamics is originally restricted to transitions between
equilibrium states. The generalized second law of the maximum work formulation isvalid for transitions between nonequilibrium states. The maximum work extractablefrom a nonequilibrium initial state is described by the minimum value of KullbackLeibler (KL) divergence between the nonequilibrium initial distribution of the systemand the corresponding canonical distribution, multiplied by temperature. Thus, themaximum work formulation is reduced to an optimization problem of the scaled KL
divergences. KL divergence plays a central role in information geometry because it
satisfies invariance and is a kind of Bregman divergence. To solve the optimization
problem in information geometry, Amari’s generalized Pythagorean theorem isprovided for a Bregman divergence.
It is well known that Amari’s generalized Pythagorean theorem for three KL
divergences gives the maximum entropy principle. For the maximum work formulation, a temperature parameter provides an energy dimension if we set Boltzmann’sconstant to 1. In applying information geometry to the generalized second law,
we need the divergence scaled by the temperature. We obtain the maximum work
formulation as a consequence of the generalized Pythagorean theorem. The scaled
divergence has dimension, while typical divergences in information geometry are
dimensionless. The metric of the scaled KL divergence breaks the invariance ofFisher’s information matrix of the original KL divergence. In applying the generalized Pythagorean theorem, the temperature line and the isentropic surface areorthogonal for dimensional scaled KL divergences, while the inverse temperatureline and the isoenergetic surface are orthogonal for the original KL divergences.
Temperature and entropy are related to each other by the Legendre transformation,
where the Bregman divergence for the free energy as a convex function leads to thisdimensional divergence in the space of canonical distributions.
2

ABSTRACTWe also discuss the dual structure of the scaled KL divergence. An important
property of information geometry is that divergence is not a mathematical distancein general. The mathematical distance of two points is symmetric for the exchangeof the two points. The divergence of two distributions is generally not symmetricfor the exchange of the two distributions. The divergence exchanged of the two
distributions is the Legendre transformed divergence. This dual structure leads
to the other important scaled divergence. We propose a divergence scaled by the
inverse of the energy that is a dual parameter of the temperature. We expect
this scaled divergence to be useful in considerations of heat engines in contact
with finite high/low-temperature heat baths. These three divergences: the original
dimensionless KL divergence, the KL divergence scaled by temperature, and the KL
divergence scaled by the inverse of energy, have a symmetric structure based on the
dual structure of information geometry. A list of these divergences is provided inthis paper.
Finally, we extend our arguments based on statistical mechanics of exponential
distributions, such as canonical distribution, to the statistical mechanics of power
distributions. In Tsallis’s 𝑞-statistics, we can extend our arguments using AmariOhara’s normalized 𝑞-geometry. Dimensional normalized 𝑞-divergences give us the
maximum 𝑞-work formulation and its geometric structures.

3

ACKNOWLEDGEMENTS

Acknowledgements
I am deeply grateful to my supervisor Professor Hiroshi H. Hasegawa for his guidance, encouragement, and support. He always listened to me and gave me suggestions whenever I had problems. In addition, he was always very caring about myhealth. I also thank Naomi Hasegawa for her advice on English writing.
I would like to express my gratitude to my collaborator, Dr. Dean J. Driebe at theCenter for Complex Quantum Systems, University of Texas. I got a lot of insights
and a deep understanding through discussions with him. I would also like to express
my gratitude to Professor Shun-ichi Amari at RIKEN Brain Science Institute for his
useful suggestions.
In my doctoral degree examination process, I would like to thank Professor
Takehisa Hasegawa, Professor Hiraku Nishimori, Professor Tatsuaki Wada, andProfessor Shinya Watanabe for their encouragement and comments. I also would
like to thank all professors of the mathematics and informatics and mathematics
courses, Hasegawa laboratory’s members, and the administrative staff of Ibaraki
University for their great support.
Finally, I am deeply thankful to my family. They have been a great support
throughout my undergraduate and graduate studies.
This work was supported by a Grant-in-Aid for JSPS Research Fellow (No.
20J13492) from the Japan Society for the Promotion of Science (JSPS).

4

CONTENTS

Contents
1

2

3

Introduction

6

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2

Outlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

The generalized second law

13

2.1

The second law of thermodynamics . . . . . . . . . . . . . . . . . 13

2.2

Generalized second law for a thermally isolated system . . . . . . . 15

2.3

Generalized second law for a system in contact with heat reservoir . 22

Reconsideration of the generalized second law based on informationgeometry

4

27

3.1

The case of cyclic processes . . . . . . . . . . . . . . . . . . . . . 27

3.2

Geometric structure of the scaled KL divergence . . . . . . . . . . . 32

3.3

Structural changes due to scaling divergence . . . . . . . . . . . . . 39

3.4

The case of non-cyclic processes . . . . . . . . . . . . . . . . . . . 42

3.5

Example 1: thermally isolated ideal gas . . . . . . . . . . . . . . . 44

3.6

Example 2: two-level quantum system . . . . . . . . . . . . . . . . 47
3.6.1

A spin-1/2 particle operated by a magnetic field . . . . . . 47

3.6.2

A vector representation of a quantum state . . . . . . . . . . 49

3.6.3

Optimization of the rotation rate . . . . . . . . . . . . . . . 53

Concluding remarks

59

A Extension to the q-geometry62

A.1 Tsallis’s and Amari-Ohara’s 𝑞-divergence . . . . . . . . . . . . . . 62
A.2 maximum 𝑞-work formulation . . . . . . . . . . . . . . . . . . . . 64

5

CONTENTS

A.3 The scaled 𝑞-divergence . . . . . . . . . . . . . . . . . . . . . . . 66
B Non-negativity of the KL divergence

69

C Concavity of W𝐿𝐵 with respect to the temperature

72

D The generalized Pythagorean theorem based on the KL divergence73

References

76

6

1

1

Introduction

1.1

Background

INTRODUCTION

The chief concern of thermodynamics is the interconversion of heat and work.
Whereas the first law of thermodynamics puts these two forms of energy on an equal
footing, the second law expresses a dissymmetry between them. Work can be turned
into heat without restriction, but heat can only be harnessed to do work with limited
efficiency [1, 2].
We have recognized for some time that information processing can play animportant role in a deeper understanding of the second law, an idea further exemplified through Maxwell’s thought experiment [3–5]. By introducing a being whose
sharpened faculties enabled it to perceive and manipulate individual molecules,
Maxwell showed that appropriate operation based on information provides energy.
Landauer [6], Bennett [7], and others [8–10], also pointed out that information processing itself may be accompanied by thermal effects, sometimes in subtle ways. For
example, Landauer’s principle dictates that erasing one bit requires at least 𝑘 T log 2of energy (𝑘: Boltzmann’s constant, T : temperature). These considerations have
led researchers to recognize information as a resource on par with heat and work informulating the thermodynamics of information [11–15].
Thermodynamics and information theory are closely related, even in light ofstatistical mechanics. Statistical mechanics applies both probability theory andmechanics at the microscopic level to the study of thermodynamic behavior of manybody systems. In these cases, the state of the system is represented by a probability
distribution. The study of statistical mechanics was initiated by Boltzmann at theend of the nineteenth century. With his contribution, Maxwell and Gibbs later
established the study of statistical mechanics for the equilibrium state.

7

1

INTRODUCTION

Recently, there has been a gradual emergence of attempts to apply informationgeometry [16, 17], a branch of information theory, to thermodynamics. Informationgeometry is a differential geometric theory for the space of probability distributions.
It has received a lot of attention in a variety of fields, such as machine learning,
neuroscience, and optimal transport problem. It is written in terms of divergencesuch as the Kullback–Leibler (KL) divergence (also known as relative entropy inphysics) in the Riemann geometry. Probability distributions are the points of a
Riemannian manifold and the Fisher information metric provides the Riemannian
metric. KL divergence is not a true metric for distance in mathematics as it is not
symmetric when replaced by two probabilities. Even so, Amari et. al. showed thegeneralized Pythagorean theorem for a "right-angled triangle" of three probability
densities and introduced "projection," an important concept in information geometry.
A unified understanding of nature based on geometry is the ultimate goal in Einstein’s ideas. The general theory of relativity and gauge theory are representative
examples. However, thermal and statistical mechanics, which deals with phenomena
in a many-body system, has not been regarded as a target of this geometrization.
Although Weinhold [18], Ruppeiner [19], Crooks [20, 21], and others [22–24] havediscussed statistical mechanics from the differential geometrical view of point, it
is still incompletely understood. They introduced the concept of "thermodynamic
length", but they only took into account symmetrical distances. Most of their discussions can only be applied to near-equilibrium states or linear-response regimes.
After the development of information geometry, which takes information as theobject of geometry, the geometrical interpretation of thermal and statistical mechanics is now a possibility. A recent study of thermodynamics based on informationgeometry has been made by Ito et. al. [25, 26] in the context of stochastic thermodynamics [27–29]. They showed that in a Markov process, the positivity of entropy
production is given by the optimized KL divergence between a discrete distributionof the system and the backward manifold. The optimization is done by the general8

1

INTRODUCTION

ized Pythagorean theorem and the positivity is valid since KL divergence is always
nonnegative. They also pointed out a relationship between information geometry
and the Glansdorff-Prigogine criterion for stability [30].
In this thesis, we consider a Hamiltonian system. The total information, GibbsShannon (GS) entropy for the whole system, is rigorously conserved in a Hamiltonian system. Change in entropy means loss of information. We can understandhow entropy increases in a Markov process at the level of underlying Hamiltonian
dynamics.
In a thermally isolated Hamiltonian system, we reconsider the maximum work
formulation of the second law generalized to transitions between nonequilibrium
states from the point of view of information geometry [31]. The original second law of thermodynamics is restricted to transitions between two equilibrium
states. Independently of one another, Takara et. al. [32–35] and Esposito and
Van den Broeck [36, 37] proposed the maximum work formulation generalized fora nonequilibrium initial state (nonequilibrium maximum work formulation). The
nonequilibrium maximum work formulation is based on two fundamental properties: entropy conservation in Hamiltonian dynamics and the non-negativity of the
KL divergence in information theory. It is a universal law and valid even for an integrable system. This fact means that it does not have anything to do with dynamical
properties such as relaxation to an equilibrium state.
The generalized second law is stated in terms of the initial and final KL divergences between each nonequilibrium state and corresponding canonical states, aswell as the Helmholtz free energy of the canonical states. The lower bound of work
is given by the KL divergence between the initial state and corresponding canonicalstate. It is intuited as a change of "nonequilibriumness" since KL divergence gives
the difference between the nonequilibrium probability density and the equilibrium
canonical distribution. If the process is cyclic and the initial state is an equilibrium
canonical state, KL divergence becomes zero and no more work can be derived from9

1

INTRODUCTION

the system. This is the original second law.
In this thesis, we discuss the geometrical structure behind this generalized second law based on information geometry, especially the generalized Pythagorean
theorem [31] developed by Amari et. al. [16, 17]. It is well known that the generalized Pythagorean theorem for three KL divergences gives the maximum entropy
principle. For the maximum work formulation, a temperature parameter provides
an energy dimension if we set Boltzmann’s constant to 1. In applying information geometry to the generalized second law, we need the divergence scaled by thetemperature. We obtain the maximum work formulation as a consequence of thegeneralized Pythagorean theorem. The scaled divergence has dimension, while typical divergences in information geometry are dimensionless. The metric of the scaled
KL divergence breaks the invariance of Fisher’s information matrix of the original
KL divergence. However, the scaled KL divergence has a dually flat structure that fits
the maximum work formulation. In applying the generalized Pythagorean theorem,
the temperature line and the isentropic surface are orthogonal for the dimensional
scaled KL divergences, while the inverse temperature line and the isoenergetic surface are orthogonal for the original KL divergences. Temperature and entropy are
related to each other by the Legendre transformation, where the Bregman divergencefor free energy as a convex function leads to this dimensional divergence in the spaceof canonical distributions.
We also discuss the dual structure of the scaled KL divergence. An important
property of information geometry is that divergence is not a mathematical distancein general. The mathematical distance of two points is symmetric for the exchangeof the two points. The divergence of two distributions is generally not symmetricfor the exchange of the two distributions. The divergence exchanged of the two
distributions is the Legendre transformed divergence. This dual structure leads
to the other important scaled divergence. We propose a divergence scaled by the
inverse of the energy that is a dual parameter of the temperature. We expect10

1

INTRODUCTION

this scaled divergence to be useful in considerations of heat engines in contact
with finite high/low-temperature heat baths. These three divergences: the original
dimensionless KL divergence, the KL divergence scaled by temperature, and the KL
divergence scaled by the inverse of energy, have a symmetric structure based on the
dual structure of information geometry. A list of these divergences is provided inthis paper.
Finally, we extend our argument for a system obeying power law such as Tsallis’s
statistics [38–41]. Systems obeying power law appear when there is a long-time correlation between particles and are found in many situations such as thermodynamics,
network science, and economics. Tsallis used parameter 𝑞 and introduced his 𝑞exponential and 𝑞-logarithmic functions, 𝑞-entropy, 𝑞-density, etc. Furthermore,he pointed out the relationship between Tsallis statistics and information geometry [42,43]. The 𝑞-divergence in Tsallis’s statistics and the 𝑞-divergence in information geometry were related by 𝛼 = 1 − 2𝑞. After that, Amari-Ohara introduced the
Bregman divergence, which is defined as Tsallis’s 𝑞-divergence divided by the conformal factor [44, 45]. Since Bregman divergence is a key concept in the structureof information geometry, the generalized Pythagorean theorem for this divergencealso holds. We show that the maximum work formulation is extended to the maximum 𝑞-work formulation. Then we introduce Amari-Ohara’s 𝑞-divergence scaledby temperature. By applying Amari-Ohara’s generalized Pythagorean theorem forscaled Amari-Ohara’s 𝑞-divergence, we obtain the maximum 𝑞-work formulationand its geometrical structure.

1.2

Outlines

In the next section, we start by recalling the maximum work formulation of thegeneralized second law. First, in subsection 2.1, we briefly review the normal
second law for the transition between two equilibrium states. This thermodynamic
11

1

INTRODUCTION

formulation of the second law has the advantage of being stated just in terms ofenergetic quantities: the work and the change in the Helmholtz free energy. It saysthat for a system in contact with a heat reservoir the work that can be extracted fromthe system is at most the decrease in its Helmholtz free energy. In subsection 2.2,
we give a derivation of the generalized second law for a thermally isolated system
and a system in contact with a heat reservoir. To consider transitions between
nonequilibrium states we go to the finer level of description of the system provided
by statistical mechanics. The nonequilibrium state is described by a distributionfunction in classical mechanics or a density matrix in quantum mechanics. In thelast subsection, we consider the generalized second law for a system in contact witha heat reservoir.
In Sections 3, we reconsider the generalized second law based on informationgeometry. In the first subsection, we consider the case of a cyclic operation. Since
there is no change in the Helmholtz free energy for a cyclic operation, the work is
written only by the KL divergences scaled by temperature. "Information distance"measured by the KL divergence is dimensionless. We need energy dimensional"distance" to measure extractable work from a nonequilibrium state in the generalized second law. This energy dimensional "distance," hereafter referred to as"thermodynamic distance," is the KL divergence scaled by a temperature.
The work is bounded from below by the scaled KL divergence between a nonequilibrium initial state and a canonical state. The canonical state is parameterized bytemperature. The maximum work is determined by minimizing the "thermodynamic
distance." Amari’s generalized Pythagorean theorem for the scaled KL divergences
gives orthogonality between the one-parameter line of canonical states and the isentropic surface of state space. The maximum work is given by the canonical state thatintersects the surface, with the temperature determined by the isentropic condition.
In subsection 3.2, we briefly explain the basic concepts of information geometry
such as Bregman divergence and the proof of Amari’s generalized Pythagorean
12

1

INTRODUCTION

theorem. Bregman divergence is an important type of divergence which leads tothe dually flat structure. We show that the scaled KL divergence is a new kind of
Bregman divergence derived from using the free energy as a convex function. In
subsection 3.3, we discuss the scaling of the KL divergence between more generalexponential distributions. We find that the temperature scaling that appears in thegeneralized second law was a natural scaling for the canonical distribution. We alsofind that the scaling with the inverse of the energy appears naturally as well. These
three divergences have a symmetrical relationship.
We extend our arguments to the case of a general non-cyclic operation in subsection 3.4. We also illustrate a concrete simple example of Amari’s generalized
Pythagorean theorem for the scaled KL divergences using the adiabatic expansion of
the ideal gas in subsection 3.5. Finally, in subsection 3.6, we apply the geometricalinterpretation of the generalized maximum work formulation to a simple two-level
quantum system. The optimal cyclic operation to extract work from a nonequilibrium state is determined by minimalizing the scaled KL divergence between the final
state and the final canonical state. The geometrical interpretation of the generalized
maximum work formulation gives us a systematic method to figure out a protocol torealize the optimal operation.
In Appendix A, a geometrical interpretation of thermodynamics for Tsallis statistics is presented. The discussion is completely parallel to the generalized second law
based on the KL divergence, although all functions have been extended by parameter
𝑞. Other details of the calculations that were not mentioned in the main text are also
supplemented in the appendices.
This thesis relies on references [31, 35]. Several statements, equations, andfigures are mentioned here without reference to these references.

13

2

THE GENERALIZED SECOND LAW

2The generalized second law

2.1

The second law of thermodynamics

We start with the maximum work formulation of the second law for transitions
between equilibrium states. Consider a system in thermal contact with a heat
reservoir at temperature T and able to perform work through the change of externalparameters. This is the ordinary situation considered in elementary thermodynamics.
The changing parameter performing work is, for example, a movable wall of acontainer [46] or a magnetic field interacting with the system of interest [47].
The first law of thermodynamics for our system of interest undergoing a processis
Δ𝐸 = 𝑊 + 𝑄,

(2.1)

where Δ𝐸 ≡ 𝐸𝑇 − 𝐸 0 is the change in internal energy of the system from its initialstate at time 𝑡 = 0 to its final state at time 𝑡 = 𝑇, 𝑊 is the work done on the system
and 𝑄 is the heat added to the system from the reservoir. Our interest is in work
extraction from the system so we rewrite the first law as
𝑊 = Δ𝐸 − 𝑄.

(2.2)

Since a positive value for 𝑊 means that energy is added to the system, extraction ofenergy in the form of work occurs when 𝑊 is negative.
A convenient thermodynamic potential to use for a system in contact with a heat
reservoir is the Helmholtz free energy, which is given by
𝐹 = 𝐸 − T 𝑆,

(2.3)

where 𝑆 is the thermodynamic entropy of the system and T is the absolute tem14

2

THE GENERALIZED SECOND LAWperature. The change in the free energy during the process we are consideringis
Δ𝐹 = Δ𝐸 − T Δ𝑆.

(2.4)

Using this in our expression Eq. (2.2) for the work gives
𝑊 = Δ𝐹 + T Δ𝑆 − 𝑄.

(2.5)

Now, we bring in the second law of thermodynamics in the form of the Clausius
inequality
T Δ𝑆 ≥ 𝑄,(2.6)

which, when used in Eq. (2.5), gives us the work inequality
𝑊 ≥ Δ𝐹.

(2.7)

This is the maximum work formulation of the second law. It tells us that the mostwork that can be extracted from the system in a transition between two equilibrium
states is the decrease in the free energy of the system. Most work is obtained in adissipationless process.
For the case of a cyclic transition, when the system returns to its initial configuration so that the free energy returns to its initial value, the work inequality
becomes𝑊 ≥ 0.

(2.8)

This tells us that we can’t get work out of a single heat reservoir by a process that
returns the system to its original configuration, i.e., thermodynamical considerations
prohibit the construction of a perpetual motion machine (of the second kind).
Perhaps the simplest application of the maximum work formulation of the second
law is to an ideal gas, in contact with a heat reservoir at temperature T , in a container
15

2

THE GENERALIZED SECOND LAWwith a movable wall, such as a piston, that performs work. This is a standard example
considered in all textbooks but we briefly recall it here for reference. Work is done by
the system when the container expands and the most work is done in a dissipationlessquasi-static expansion. The work, in this case, is given by∫ 𝑉𝑇
𝑊𝑞𝑠 = −∫ 𝑉𝑇
𝑝𝑑𝑉 = −𝑁 𝑘T

𝑉0

𝑉0

𝑉𝑇
𝑑𝑉
= −𝑁 𝑘T log ,
𝑉
𝑉0

(2.9)

where 𝑉𝑡 is the volume at time 𝑡, 𝑝 is the pressure, 𝑁 is the number of particles
and 𝑘 is the Boltzmann’s constant. We also used the equation of state for the ideal
gas: 𝑝𝑉 = 𝑁 𝑘T . Since the internal energy of the ideal gas doesn’t change when its
temperature remains the same, the work done is equal to the heat received from thebath. The heat flow is responsible for the entropy increase of the gas given by
Δ𝑆 =

𝑊𝑞𝑠
𝑄
=−
.
T
T

(2.10)

So, the change in the Helmholtz free energy of the gas is precisely the work done by
the gas found in Eq. (2.9) above,
Δ𝐹 = Δ𝐸 − T Δ𝑆 = 𝑊𝑞𝑠 .

(2.11)

Since the expansion is dissipationless the work inequality (2.7) is saturated.

2.2

Generalized second law for a thermally isolated system

For the generalization of the second law, we need to go beyond the purely thermodynamical description in the previous subsection to a statistical mechanics description.
We first consider a thermally isolated Hamiltonian system and focus on the periodof the operation starting at 𝑡 = 0 and finishing at 𝑡 = 𝑇. The state is describedusing the probability density, 𝜌(𝑥, 𝑡) (or 𝜌𝑡 in abbreviated form), at time 𝑡 and16

2

THE GENERALIZED SECOND LAW

phase-space point 𝑥, which is either specified or is obtained by dynamical evolution
from a previously specified state. The dynamics of the system is governed by its
Hamiltonian, 𝐻 (𝑥, 𝜅(𝑡)) (abbreviated as 𝐻𝑡 ), that has an explicit time dependence
due to the external parameters, 𝜅(𝑡), under our control. Hereafter we abbreviate afunction of time 𝐴(𝑡) as 𝐴𝑡 .
The time evolution of the state is written using the time evolution operator 𝑈 as𝜌𝑇 = 𝑈 𝑇 𝜌0 .

(2.12)

where 𝑈 𝑇 is defined as

𝑈 = 𝑇ˆ exp −i



∫ 𝑇

𝑇

𝐿 (𝜅(𝑡))𝑑𝑡 ,(2.13)

0where 𝑇ˆ is the time-ordered product and 𝐿 is the Liouvillian for the Hamiltonian.
We employ the bracket notation to describe the expectation value of an observable
𝐴(𝑥) as,∫h𝐴|𝜌i =𝐴∗ (𝑥) 𝜌(𝑥)𝑑𝑥,(2.14)

Γ

where Γ is the phase space of the Hamiltonian system and 𝐴∗ is the (transposed)
complex conjugate of 𝐴. We employ the bracket notation for a quantum system
as h𝐴|𝜌i = Tr[ 𝐴𝜌]. This quantum version will be used later when dealing withconcrete examples of a simple two-level quantum system.
Thus the expectation value of the Hamiltonian of the system at time 𝑡 is given
by h𝐻𝑡 |𝜌𝑡 i. The work done on this thermally isolated system is given just by thechange in the internal energy of the system, i.e.,
𝑊 = 𝐸𝑇 − 𝐸 017(2.15)

2

THE GENERALIZED SECOND LAW

where the internal energy of the system at time 𝑡 is
𝐸 𝑡 = h𝐻𝑡 |𝜌𝑡 i.

(2.16)

The canonical distribution of the system with respect to the Hamiltonian andwith a parameter 𝛼 will play an important role:𝑒 −𝛼𝐻.
𝜌can (𝛼) =𝑍 (𝛼)(2.17)

𝑍 (𝛼) = 1|𝑒 −𝛼𝐻 .

(2.18)

Here, the partition function is

We are going to rewrite by using its relation to the Helmholtz free energy
𝐹 (𝛼) = −𝛼−1 log 𝑍 (𝛼),(2.19)

so that the canonical distribution may be expressed simply as the exponential function
𝜌can (𝛼) = 𝑒 𝛼{𝐹 (𝛼)−𝐻} .

(2.20)We also will employ the Gibbs–Shannon (GS) entropy of the system as𝑆[𝜌] = −hlog 𝜌|𝜌i

(2.21)

where we set the Boltzmann constant 𝑘 = 1 to make the thermodynamic entropy
compatible with the dimensionless entropy in information theory.
The cross entropy appearing in information theory is defined as
𝑆𝐶 [𝜌 𝐴 , 𝜌 𝐵 ] = −hlog 𝜌 𝐵 |𝜌 𝐴 i.
18

(2.22)

2

THE GENERALIZED SECOND LAW

Specifically, the cross entropy between a probability density and a canonical distribution is given as,
𝑆𝐶 [𝜌, 𝜌can (𝛼)] = −𝛼𝐹 (𝛼) + 𝛼𝐸

(2.23)

where we used the property of a normalized distribution, < 1|𝜌 >= 1.
Of chief importance for the generalized second law is the KL divergence between
two states, 𝜌 𝐴 and 𝜌 𝐵 , given by𝜌𝐴
𝐷 𝐾 𝐿 [𝜌 𝐴 k 𝜌 𝐵 ] ≡ log
𝜌 𝐴 = hlog 𝜌 𝐴 |𝜌 𝐴 i − hlog 𝜌 𝐵 |𝜌 𝐴 i
𝜌𝐵(2.24)= −𝑆[𝜌 𝐴 ] + 𝑆𝐶 [𝜌 𝐴 , 𝜌 𝐵 ].

(2.25)The KL divergence measures the "distance" between the two states. Its most important property in the derivation of the generalized second law will be its nonnegativity:
𝐷 𝐾 𝐿 [𝜌 𝐴 k 𝜌 𝐵 ] ≥ 0.

(2.26)The KL divergence is zero only when 𝜌 𝐴 = 𝜌 𝐵 . See Appendix B for proofs of thesefacts. In Appendix B, we also show the non-negativity of KL divergence in thequantum case.
The basic quantitative relationship we need to obtain the generalized second law
emerges when we take the KL divergence of an arbitrary state, 𝜌, with respect to a
canonical state, 𝜌can (𝛼). This gives
𝐷 𝐾 𝐿 [𝜌 k 𝜌can (𝛼)] = hlog 𝜌|𝜌i − hlog 𝜌can (𝛼)|𝜌i(2.27)= −𝑆[𝜌] − h𝛼(𝐹 (𝛼) − 𝐻)|𝜌i.

(2.28)

Now, 𝜌 is a normalized distribution and 𝐹 (𝛼) does not depend on the phase space

19

2

THE GENERALIZED SECOND LAW

variables so it has nothing to integrate. Thus,
𝐷 𝐾 𝐿 [𝜌 k 𝜌can (𝛼)] = −𝑆[𝜌] − 𝛼𝐹 (𝛼) + 𝛼𝐸.

(2.29)The KL divergence of an arbitrary state with a canonical state gives us a sum ofterms involving the entropy of the arbitrary state, the free energy of the canonicalstate and the internal energy of the arbitrary state.
Using the above relation, the internal energy of the state at time 𝑡 is written as
𝐸 𝑡 = 𝐹𝑡 (𝛼) + 𝛼−1 𝑆[𝜌𝑡 ] + 𝛼−1 𝐷 𝐾 𝐿 [𝜌𝑡 k 𝜌can,𝑡 (𝛼)].

(2.30)

Since the work is given by the change in the internal energy as in Eq. (2.15), we
obtain𝑊 = Δ𝐹 (𝛼) + 𝛼−1 𝐷 𝐾 𝐿 [𝜌𝑇 k 𝜌can,𝑇 (𝛼)] − 𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 (𝛼)](2.31)

where we used the conservation of the GS entropy in a thermally isolated Hamiltonian system, i.e., 𝑆[𝜌𝑇 ] = 𝑆[𝜌0 ]; and Δ𝐹 (𝛼) ≡ 𝐹𝑇 (𝛼) − 𝐹0 (𝛼) is the change inthe Helmholtz free energy of the system. Since there is no heat bath and we do
not have an a priori temperature, we call the parameter corresponding to an inverse
temperature as 𝛼.
The work equality, Eq. (2.31), is just a statement of energy accounting when
we know the initial and final state of the system as well as the initial and final
Hamiltonians. Let us suppose that a nonequilibrium initial state is just given andassociated Hamiltonian with external parameters are under our control. Until wechoose a schedule for the parameter control of the Hamiltonian the final state is
not determined. Due to the non-negativity of the KL divergence term involving the
unknown final state, Eq. (2.31) gives us under the condition of a known initial state20

2

THE GENERALIZED SECOND LAW

the following work inequality:
𝑊 ≥ Δ𝐹 (𝛼) − 𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 (𝛼)] ≡ W𝐿𝐵 (𝛼)(2.32)

where we denote the lower bound for the work as W𝐿𝐵 (𝛼).
First, we note that the inequality (2.32) is true for any value of 𝛼. Second,
W𝐿𝐵 (𝛼) is a concave function of 𝛼−1 , which means that it has a global maximum
as a function of 𝛼 as discussed in Appendix C. Thus, there exists a best value for𝛼; namely, the value for which W𝐿𝐵 (𝛼) is a maximum. This is so because themaximum of W𝐿𝐵 (𝛼) is a value for the work that may be achieved. A lesser value
for W𝐿𝐵 (𝛼) (i.e., a more negative value when considering work extraction) while
giving a true inequality would not be an achievable value for the work since thisvalue would not satisfy the inequality of 𝑊 ≥ max.[W𝐿𝐵 (𝛼)], which must be true.
To find the maximum of W𝐿𝐵 (𝛼) we make its dependence on 𝛼 explicit bywriting the KL divergence, as in Eq. (2.29), in terms of the entropy, free energy andinternal energy as
W𝐿𝐵 (𝛼) = 𝐹𝑇 (𝛼) + 𝛼−1 𝑆[𝜌0 ] − 𝐸 0 .

(2.33)

Now, taking the derivative with respect to 𝛼, and using the fact that the derivativeof the free energy with respect to temperature is minus the entropy gives us
11
𝜕W𝐿𝐵 (𝛼)= 2 𝑆[𝜌can,𝑇 (𝛼)] − 2 𝑆[𝜌0 ].
𝜕𝛼𝛼𝛼(2.34)The value of 𝛼 = 𝛽˜ where this equals zero we call the effective (inverse) temperature.
This gives us the isentropic condition of˜ = 𝑆[𝜌0 ].
𝑆[𝜌can,𝑇 ( 𝛽)](2.35)

So, the effective temperature is determined as the temperature of the canonical21

2

THE GENERALIZED SECOND LAW

distribution with respect to the final Hamiltonian that has the same entropy as theinitial nonequilibrium state. This condition is entirely reasonable as the dynamics
preserves the entropy and the final distribution being canonical means that the
maximum energy permissible has been extracted. The isentropic condition is also
naturally derived from Amari’s generalized Pythagorean theorem in informationgeometry, which will be discussed in the next section.
The generalized second law for a nonequilibrium initial state in a thermallyisolated system is thus,˜ − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)],˜𝑊 ≥ Δ𝐹 ( 𝛽)(2.36)

where the effective temperature, 𝛽˜−1 , is determined by the isentropic condition,
Eq. (2.35).
If the Hamiltonian is changed back to its original form, as in a cyclic process,then Δ𝐹 = 0 and the work on the system from the initial nonequilibrium state is just˜
𝑊 ≥ − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)].

(2.37)

In both cases the work extracted from the system is −𝑊.
The KL divergence in the right-hand-side of Eq. (2.37) measures the "information distance" corresponding to the available informational resource. The effective
temperature converts the available informational resource to the extractable work.
Considering the Kelvin principle in the context of a thermally isolated system leads
us to regard a canonical state as an equilibrium state for such a system as well [48].
At the end of this subsection, we comment on the physical depiction of work
equality, Eq. (2.31) and consider the generalized second law to formulate a protocol
for work extraction. To extract maximum work as in Eq. (2.36) or Eq. (2.37), it
is best that the final state of the system is in equilibrium. The relative entropy of22

2

THE GENERALIZED SECOND LAW

a nonequilibrium initial state with the canonical state corresponding to the initial
Hamiltonian of the system quantifies the maximum extractable work from such anonequilibrium state as seen in Eq. (2.37). We know in general that dissipation is associated with nonequilibrium states so in order to get the maximum extractable work
we should immediately stop the evolution of the nonequilibrium state and make it anequilibrium state for a new Hamiltonian. This new Hamiltonian should be reachable
from the initial Hamiltonian through the time-dependent external parameters thatare under our control. Once the state is stabilized and made an equilibrium state aprocess (in general quasi-static) is performed by changing the external parametersand putting the system into its final state while work is extracted. In general work
extraction may occur in either or both of the stabilization and restoration processes.
With such an operation, entropy is conserved as in Eq. (2.35) and maximum work
is achieved as in Eq. (2.36) or Eq. (2.37).

2.3

Generalized second law for a system in contact with heat
reservoir

Based on the generalized second law for thermally isolated systems, we derive thegeneralized second law for the case where the system is in contact with a heat
reservoir with inverse temperature 𝛽. First, we consider the whole system as athermally isolated system and divide it into two subsystems. The isothermal system
in contact with a heat reservoir is then realized by taking the thermodynamic limit
for one of the two subsystems.
Consider a situation where a finite system of coupled with a finite heat reservoir.
We represent functions related to the system of interest and heat reservoir withsuperscript (𝑆) and (𝑅), respectively. They are interacting and exchanging energy.
The state is described using the joint probability, 𝜌𝑡 (𝑋) = 𝜌𝑡 (𝑥, 𝑦) at time 𝑡 where
the phase-space point is 𝑋 = (𝑥, 𝑦), 𝑥 is a point with respect to the system and 𝑦 is23

2

THE GENERALIZED SECOND LAW

0<𝑡<𝑇

𝑡≤0

𝑇≤𝑡

energySystem(𝑆)

Reservoir(𝑅)

System(𝑆)

System(𝑆)

Reservoir(𝑅)

Reservoir(𝑅)

𝐻 ! 𝑥, 𝑦 = 0

𝐻 ! 𝑥, 𝑦 ≠ 0

𝐻 ! 𝑥, 𝑦 = 0

The two systems are far apart.

The two systems are interacting.

The two systems are far apart again.

Figure 1: Image of a situation where a system and a heat reservoir interact eachother. The interaction occurs during 0 < 𝑡 < 𝑇, and the rest of the time they are farapart.
a point with respect to the heat reservoir. The total Hamiltonian consists of threeparts:𝐻𝑡 (𝑋) = 𝐻 (𝑋, 𝜅(𝑡))= 𝐻 (𝑆) (𝑥, 𝜅 𝑆 (𝑡)) + 𝐻 (𝑅) (𝑦, 𝜅 𝑅 (𝑡)) + 𝐻 (𝐼) (𝑥, 𝑦)= 𝐻𝑡(𝑆) (𝑥) + 𝐻𝑡(𝑅) (𝑦) + 𝐻 (𝐼) (𝑥, 𝑦)
where 𝐻𝑡(𝑆) (𝑥)= 𝐻 (𝑆) (𝑥, 𝜅𝑆 (𝑡))

and 𝐻𝑡(𝑅) (𝑦)= 𝐻 (𝑅) (𝑦, 𝜅(2.38)
𝑅 (𝑡))

are the Hamil-

tonian for the system and reservoir respectively, and 𝐻 (𝐼) (𝑥, 𝑦) is the interaction
Hamiltonian between them. We assume that the interaction occurs during 0 < 𝑡 < 𝑇.
So the interaction Hamiltonian is

𝐻 (𝐼) (𝑥, 𝑦)




= 0


(𝑡 ≤ 0, 𝑇 ≤ 𝑡)



≠ 0


(0 < 𝑡 < 𝑇)

as described in Figure 1.

24(2.39)

2

THE GENERALIZED SECOND LAW

The two systems are far apart and independent at time 𝑡 ≤ 0, so the initial joint
probability of the whole system is written as a product of two marginal distributions:𝜌0 (𝑥, 𝑦) = 𝜌0(𝑆) (𝑥) 𝜌0(𝑅) (𝑦).

(2.40)The initial and final canonical distributions are also written in a simple product since
their Hamiltonians are additive,
𝜌can,0/𝑇 (𝛼) = exp 𝛼 𝐹0/𝑇 (𝛼) − 𝐻0/𝑇
o
n 
(𝑅)
(𝑆)
(𝑅)
(𝑆)
− 𝐻0/𝑇(𝛼) − 𝐻0/𝑇(𝛼) + 𝐹0/𝑇
= exp 𝛼 𝐹0/𝑇
n 
o
n 
o
(𝑆)
(𝑆)
(𝑅)
(𝑅)
= exp 𝛼 𝐹0/𝑇 (𝛼) − 𝐻0/𝑇 exp 𝛼 𝐹0/𝑇 (𝛼) − 𝐻0/𝑇
(𝑆)(𝑅)
= 𝜌can,0/𝑇(𝛼) 𝜌can,0/𝑇(𝛼),(2.41)

where 0/𝑇 means 0 or 𝑇 and we used the additivity of free energy 𝐹0/𝑇 (𝛼) =
(𝑆)
(𝑅)
𝐹0/𝑇
(𝛼)+𝐹0/𝑇(𝛼) which is also easily derived from the additivity of the Hamiltonian.

The final state, however, are dependent. We only decompose it as𝜌𝑇 (𝑥, 𝑦) = 𝜌𝑇(𝑆) (𝑥) 𝜌𝑇(𝑅) (𝑦|𝑥).

(2.42)

Since the whole system is assumed to be a thermally isolated system, we apply thegeneralized maximum work formulation (2.36) for it under the isentropic condition(2.35). Independency of Eqs. (2.40) and (2.41) gives us
˜ − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)]˜𝑊 ≥ Δ𝐹 ( 𝛽)˜ + Δ𝐹 (𝑅) ( 𝛽)˜ − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌 (𝑅) k 𝜌 (𝑅) ( 𝛽)]˜˜ − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌 (𝑆) k 𝜌 (𝑆) ( 𝛽)]= Δ𝐹 (𝑆) ( 𝛽)
0
can,0
0
can,0
(2.43)

25

2

THE GENERALIZED SECOND LAW

and the isentropic condition is(𝑆)
˜ + 𝑆[𝜌 (𝑅) ( 𝛽)]˜ = 𝑆[𝜌 (𝑆) ] + 𝑆[𝜌 (𝑅) ].
𝑆[𝜌can,𝑇
( 𝛽)]
can,𝑇
0
0

(2.44)

We assume that the heat reservoir is initially canonical distribution with a temperature 𝛽, i.e.,(𝑅)(𝛽).
𝜌0(𝑅) = 𝜌can

(2.45)

Eq. (2.43) is transformed using Eqs. (2.44) and (2.45) as follows,(𝑆)˜ k 𝜌can,𝑇 (𝛽)].
(𝛽)]+𝛽−1 𝐷 𝐾 𝐿 [𝜌can,𝑇 ( 𝛽)𝑊 ≥ Δ𝐹 (𝑆) (𝛽)+Δ𝐹 (𝑅) (𝛽)−𝛽−1 𝐷 𝐾 𝐿 [𝜌0(𝑆) k 𝜌can,0(2.46)
We put(𝑆)
𝑊𝑂(𝑆) = −Δ𝐹 (𝑆) (𝛽) + 𝛽−1 𝐷 𝐾 𝐿 [𝜌0(𝑆) k 𝜌can,0(𝛽)](2.47)

which is a quantity that only depends on the system, and assume that no operations
are performed on the reservoir (i.e. there is no change in the Hamiltonian of thereservoir) and that Δ𝐹 (𝑅) (𝛽) = 0. Then we have˜ k 𝜌can,𝑇 (𝛽)].
𝑊 ≥ −𝑊𝑂(𝑆) + 𝛽−1 𝐷 𝐾 𝐿 [𝜌can,𝑇 ( 𝛽)(2.48)

Finally, we consider the thermodynamic limit where the heat reservoir is infinitely
large. The effective temperature becomes the temperature of reservoir, 𝛽˜ = 𝛽 inthis case. It is justified by the isentropic condition (2.44) and the fact that entropy
is a quantitative quantity and that entropy is a monotonic function of temperature.
Thus, the KL divergence in Eq. (2.48) vanishes and we obtain the maximum work
formulation for the isothermal process as(𝑆)
𝑊 ≥ Δ𝐹 (𝑆) (𝛽) − 𝛽−1 𝐷 𝐾 𝐿 [𝜌0(𝑆) k 𝜌can,0
(𝛽)].

26

(2.49)

2

THE GENERALIZED SECOND LAW

The difference from Eq. (2.36) in the thermally isolated system is that the effective
temperature has been changed to the temperature of the heat reservoir.
The right-hand-side of Eq. (2.49) is the maximum work that can be extracted
when we are given only the nonequilibrium initial state of the system. It is again
generally obtained by instantaneous stabilization, which stops the relaxation of
the system and eliminates the temperature difference with the heat reservoir, andby quasi-static operations, which conserve entropy. If the initial distribution of
the system is a canonical distribution with the same temperature as the reservoir,(𝑆)(𝛽), the extractable work is the difference in free energy of the system,
𝜌0(𝑆) = 𝜌can,0

reproducing Kelvin’s principle.
In this section, we have shown the derivation of the generalized second law
extended to transitions between nonequilibrium states. It was derived from twobasic properties: the conservation of GS entropy in Hamiltonian dynamics and thenon-negativity of KL divergence. It is important to note that the maximum workthat can be extracted from the system is determined by the "nonequilibriumness"measured by the KL divergence. In the next section, we will consider the geometricalstructure of this generalized second law on the basis of the geometry of the KLdivergence given by information geometry.

27

3

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

Reconsideration of the generalized second
law based on information geometry

3.1

The case of cyclic processes

As shown in Eq. (2.31), the extractable work for a thermally isolated system is given
as the difference between two KL divergences scaled by a temperature 𝛼−1 (and thechange in the Helmholtz free energy). This work equality suggests that there is an
information-geometric foundation of the generalized second law. In this subsection,
we consider a cyclic operation because this case has a clearer geometric structure
than the case of a non-cyclic operation, which we will consider in subsection 3.4.
Information geometry considers the geometric structure in parameter space offamilies of probability distributions [16, 17]. A parameter (vector) 𝜃 specifies the
distribution 𝜌(𝜃) and the notion of "distance" between two distributions is provided
by a divergence function 𝐷 [𝜌(𝜃 𝐴 ) k 𝜌(𝜃 𝐵 )], which satisfies the following three
conditions:
1. Non-negativity: 𝐷 [𝜌(𝜃 𝐴 ) k 𝜌(𝜃 𝐵 )] ≥ 0.
2. Uniqueness: 𝐷 [𝜌(𝜃 𝐴 ) k 𝜌(𝜃 𝐵 )] = 0, if and only if 𝜃 𝐴 = 𝜃 𝐵 .
3. Positive-definiteness of the metric tensor 𝑔𝑖 𝑗 :𝐷 [𝜌(𝜃) k 𝜌(𝜃 + 𝑑𝜃)] =Õ(𝑔𝑖 𝑗 /2)𝑑𝜃 𝑖 𝑑𝜃 𝑗 .

The geometric structure is determined by the metric tensor appearing in the above
local divergence. The KL divergence satisfies the above conditions and its metric28

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

tensor is the well-known Fisher information matrix,
𝑔𝑖 𝑗 = h𝜕 log 𝜌(𝜃) 𝜕 log 𝜌(𝜃)|𝜌(𝜃)i.
𝜕𝜃 𝑖
𝜕𝜃 𝑗

(3.1)

Now we define the key concept, "thermodynamic distance." The "thermodynamic
distance" is the KL divergence between an arbitrary probability distribution anda canonical distribution scaled by the temperature of the canonical distribution,
𝛼−1 𝐷 𝐾 𝐿 [𝜌 k 𝜌can (𝛼)]. A divergence scaled by a positive parameter satisfies theabove three conditions so that the scaled divergence is also a divergence. This scaled
KL divergence has an energy dimension and measures the extractable work in thegeneralized second law as a "thermodynamic distance." The metric tensor of thescaled KL divergence is the Fisher information matrix scaled by the temperature.
The different metric tensor gives us a completely different geometric structure as
will be shown in the next subsection.
The maximum extractable work for a cyclic process is determined by the following inequality obtained from Eq. (2.32),
𝑊 ≥ −𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 (𝛼)](3.2)

where we used Δ𝐹 (𝛼) = 0 and the non-negativity of the KL divergence involving thefinal distribution. In order to find the lower bound of 𝑊 in Eq. (3.2) (i.e., the upperbound for the extractable work), we have to find the value of 𝛼 that makes the righthand-side of Eq. (3.2) maximum. In the previous section we derived the conditionof the effective temperature by differentiating the right-hand-side of Eq. (3.2). Now
we use the information-geometric structure to derive the same condition.
The minimum (or shortest) distance from a point to a plane is obtained by the
Pythagorean theorem in elementary geometry. Similarly, the minimum divergencefrom a probability distribution to a curved surface is obtained by the generalized29

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

Pythagorean theorem in information geometry. Hereafter, for the sake of brevity, we
will write it as GPT. The GPT is based on three divergences. Suppose that a point 𝑃
is a probability distribution and S is a curved surface which does not include point
𝑃 in the space of probability distribution. When there exists a point 𝑄 within S suchthat the geodesic line from 𝑃 to 𝑄 is orthogonal to S as illustrated in Figure 2, theGPT holds as
𝐷 [𝑃 k 𝑅] = 𝐷 [𝑃 k 𝑄] + 𝐷 [𝑄 k 𝑅]

for ∀ 𝑅 ∈ S.

(3.3)

From the non-negativity of the divergence, we obtain
𝐷 [𝑃 k 𝑅] ≥ 𝐷 [𝑃 k 𝑄]

for ∀ 𝑅 ∈ S(3.4)

which means that the divergence between 𝑃 and 𝑄 is minimum [16, 17].

#![# ∥ %]S![# ∥ ']

<latexit sha1_base64="4nJNPlzN8M9mnW4GD/Fe9S2os0Q=">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</latexit><latexit%'![' ∥ %]

Figure 2: The image of the GPT.
The GPT based on three bare KL divergences has been well studied in information

30

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

geometry. The validity condition of the GPT based on three bare KL divergencesis the isoenergetic condition. It determines the minimum KL divergence from aprobability distribution to a set of canonical distributions (one-parameter geodesic
line of canonical distribution). The principle of maximum entropy is derived fromthe GPT based on three bare KL divergences. We show the details in Appendix D.
Since the (information) entropy is dimensionless and the principle of maximum entropy is based on dimensionless KL divergences, we expect that the maximum work formulation is based on energy-dimensional divergences. Our energydimensional divergence is equal to (the minus of) the right-hand-side of Eq. (3.2)
for the pair of the initial probability distribution and the initial canonical distribution. The change of scaled KL divergences is the (dissipative) work in Eq. (2.31)
for a cyclic process. The minimization of this scaled KL divergence means the
maximization of the work in Eq. (3.2).
The GPT based on scaled KL divergences holds as the following theorem:
Theorem 1. If the isentropic condition is satisfied, i.e.,˜ = 𝑆[𝜌0 ],𝑆[𝜌can,0 ( 𝛽)](3.5)

then the GPT holds,−1˜
˜ k 𝜌can,0 (𝛼)].
𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 (𝛼)] = 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)]+𝛼
𝐷 𝐾 𝐿 [𝜌can,0 ( 𝛽)

(3.6)

31

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

Proof. The derivation is straightforward,
˜ − 𝛼−1 𝐷 𝐾 𝐿 [𝜌can,0 ( 𝛽)˜ k 𝜌can,0 (𝛼)]𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 (𝛼)] − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)]˜ − h𝐻0 |𝜌0 i= −𝛼−1 𝑆[𝜌0 ] − 𝐹0 (𝛼) + h𝐻0 |𝜌0 i + 𝛽˜−1 𝑆[𝜌0 ] + 𝐹0 ( 𝛽)˜ + 𝐹0 (𝛼) − h𝐻0 |𝜌can,0 ( 𝛽)i˜+ 𝛼−1 𝑆[𝜌can,0 ( 𝛽)]˜ + 𝛼−1 𝑆[𝜌can,0 ( 𝛽)]˜ − h𝐻0 |𝜌can,0 ( 𝛽)i.
˜
= −𝛼−1 𝑆[𝜌0 ] + 𝛽˜−1 𝑆[𝜌0 ] + 𝐹0 ( 𝛽)(3.7)

Substituting the following relation into Eq. (3.7),
˜ − h𝐻0 |𝜌can,0 ( 𝛽)i˜ = − 𝛽˜−1 𝑆[𝜌can,0 ( 𝛽)],˜𝐹0 ( 𝛽)(3.8)

we obtain
˜ − 𝛼−1 𝐷 𝐾 𝐿 [𝜌can,0 ( 𝛽)˜ k 𝜌can,0 (𝛼)]𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 (𝛼)] − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)]˜ − 𝛽˜−1 𝑆[𝜌can,0 ( 𝛽)]˜= −𝛼−1 𝑆[𝜌0 ] + 𝛽˜−1 𝑆[𝜌0 ] + 𝛼−1 𝑆[𝜌can,0 ( 𝛽)]˜= 𝛽˜−1 − 𝛼−1 𝑆[𝜌0 ] − 𝑆[𝜌can,0 ( 𝛽)]
= 0,

(3.9)

where we used the isentropic condition (3.5) in the last line.



From Eq. (3.6) and the non-negativity of the scaled KL divergence, we immediately obtain the following inequality,˜−𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 (𝛼)] ≤ − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)].

(3.10)

Since Eq. (3.2) is valid for any 𝛼, the right-hand-side of Eq. (3.10) provides thegreatest lower bound of 𝑊 consistent with the isentropic condition (3.5); hence, its
negative gives the maximum extractable work from the system.
The geometric image of the GPT based on scaled KL divergences is illustrated
32

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

in Figure 3. The set of canonical distributions is drawn as a line parametrized by
˜ on the isentropic
the temperature. The (geodesic) line connecting 𝜌0 and 𝜌can,0 ( 𝛽)surface is orthogonal to the parametric line of canonical distributions in terms of thescaled KL divergence. The difference in structure between using dimensionless KL
divergences and scaled KL divergences, in particular the change of metric and thedually flat structure with Bregman divergence in information geometry, are discussed
in detail in subsection 3.2 and 3.3.
Isentropic
surface!"( ,- ./0 [!" ∥ !#$%," ( ]+* ,- ./0 [!" ∥ !#$%," +* ]!#$%," (()*!#$%," (+)

Set of canonicaldistribution( ,- ./0 [!#$%," +* ∥ !#$%," (()]

Figure 3: The image of the GPT based on the scaled KL divergences.

3.2

Geometric structure of the scaled KL divergence

In this subsection, we show the new geometric structure that scaled KL divergenceleads to. First, we briefly explain the Bregman divergence and the dually flat
structure in information geometry. The KL divergence is a Bregman divergence with−𝛼𝐹 (𝛼) as a convex function and its Riemannian metric is the Fisher information
matrix. On the other hand, we show that the scaled KL divergence is the Bregman
divergence with −𝐹 (𝛼) as a convex function. The Riemannian metric is also scaled33

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

by temperature and the way of taking the coordinates in the parameter space haschanged.
The space of interest in information geometry is a statistical manifold constructedfrom probability distributions. It is assumed that a point in 𝑛-dimensional manifold
is represented using 𝑛-dimensional coordinates. One example of such a manifold isa family of one-dimensional Gaussian distributions. Since the Gaussian distribution
is uniquely determined by the mean 𝜇 and the variance 𝜎 2 as(𝑥 − 𝜇) 2,
exp −𝑝(𝑥; 𝜇, 𝜎 ) = √
2𝜎 2
2𝜋𝜎 2
2



1

(3.11)

the family of Gaussian distributions can be regarded as a two-dimensional manifold
if we take (𝜇, 𝜎 2 ) as the coordinate system.
In information geometry, the exponential family plays an important role. The
exponential family consists of exponential distributions in the form of
𝜌(𝑥; θ) = exp {θ · H (𝑥) − 𝜓(θ)}(3.12)

θ = (𝜃 (0) , 𝜃 (1) , · · · , 𝜃 (𝑛−1) ),(3.13)

H (𝑥) = (H (0) (𝑥), H (1) (𝑥), · · · , H (𝑛−1) (𝑥)),(3.14)

whereand 𝜓(θ) is the potential function determined by the normalization condition as
𝜓(θ) = log h1| exp (θ · H )i .

(3.15)The parameter that identifies the distribution is θ. Assuming that −H (0) is the
Hamiltonian of the system, 𝐻, then the parameter θcan = (𝛼, 0, · · · , 0) identifies the34

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

canonical distributions
𝜌(𝑥; θcan ) = exp {𝛼(𝐹 (𝛼) − 𝐻 (𝑥))} = 𝜌can (𝑥; 𝛼)(3.16)

where the potential function and the free energy are connected by the following
relationship: 𝜓(θcan ) = −𝛼𝐹 (𝛼).
We then introduce a Bregman divergence between two exponential distributions.
It is derived naturally from the convex function and we use 𝜓(θ) as the convex
function. The convexity of 𝜓(θ) is guaranteed by the fact that its Hessian is apositive definite matrix because a simple calculation shows that
D


E
𝜕2
(𝑖)
(𝑖)
( 𝑗)
( 𝑗)
𝜓(θ)
=
H
−
hH
|𝜌(θ)iH−hH|𝜌(θ)i|𝜌(θ)
𝜕𝜃 (𝑖) 𝜕𝜃 ( 𝑗)(3.17)

which means that the Hessian is a variance-covariance matrix and it is positive
definite in general. The Bregman divergence is defined as follows:
𝐷 [θ 𝐴 k θ𝐵 ] = 𝜓(θ 𝐴 ) − 𝜓(θ𝐵 ) − ∇𝜓(θ𝐵 ) · (θ 𝐴 − θ𝐵 ).
where we note that ∇ ≡𝜕/𝜕𝜃 (0) , 𝜕/𝜕𝜃 (1) , · · · , 𝜕/𝜕𝜃 (𝑛−1)(3.18). The intuitive image of

Bregman divergence in the one-dimensional case is shown in Figure 4. The Bregman
divergence from 𝜌(θ 𝐴 ) to 𝜌(θ𝐵 ) is defined by how far the potential function at θ 𝐴
is from the tangent plane at θ𝐵 . Eq. (3.18) is guaranteed to be always positive due tothe convexity of 𝜓, and it also satisfies all three conditions of divergence written at
the beginning of the previous subsection. Referring only to condition (3), the metrictensor 𝑔𝑖, 𝑗 corresponds to the Hessian of 𝜓. Eq. (3.17) is another way of writing the
Fisher information matrix, which is a metric tensor of KL divergence.
The Bregman divergence leads to a dual structure. Legendre transformationon the previous coordinates θ with a convex function 𝜓 gives us a new coordinate35

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY𝑧𝑧 = 𝜓(𝜽)𝜓 𝜽"𝐷[𝜽" ∥ 𝜽! ]

𝜓 𝜽! + 𝛻𝜓 𝜽! ⋅ (𝜽" − 𝜽! )

𝜽!

𝜽"

𝜽

Figure 4: One-dimensional picture of the Bregman divergence derived from theconvex function 𝜓(θ) and its tangent plane.
system θ ∗ as
θ ∗ = ∇𝜓(θ).

(3.19)

This dual coordinate θ ∗ is often referred to as η in information geometry. By
defining dual convex function for θ ∗ as
𝜓 ∗ (θ ∗ ) = max {θ · θ ∗ − 𝜓(θ)} ,θ(3.20)

we obtain two coordinate systems. These are transformed to and from each other byLegendre transformation:θ ∗ = ∇𝜓(θ),

𝜓 ∗ (θ ∗ ) = θ · θ ∗ − 𝜓(θ)(3.21)

θ = ∇𝜓 ∗ (θ ∗ ),𝜓(θ) = θ ∗ · θ − 𝜓 ∗ (θ ∗ )(3.22)The dual Bregman divergence is derived from the dual coordinate θ ∗ and dual

36

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

convex function 𝜓 ∗ ,𝐷 ∗ [θ ∗𝐴 k θ𝐵∗ ] = 𝜓 ∗ (θ ∗𝐴 ) − 𝜓 ∗ (θ𝐵∗ ) − ∇𝜓 ∗ (θ𝐵∗ ) · (θ ∗𝐴 − θ𝐵∗ )= θ 𝐴 · θ ∗𝐴 − 𝜓(θ 𝐴 ) − θ𝐵 · θ𝐵∗ + 𝜓(θ𝐵 ) − θ𝐵 · (θ ∗𝐴 − θ𝐵∗ )= θ 𝐴 · ∇𝜓(θ 𝐴 ) − 𝜓(θ 𝐴 ) + 𝜓(θ𝐵 ) − θ𝐵 · ∇𝜓(θ 𝐴 )= 𝜓(θ𝐵 ) − 𝜓(θ 𝐴 ) − ∇𝜓(θ 𝐴 ) · (θ𝐵 − θ 𝐴 )= 𝐷 [θ𝐵 k θ 𝐴 ].

(3.23)We have mentioned that there is no symmetry in divergence, but when we consider
the dual coordinate, the dual structure is established: a Bregman divergence with adifferent order of two arguments becomes a dual Bregman divergence
𝐷 ∗ [θ ∗𝐴 k θ𝐵∗ ] = 𝐷 [θ𝐵 k θ 𝐴 ],which is the most interesting and important result of information geometry.
Next, we give a proof of the generalized Pythagorean theorem (GPT) and its
geometric structure based on the Bregman divergence. Suppose there exist a flat
coordinate system (the affine coordinate system) θ. We refer to the curved line
θ(𝑢) = a𝑢 + b,

(3.24)

as geodesic line, where a and b are the constant vectors and 𝑢 is the real parameter.
For the dual coordinates θ ∗ , we also refer to the curved lineθ ∗ (𝑢) = c𝑢 + d,

(3.25)as dual geodesic line, where c and d are the constant vectors. Of course, the geodesic
line and dual geodesic line are different and they are not straight lines from the point37

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

of view of different coordinate systems. Two geodesic lines (or dual geodesic lines)
are orthogonal if their tangent vectors are orthogonal to each other.
The GPT
𝐷 [𝑃 k 𝑅] = 𝐷 [𝑃 k 𝑄] + 𝐷 [𝑄 k 𝑅],(3.26)

holds when the geodesic line connecting 𝑃 and 𝑄 and the dual geodesic line connecting 𝑄 and 𝑅 are orthogonal. Let the coordinates of points 𝑃, 𝑄 and 𝑅 be θ𝑃 ,
θ𝑄 and θ 𝑅 respectively, and using the definition of Bregman divergence (3.18), we
obtain
𝐷 [𝑃 k 𝑄] + 𝐷 [𝑄 k 𝑅] − 𝐷 [𝑃 k 𝑅] = (θ𝑃 − θ𝑄 ) · (θ 𝑅∗ − θ𝑄∗ ).

(3.27)The geodesic line connecting 𝑃 and 𝑄 is written asθ(𝑢) = 𝑢θ𝑃 + (1 − 𝑢)θ𝑄 ,(3.28)

𝜕
θ(𝑢) = θ𝑃 − θ𝑄 .
𝜕𝑢(3.29)

andThe dual geodesic line connecting 𝑄 and 𝑅 is also written asθ ∗ (𝑢) = (1 − 𝑢)θ𝑄∗ + 𝑢θ 𝑅∗(3.30)𝜕 ∗θ (𝑢) = θ 𝑅∗ − θ𝑄∗ .
𝜕𝑢

(3.31)

andSince these are orthogonal we get(θ𝑃 − θ𝑄 ) · (θ 𝑅∗ − θ𝑄∗ ) = 0(3.32)

which means that the right-hand side of Eq. (3.27) is zero. We note that the dot
38

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

product in Eq. (3.32) cannot be defined without the Fisher metric 𝑔𝑖 𝑗 mentioned
earlier since it is an inner product on a statistical manifold which is a Riemannian
manifold.
It is already known that the KL divergence is the Bregman divergence of the
exponential family, where the coordinate system is θ and the convex function is𝜓(θ). The dual coordinate isθ ∗ = ∇𝜓(θ) = −hH |𝜌(θ)i,(3.33)and the dual convex function is𝜓 ∗ (θ ∗ ) = −𝑆[𝜌(θ)].

(3.34)The scaled KL divergence introduced in the previous subsection is also derived
as a Bregman divergence when considering two canonical distributions. In thiscase, we take (the minus of) the free energy −𝐹 (𝛼) of the canonical distribution
as a convex function and 𝛼−1 as the coordinate system instead of 𝛼(= −𝜃 (0) ). The
Bregman divergence is as follows:−1−1−1
𝐷 [𝛼−1𝐴 k 𝛼 𝐵 ] = −𝐹 (𝛼 𝐴 ) + 𝐹 (𝛼 𝐵 ) + ∇𝛼−1 𝐹 (𝛼 𝐵 ) · (𝛼 𝐴 − 𝛼 𝐵 )𝐵−1= −𝐹 (𝛼 𝐴 ) + 𝐹 (𝛼𝐵 ) − 𝑆[𝜌can (𝛼𝐵 )] (𝛼−1𝐴 − 𝛼𝐵 )= −𝐹 (𝛼 𝐴 ) + h𝐻|𝜌can (𝛼𝐵 )i − 𝛼−1
𝐴 𝑆[𝜌can (𝛼 𝐵 )]−1
= −𝛼−1
𝐴 hlog 𝜌can (𝛼 𝐴 )|𝜌can (𝛼 𝐵 )i + 𝛼 𝐴 hlog 𝜌can (𝛼 𝐵 )|𝜌can (𝛼 𝐵 )i

= 𝛼−1
𝐴 𝐷 𝐾 𝐿 [𝜌can (𝛼 𝐵 ) k 𝜌can (𝛼 𝐴 )](3.35)The right-hand-side of Eq. (3.35) is the scaled KL divergence itself. The dual coordinate is entropy 𝑆[𝜌can (𝛼)] and the dual convex function is energy 𝐸 = h𝐻|𝜌can (𝛼)i.
Therefore, the condition of the GPT in Eq. (3.9) is in the form of a product of39

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

difference of temperature and difference of entropy. This is a completely different
geometrical structure to that of the bare KL divergence as you can see by comparing
Figure 3 and Figure 12 in Appendix D.
The differences in structure is seen from considering local KL divergence for acanonical distribution. In the case of bare KL divergence,
1𝑑𝛼Δ𝐸 2 𝑑𝛼
2
1 𝑑 (−𝐸)
𝑑𝛼
= 𝑑𝛼
2
𝑑𝛼
1
= 𝑑𝛼𝑑 (−𝐸)2

𝐷 𝐾 𝐿 [𝜌can (𝛼) k 𝜌can (𝛼 + 𝑑𝛼)] =(3.36)

where Δ𝐸 2 is the Fisher information matrix. On the other hand, the scaled local KL
divergence is also written as
1 Δ𝐸 2
𝑑𝛼
𝑑𝛼
2
𝛼
1
𝑑𝑆
= 𝑑 (𝛼−1 ) 𝑑𝛼2𝑑𝛼1
= 𝑑 (𝛼−1 )𝑑𝑆.
2

𝛼−1 𝐷 𝐾 𝐿 [𝜌can (𝛼) k 𝜌can (𝛼 + 𝑑𝛼)] =(3.37)The dual orthogonality, 𝑑𝛼𝑑 (−𝐸) is changed to 𝑑 (𝛼−1 )𝑑𝑆 by the scale transformation.

3.3

Structural changes due to scaling divergence

More generally, we consider the divergence in the form of the KL divergence between
𝜌(θ 𝐴 ) and 𝜌(θ𝐵 ) divided by the scalar quantity 𝜆 𝐵 which depends on 𝜌(θ𝐵 ), i.e.,40

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

𝜆−1
𝐵 𝐷 𝐾 𝐿 [𝜌(θ 𝐴 ) k 𝜌(θ 𝐵 )]. The condition under which the GPT holds is1
1
1
𝐷 𝐾 𝐿 [𝜌(θ 𝐴 ) k 𝜌(θ𝐶 )] =𝐷 𝐾 𝐿 [𝜌(θ 𝐴 ) k 𝜌(θ𝐵 )] +𝐷 𝐾 𝐿 [𝜌(θ𝐵 ) k 𝜌(θ𝐶 )]
𝜆𝐶
𝜆𝐵
𝜆𝐶




1
θ𝐵 θ𝐶
1
⇔ (𝑆[𝜌(θ 𝐴 )] − 𝑆[𝜌(θ𝐵 )]) ·−+ (∇𝜓(θ 𝐴 ) − ∇𝜓(θ𝐵 )) ·−=0𝜆 𝐵 𝜆𝐶𝜆 𝐵 𝜆𝐶θ𝐵 θ𝐶=.
(3.38)⇔ 𝑆[𝜌(θ 𝐴 )] = 𝑆[𝜌(θ𝐵 )]and𝜆 𝐵 𝜆𝐶
Fig. 5 shows an image of this situation. The KL divergence scaled by temperature,𝑝(𝜽! )

Isentropic surface
𝑝(𝜽# )𝑝(𝜽" )

𝜽# = 𝜽"𝜆#𝜆"Figure 5: The visual image of Eq. (3.38). The GPT holds when the isentropic
condition and the scaling condition θ𝐵 /𝜆 𝐵 = θ𝐶 /𝜆𝐶 are satisfied.
introduced in subsection 3.1, satisfies this latter condition θ𝐵 /𝜆 𝐵 = θ𝐶 /𝜆𝐶 because
the second argument (distribution) of the divergence was restricted to the canonicaldistribution. Thus, only the isentropic condition is needed. The set of canonicaldistributions is a kind of special subspace that satisfies the condition θ𝐵 /𝜆 𝐵 = θ𝐶 /𝜆𝐶 ,and the scaled KL divergence is one of the examples of this geometric structure.
In the above setup, we scaled by the quantity associated with the second argumentof KL divergence. Now, let us consider scaling with the quantity associated with the41

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

first argument, i.e., 𝜆−1
𝐴 𝐷 𝐾 𝐿 [𝜌(θ 𝐴 ) k 𝜌(θ 𝐵 )]. The condition under which the GPTholds is1
1
1
𝐷 𝐾 𝐿 [𝜌(θ 𝐴 ) k 𝜌(θ𝐶 )] =𝐷 𝐾 𝐿 [𝜌(θ 𝐴 ) k 𝜌(θ𝐵 )] +𝐷 𝐾 𝐿 [𝜌(θ𝐵 ) k 𝜌(θ𝐶 )]
𝜆𝐴
𝜆𝐴
𝜆𝐵

 

1
1
∇𝜓(θ𝐵 ) ∇𝜓(θ 𝐴 )⇔ (𝜓(θ𝐶 ) − 𝜓(θ𝐵 )) ·−+−· (θ𝐶 − θ𝐵 ) = 0
𝜆 𝐴 𝜆𝐵
𝜆𝐵
𝜆𝐴∇𝜓(θ𝐵 ) ∇𝜓(θ 𝐴 )⇔ 𝜓(θ𝐵 ) = 𝜓(θ𝐶 )and=.
(3.39)𝜆𝐵
𝜆𝐴
Fig. 6 shows an image of this situation. If we restrict to the space of canonical𝜌(𝜽! )
isopotential surface∇𝜓 𝜽! =

𝜌(𝜽# )𝜆!
∇𝜓 𝜽"𝜆"𝜌(𝜽" )

Figure 6: The visual image of Eq. (3.39). The GPT holds when the isopotentialcondition and the scaling condition ∇𝜓(θ𝐵 )/𝜆 𝐵 = ∇𝜓(θ 𝐴 )/𝜆 𝐴 are satisfied.
distributions, 𝜓 = −𝛼𝐹 (𝛼) and ∇𝜓 = −𝐸. Therefore, if 𝜌(θ 𝐴 ) and 𝜌(θ𝐵 ) are
canonical distributions and 𝜆 is taken by energy, only the isopotential condition
remains. This is the KL divergence scaled by the inverse of the energy. For thisscaled divergence, the space of canonical distributions is also a special subspace.
Scaling by a quantity related to the first argument of divergence appears when
considering the "cost performance" of a system in contact with a finite heat bath.
"Cost performance" is defined by the work obtained in ratio to the size of the finite42

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

heat bath. Therefore, the KL divergence scaled by 𝐸 −1 may be useful in consideringthe "cost performance" of heat engines in the future.
Table 1 summarizes the dual structure of the three divergences we have obtainedso far. We use these three divergences depending on what we want to keep constant
and what we want to optimize. First, the optimization of KL divergence leads tothe maximum entropy principle under constant energy. Second, the optimization of
KL divergence scaled by 𝛼−1 leads to the maximum work under constant entropy.
Finally, the optimization of KL divergence scaled by 𝐸 −1 leads to the maximum
temperature under constant potential.
Table 1: Dual structure of the three divergences.
convexdual
Bregman divergence coordinates
function
coordinates
KL divergence𝛼−𝛼𝐹 (𝛼)−𝐸
KL divergence
𝛼−1−𝐹 (𝛼)
𝑆
scaled by 𝛼−1
KL divergence
𝐸 −1−𝐸 −1 𝑆𝛼𝐹 (𝛼)
scaled by 𝐸 −1

3.4

dual convex
function−𝑆𝐸
𝛼The case of non-cyclic processes

The maximum work formulation of the generalized second law for a non-cyclic
operation (Δ𝐹 (𝛼) ≠ 0) may also be obtained by the GPT based on the scaled KL
divergences. The work inequality is written as Eq. (2.32),
𝑊 ≥ Δ𝐹 (𝛼) − 𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 (𝛼)].
When 𝛽˜ is the inverse effective temperature determined by the isentropic condition(2.35), i.e.,˜ = 𝑆[𝜌0 ],𝑆[𝜌can,𝑇 ( 𝛽)]433RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

the following inequality holds for any 𝛼,˜ − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)]˜ ≥ Δ𝐹 (𝛼) − 𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 (𝛼)]Δ𝐹 ( 𝛽)(3.40)

which means that the inverse effective temperature maximizes the right-hand-sideof Eq. (2.32).
We prove Eq. (3.40) as follows. First, we rewrite the difference between the twosides of Eq. (3.40) in terms of scaled KL divergences as,
˜ − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)]˜ − Δ𝐹 (𝛼) + 𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 (𝛼)]Δ𝐹 ( 𝛽)˜ − 𝐹0 ( 𝛽)˜ + 𝛽˜−1 𝑆[𝜌0 ] + 𝐹0 ( 𝛽)˜ − h𝐻0 |𝜌0 i= 𝐹𝑇 ( 𝛽)− 𝐹𝑇 (𝛼) + 𝐹0 (𝛼) − 𝛼−1 𝑆[𝜌0 ] − 𝐹0 (𝛼) + h𝐻0 |𝜌0 i˜ 𝑇 ( 𝛽)˜ + 𝛽h𝐻
˜ 𝑇 |𝜌0 i + 𝛼−1 {−𝑆[𝜌0 ] − 𝛼𝐹𝑇 (𝛼) + 𝛼h𝐻𝑇 |𝜌0 i}= − 𝛽˜−1 −𝑆[𝜌0 ] − 𝛽𝐹
˜ + 𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,𝑇 (𝛼)].
= − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,𝑇 ( 𝛽)](3.41)

Then the next theorem holds:
Theorem 2. When the isentropic condition (2.35) is given, the GPT based on thescaled KL divergences holds as−1
˜
˜ k 𝜌can,𝑇 (𝛼)].
𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,𝑇 (𝛼)] = 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,𝑇 ( 𝛽)]+𝛼
𝐷 𝐾 𝐿 [𝜌can,𝑇 ( 𝛽)(3.42)

44

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

Proof. Eq. (3.42) is derived straightforwardly,
˜ − 𝛼−1 𝐷 𝐾 𝐿 [𝜌can,𝑇 ( 𝛽)˜ k 𝜌can,𝑇 (𝛼)]𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,𝑇 (𝛼)] − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,𝑇 ( 𝛽)]˜ − h𝐻𝑇 |𝜌0 i= −𝛼−1 𝑆[𝜌0 ] − 𝐹𝑇 (𝛼) + h𝐻𝑇 |𝜌0 i + 𝛽˜−1 𝑆[𝜌0 ] + 𝐹𝑇 ( 𝛽)˜ + 𝐹𝑇 (𝛼) − h𝐻𝑇 |𝜌can,𝑇 ( 𝛽)i˜+ 𝛼−1 𝑆[𝜌can,𝑇 ( 𝛽)]˜ + 𝛼−1 𝑆[𝜌can,𝑇 ( 𝛽)]˜ − h𝐻𝑇 |𝜌can,𝑇 ( 𝛽)i˜= −𝛼−1 𝑆[𝜌0 ] + 𝛽˜−1 𝑆[𝜌0 ] + 𝐹𝑇 ( 𝛽)˜ − 𝛽˜−1 𝑆[𝜌can,𝑇 ( 𝛽)]˜= −𝛼−1 𝑆[𝜌0 ] + 𝛽˜−1 𝑆[𝜌0 ] + 𝛼−1 𝑆[𝜌can,𝑇 ( 𝛽)]˜= ( 𝛽˜−1 − 𝛼−1 ) 𝑆[𝜌0 ] − 𝑆[𝜌can,𝑇 ( 𝛽)]= 0,(3.43)

where we used the relation,
˜ − h𝐻𝑇 |𝜌can,𝑇 ( 𝛽)i˜ = − 𝛽˜−1 𝑆[𝜌can,𝑇 ( 𝛽)],˜𝐹𝑇 ( 𝛽)and the isentropic condition Eq. (2.35).

(3.44)
The GPT (3.42) is rearranged to yield the right-hand-side of Eq. (3.41),
˜ + 𝛼−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,𝑇 (𝛼)]− 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,𝑇 ( 𝛽)]˜ k 𝜌can,𝑇 (𝛼)] ≥ 0,
= 𝛼−1 𝐷 𝐾 𝐿 [𝜌can,𝑇 ( 𝛽)(3.45)

guaranteeing the non-negativity of the left-hand-side of Eq. (3.41) and the validityof Eq. (3.40).

3.5

Example 1: thermally isolated ideal gas

A simple example that illustrates the GPT is a thermally isolated ideal gas that isconfined to a container with a movable wall. Suppose that initially the volume of gas
is 𝑉𝑖 and the temperature is 𝛽𝑖−1 . The wall is then adiabatically moved, expanding45

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

the volume to 𝑉 𝑓 as work is extracted from the gas. This is a non-cyclic process.
The GPT for scaled KL divergences, in this case, is given by Eq. (3.42). We
describe the initial state and the final canonical state using a Maxwellian velocity
distribution 𝜌Max as 𝜌0 = 𝜌Max (𝛽𝑖 , 𝑉𝑖 ) and 𝜌can,𝑇 (𝛼) = 𝜌Max (𝛼, 𝑉 𝑓 ), where


3𝑁
Ö
𝑝 2𝜈
1
exp −𝛼𝜒𝑉
𝜌Max (𝛼, 𝑉) =𝑍 (𝛼, 𝑉) 𝜈=12𝑚(3.46)

where 𝑁 is the number of particles and 𝑝 𝜈 and 𝑚 are momentum and mass of aparticle, respectively. The indicator function 𝜒𝑉 serves to indicate the configuration
space dependence of the distribution and 𝑍 is the partition function, 3𝑁/2
𝑉 𝑁 2𝜋𝑚
𝑍 (𝛼, 𝑉) =𝑁! ℎ2 𝛼(3.47)

where ℎ represents Planck constant.
The GPT is then written as
𝛼−1 𝐷 𝐾 𝐿 [𝜌Max (𝛽𝑖 , 𝑉𝑖 ) k 𝜌Max (𝛼, 𝑉 𝑓 )]˜ 𝑉 𝑓 )] + 𝛼−1 𝐷 𝐾 𝐿 [𝜌Max ( 𝛽,˜ 𝑉 𝑓 ) k 𝜌Max (𝛼, 𝑉 𝑓 )],= 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌Max (𝛽𝑖 , 𝑉𝑖 ) k 𝜌Max ( 𝛽,(3.48)
which holds for the effective temperature 𝛽˜ determined by the following condition,
𝛽˜ =



𝑉𝑓
𝑉𝑖

 2/3

46

𝛽𝑖 .

(3.49)

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

The proof is straightforward using Eqs. (3.46) and (3.47) as
𝛼−1 𝐷 𝐾 𝐿 [𝜌Max (𝛽𝑖 , 𝑉𝑖 ) k 𝜌Max (𝛼, 𝑉 𝑓 )]˜ 𝑉 𝑓 )] − 𝛼−1 𝐷 𝐾 𝐿 [𝜌Max ( 𝛽,˜ 𝑉 𝑓 ) k 𝜌Max (𝛼, 𝑉 𝑓 )]− 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌Max (𝛽𝑖 , 𝑉𝑖 ) k 𝜌Max ( 𝛽,
!




˜
𝑍
(𝛼,
𝑉
)
𝑍
(
𝛽,
𝑉
)
3
𝑓
𝑓
= 𝛼−1 log
− 𝑁 𝛼−1 − 𝛽𝑖−1 − 𝛽˜−1 log
𝑍 (𝛽𝑖 , 𝑉𝑖 )2
𝑍 (𝛽𝑖 , 𝑉𝑖 )
!


𝑍
(𝛼,
𝑉
)
3  ˜−13 𝑓+ 𝑁 𝛽 − 𝛽𝑖−1 − 𝛼−1 log+ 𝑁 𝛼−1 − 𝛽˜−1
˜ 𝑉𝑓 )2
2
𝑍 ( 𝛽,
˜ 𝑉 𝑓 ) + 𝛽˜−1 log (𝑍 (𝛽𝑖 , 𝑉𝑖 )) + 𝛼−1 log 𝑍 ( 𝛽,˜ 𝑉𝑓 )= −𝛼−1 log (𝑍 (𝛽𝑖 , 𝑉𝑖 )) − 𝛽˜−1 log 𝑍 ( 𝛽,!
˜ 𝑉𝑓 ) 𝑍 ( 𝛽,= log
𝛼−1 − 𝛽˜−1
𝑍 (𝛽𝑖 , 𝑉𝑖 )!

𝑉 𝑓𝑁 𝛽˜−3𝑁/2 
−1
−1
˜
𝛼 −𝛽
= log
𝑉𝑖𝑁 𝛽𝑖−3𝑁/2(
  3𝑁/2 ) 
 𝑁

𝑉𝑓
𝛽˜
−1
−1
˜
(3.50)
− log
𝛼 −𝛽
= log
𝑉𝑖
𝛽𝑖
where we used the relation,𝛼2−1 𝐷 𝐾 𝐿 [𝜌Max (𝛼1 , 𝑉1 ) k 𝜌Max (𝛼2 , 𝑉2 )] = 𝛼2−1 log𝑍 (𝛼2 , 𝑉2 )3 
− 𝑁 𝛼2−1 − 𝛼1−1 .
𝑍 (𝛼1 , 𝑉1 )2
(3.51)In order for Eq. (3.50) to become zero, we must have


𝑉𝑓
log
𝑉𝑖

 3𝑁/2
𝛽˜
− log
=0
𝛽𝑖
  𝑁   3𝑁/2
𝑉𝑓
𝛽˜
⇔
=
𝑉𝑖
𝛽𝑖
𝑁



(3.52)
which leads to the condition (3.49).
47

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

Eq. (3.48) guarantees that the final state of a Maxwellian velocity distribution
with the effective temperature 𝛽˜−1 gives the maximum work as discussed in thissection. The condition of effective temperature (3.49) is the well-known polytropic
process equation. Of course, it can also be derived from the isentropic condition
and the Sackur-Tetrode equation for the entropy of an ideal gas.

3.6

Example 2: two-level quantum system

We apply the GPT to an optimization problem in nonequilibrium statistical mechanics. From the generalized maximum work formulation, the maximum work is
extracted by an operation under which the final state is the canonical state with thefinal Hamiltonian and the effective temperature. The final canonical state with theeffective temperature is the ideally optimal state to extract the maximum work. The
ideally optimal state may be difficult to realize experimentally. It is important tofigure out what is the "closest" (minimum divergence) state from the ideally optimal
state in a set of realizable final states. The "closest" state is the experimentally
optimal state to extract work that satisfies the orthogonality condition.
3.6.1

A spin-1/2 particle operated by a magnetic field

We consider a thermally isolated two-level quantum system as a simple example [34].
The application of nonequilibrium statistical mechanics to few-degrees-of-freedom
quantum systems is a current topic of much interest. Time-dependent two-level
quantum systems often appear in the context of matter-field interactions or nuclear
magnetic resonance. Such systems have been used to demonstrate the validity of thefluctuation relations of modern nonequilibrium statistical mechanics [47, 49].
We consider a spin-1/2 particle embedded in a magnetic field that is controlled48

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY[50]. The Hamiltonian of the system is𝐻𝑡 =ℏ𝜔(cos 𝜙𝑡 𝜎𝑧 + sin 𝜙𝑡 𝜎𝑥 ) ,2(3.53)

where 𝜎𝑖 (𝑖 = 𝑥, 𝑦, 𝑧) are the standard Pauli matrices and we choose the directionof the magnetic field restricted to the 𝑥 − 𝑧 plane with 𝜙𝑡 the rotation-angle of themagnetic field around the 𝑦-axis at time 𝑡. The Hamiltonian is rewritten by using
rotation operators in spin space,𝐻𝑡 =ℏ𝜔 −i𝜎𝑦 𝜙𝑡 /2 i𝜎𝑦 𝜙𝑡 /2
e
𝜎𝑧 e
.
2

(3.54)The eigenvalues of the Hamiltonian are 𝐸 0 = −ℏ𝜔/2 and 𝐸 1 = ℏ𝜔/2, which we
identify as the ground and excited state energies, respectively.
The spin-1/2 particle is externally operated from 𝑡 = 0 to 𝑡 = 𝑇. The operationis represented by the time-dependent angle in the Hamiltonian. We choose a cyclic
operation, 𝜙0 = 0 at 𝑡 = 0 and 𝜙𝑇 = 2𝜋 at 𝑡 = 𝑇. The angle increases from 0 to 2𝜋
monotonically and 𝜙𝜏 = 𝜋 at 𝑡 = 𝜏. The Hamiltonian is proportional to the Paulimatrix 𝜎𝑧 at 𝑡 = 0. It is proportional to −𝜎𝑧 at 𝑡 = 𝜏. When 𝑡 = 𝑇 the Hamiltonian
returns to its original form at 𝑡 = 0.
We discuss how to extract the maximum work from a simple noneqilibrium initialstate such as a pure excited state [32–34]. The maximum work can be obtained by
the cyclic operation including two processes:
1. The short stabilization process for 𝑡 ∈ [0, 𝜏) in which the pure excited state
becomes the ground state by changing 𝐻0 to 𝐻𝜏 = −𝐻0 .
2. The restoration process for 𝑡 ∈ [𝜏, 𝑇) to the original Hamiltonian, 𝐻𝑇 = 𝐻0 ,without any transition to the excited state.
These processes are analogous to those that have been discussed on how to extract49

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

the maximum work from a noneqilibrium initial state in a thermodynamic system[32–34]. The first corresponds to the stabilization process to prevent spontaneous
relaxation in a thermodynamic system. For simplicity we consider here the sudden
process, 𝜏 → 0. The second is generally a quasi-static (isentropic) process without
dissipation. After the sudden process, the initial state does not change and the
Hamiltonian changes as 𝐻0+ = −𝐻0 . The control parameter of our operation is
the rate at which the field is rotated. We will find the optimal angular frequency
to extract work. The solutions for finite frequencies correspond to dissipationlessnon-quasi-static processes.
The Schrödinger equation isiℏℏ𝜔 −i𝜎𝑦 𝜙𝑡 /2 i𝜎𝑦 𝜙𝑡 /2𝜕|𝜓𝑡 i = 𝐻𝑡 |𝜓𝑡 i =e𝜎𝑧 e|𝜓𝑡 i.
𝜕𝑡2(3.55)We choose a simple linear time dependence for the rotation angle,
𝜙𝑡 = 𝜋 + Ω𝑡 for 0 < 𝑡 ≤ 𝑇(3.56)

where Ω = 𝜋/𝑇 is the adjustable angular frequency. Then, the Schrödinger equation
is easily solved by using the rotating amplitude ei𝜎𝑦 𝜙𝑡 /2 |𝜓𝑡 i. The solution is writtenas,|𝜓𝑡 i = 𝑢 𝑡 |𝜓0 i(3.57)

where the time evolution operator 𝑢 𝑡 is given as
𝑢 𝑡 = e−i(𝜋+Ω𝑡)𝜎𝑦 /2 e−i(𝜔𝜎𝑧 −Ω𝜎𝑦 )𝑡/2 ei𝜋𝜎𝑦 /2 .
3.6.2

(3.58)

A vector representation of a quantum state

We introduce a vector representation of a quantum state. This section 3.6.2 preparesfor the optimization of the rotation-angle frequency discussed in the next section50

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

3.6.3. A quantum state is represented as a point vector in parameter space ininformation geometry. The orthogonality condition from the GPT is written interms of these vectors [16, 17].
A quantum state is written as a 2 × 2 density matrix that is expanded as alinear combination of four Hermitian matrixes: the unit matrix 𝐼 and the three Paulimatrixes 𝜎𝑖 (𝑖 = 𝑥, 𝑦, 𝑧). Any state except pure states is written as a positive definite
density matrix,𝜌=1
1
𝐼 − 𝜂ˆ · 𝜎 tanh |𝜂|22(3.59)

where 𝜂 is a three dimensional real vector, 𝜂ˆ = 𝜂/|𝜂| is its unit vector, and 𝜎 =(𝜎𝑥 , 𝜎𝑦 , 𝜎𝑧 ). The condition of total probability is obvious, Tr[𝜌] = 1. The positive
definiteness is guaranteed from the range, | tanh |𝜂|| < 1, since the eigenvalues of𝜌 are (1 ± tanh |𝜂|)/2. A density matrix corresponding to a pure state is positive
semi-definite. We consider the pure state in the limit of | tanh |𝜂|| → 1 for |𝜂| → ∞.
The pure ground state of 𝐻0 is 𝜌 = (𝐼 − 𝜎𝑧 )/2 where 𝜂ˆ = (0, 0, 1) and the pureexcited state is 𝜌 = (𝐼 + 𝜎𝑧 )/2 where 𝜂ˆ = (0, 0, −1). Hereafter, we call the above
representation based on a linear combination of four Hermitian matrixes as the
M-representation.
The parameter vector 𝜂 naturally appears in an exponential-type representation(E-representation),
𝜌=

e−𝜂·𝜎
.
𝑍

(3.60)

where
𝑍 = Tr[e−𝜂·𝜎 ] = 2 cosh |𝜂|.

(3.61)The equivalence between two representations is confirmed by the Euler-like formula,e−𝜂·𝜎 = cosh |𝜂|𝐼 − 𝜂ˆ · 𝜎 sinh |𝜂|.

51

(3.62)

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

We write the nonequilibrium initial state at time 𝑡 = 0 as
𝜌0 =

1
1
𝐼 − 𝜂ˆ0 · 𝜎 tanh |𝜂0 |.
2
2(3.63)We choose 𝜂ˆ0 = (0, − sin 𝜑, − cos 𝜑) (|𝜑| < 𝜋/2) for the nonequilibrium initial state.
From the condition (|𝜑| < 𝜋/2), the initial energy 𝐸 0 = Tr[𝜌0 𝐻0 ] is positive so thatwork is extractable. We chose the 𝑥 component 𝜂ˆ0,𝑥 = 0 for simplicity. We may beable to adjust 𝜂ˆ0,𝑥 by an additional operation around the 𝑦-axis.
From the solution of the Schrödinger equation, the time evolution of the density
matrix is given,𝜌𝑡 = 𝑢 𝑡 𝜌0 𝑢 −𝑡 =1
1
𝐼 − 𝜂ˆ𝑡 · 𝜎 tanh |𝜂𝑡 |,22(3.64)where1
𝜂𝑡 = Tr[𝜎𝑢 𝑡 (𝜂0 · 𝜎)𝑢 −𝑡 ].
2(3.65)We used the orthogonality of Pauli matrixes, Tr[𝜎𝑖 𝜎 𝑗 ] = 2𝛿𝑖, 𝑗 . The norm |𝜂𝑡 | isconstant in time under the unitary time evolution,r|𝜂𝑡 | =

1
Tr[(𝑢 𝑡 (𝜂0 · 𝜎)𝑢 −𝑡 ) 2 ] = |𝜂0 |.
2(3.66)

The final state at time 𝑡 = 𝑇 is written as
𝜌𝑇 =

1
1
𝐼 − 𝜂ˆ𝑇 · 𝜎 tanh |𝜂𝑇 |.
2
2(3.67)

We note that the time period 𝑇 may also be considered as the control parameter inour operation, since the angular frequency Ω = 𝜋/𝑇. We will discuss the optimal
final state to extract work in the section 3.6.3.
The canonical state at time 𝑡 (𝑡 ∈ (0, 𝑇]) with the inverse of the effective

52

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RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

temperature 𝛽˜ is written in E-representation as˜e− 𝛽𝐻𝑡 e−𝜂can,𝑡 ·𝜎𝜌can,𝑡 =
=
𝑍can
𝑍can

(3.68)

˜ 𝑡 ]/2 andwhere 𝜂can,𝑡 = Tr[𝜎 𝛽𝐻
𝑍can = Tr[e

˜ 𝑡
− 𝛽𝐻


˜
𝛽ℏ𝜔
.
] = 2 cosh2

(3.69)The partition function 𝑍can is constant in time, since the trace is preserved under therotation operation. The initial canonical state before the sudden process is the sameas the final canonical state because of the cyclic operation. The initial/final state is
written in E-representation as𝜌can,0/𝑇 =e−𝜂can,0/𝑇 ·𝜎
𝑍can(3.70)

˜
where 0/𝑇 means 0 or 𝑇, |𝜂can,0/𝑇 | = 𝛽ℏ𝜔/2and 𝜂ˆcan,0/𝑇 = (0, 0, 1). Similarly, theinitial/final canonical state is written in M-representation as


˜
1
𝛽ℏ𝜔
1
𝜌can,0/𝑇 = 𝐼 − 𝜂ˆcan,0/𝑇 · 𝜎 tanh.
222

(3.71)

We note that the final canonical state does not depend on the controllable time period
𝑇, since 𝜙𝑇 = 2𝜋 for any 𝑇.
We briefly discuss the isentropic condition to determine the effective temperature,˜ = 𝑆[𝜌0 ]. The entropy of the initial state is calculated by using both E𝑆[𝜌can,𝑇 ( 𝛽)]533RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

and M-representations,𝑆[𝜌0 ] = −Tr[𝜌0 log 𝜌0 ]1
1
= −Tr[( 𝐼 − 𝜂ˆ0 · 𝜎 tanh |𝜂0 |)(− log(2 cosh |𝜂0 |)𝐼 − 𝜂0 · 𝜎)]22= log(2 cosh |𝜂0 |) − |𝜂0 | tanh |𝜂0 |.

(3.72)

Similarly, the entropy of the final canonical state is˜ = log(2 cosh |𝜂can,𝑇 |) − |𝜂can,𝑇 | tanh |𝜂can,𝑇 |𝑆[𝜌can,𝑇 ( 𝛽)](3.73)

˜
where |𝜂can,𝑇 | = 𝛽ℏ𝜔/2.
The isentropic condition is simply rewritten as |𝜂can,𝑇 | =|𝜂0 |. The effective temperature is determined from the initial condition,˜𝛽ℏ𝜔= |𝜂0 |.
2
3.6.3

(3.74)

Optimization of the rotation rate

We consider the quantum mechanical version of the work identity Eq. (2.31) for the
˜effective temperature 𝛽,
˜ − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)]˜𝑊 = 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌𝑇 k 𝜌can,𝑇 ( 𝛽)](3.75)

˜ = 0 for a cyclic operation is used. The quantum relative entropy iswhere Δ𝐹 ( 𝛽)defined as the following quantum-mechanical analog of the KL divergence,
𝐷 𝐾 𝐿 [𝜌 𝐴 k 𝜌 𝐵 ] = hlog 𝜌 𝐴 |𝜌 𝐴 i − hlog 𝜌 𝐵 |𝜌 𝐴 i

54(3.76)

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

where the bra-ket notation is defined ash𝐴|𝐵i = Tr[ 𝐴𝐵].

(3.77)We used the same notation 𝐷 𝐾 𝐿 [𝜌 𝐴 k 𝜌 𝐵 ] for the quantum relative entropy sinceEq. (3.76) is the same as Eq. (2.24).
The extractable work is the negative of the work done on the system,˜−𝑊 = 𝑊max − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌𝑇 k 𝜌can,𝑇 ( 𝛽)](3.78)

where the maximum extractable work is defined as˜ ≥ 0.
𝑊max = 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌0 k 𝜌can,0 ( 𝛽)](3.79)

Now we try to find the optimal operation with the angular frequency Ω∗ = 𝜋/𝑇 ∗ forwhich the final state satisfies the minimum scaled KL divergence,˜
𝑇 ∗ = arg min 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌𝑇 k 𝜌can,0 ( 𝛽)].

(3.80)

𝑇

where we replaced the final canonical state with the initial canonical state to makeclear the absence of 𝑇 dependence of the final canonical state, since 𝐻𝑇 = 𝐻0 for
any 𝑇.
First we consider the validity condition of the GPT Δ = 0 [51], where˜ − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌𝑇 k 𝜌𝑇 ∗ ] − 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌𝑇 ∗ k 𝜌can,0 ( 𝛽)].
˜
Δ = 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌𝑇 k 𝜌can,0 ( 𝛽)](3.81)The KL divergence may be divided into the sum of the negative entropy and thecross entropy,
𝐷 𝐾 𝐿 [𝜌 𝐴 k 𝜌 𝐵 ] = −𝑆[𝜌 𝐴 ] + 𝑆C [𝜌 𝐴 , 𝜌 𝐵 ],55

(3.82)

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

where the cross entropy is
𝑆C [𝜌 𝐴 , 𝜌 𝐵 ] = −Tr[𝜌 𝐴 log 𝜌 𝐵 ].

(3.83)

After substituting the above sum into Δ,˜ − 𝛽˜−1 𝑆C [𝜌𝑇 , 𝜌𝑇 ∗ ]Δ = 𝛽˜−1 𝑆C [𝜌𝑇 , 𝜌can,0 ( 𝛽)]˜+ 𝛽˜−1 𝑆C [𝜌𝑇 ∗ , 𝜌𝑇 ∗ ] − 𝛽˜−1 𝑆C [𝜌𝑇 ∗ , 𝜌can,0 ( 𝛽)].

(3.84)

where we used 𝑆[𝜌𝑇 ∗ ] = 𝑆C [𝜌𝑇 ∗ , 𝜌𝑇 ∗ ]. Similar to the entropy, the cross entropy
between state A and state B is calculated by using both E- and M-representations,
𝑆[𝜌 𝐴 , 𝜌 𝐵 ] = −Tr[𝜌 𝐴 log 𝜌 𝐵 ]





˜
˜
˜
𝛽ℏ𝜔
𝛽ℏ𝜔
𝛽ℏ𝜔
tanh
−
𝜂ˆ 𝐴 · 𝜂ˆ 𝐵
= log 2 cosh222

(3.85)

˜
where we chose |𝜂 𝐴 | = |𝜂 𝐵 | = 𝛽ℏ𝜔/2.
After substituting the above expression of
the cross entropy, we obtain,


˜
ℏ𝜔
𝛽ℏ𝜔
Δ=
tanh
( 𝜂ˆ𝑇 ∗ − 𝜂ˆ𝑇 ) · ( 𝜂ˆcan,0 − 𝜂ˆ𝑇 ∗ ).
2
2

(3.86)

Quantum states are represented as vectors such as 𝜂𝑇 . The validity conditionof the GPT is written as the orthogonality condition ( 𝜂ˆ𝑇 ∗ − 𝜂ˆ𝑇 ) · ( 𝜂ˆcan,0 − 𝜂ˆ𝑇 ∗ ) = 0
in the vector space. In information geometry, a state is represented as a vector
in a space of parameters. The "distance" (square of length in the sense of the
Pythagorean theorem) between two states is measured through the divergence. In
nonequilibrium statistical mechanics, the energy-dimensional "distance" between
two states is measured through the scaled KL divergence.
Suppose 𝜂ˆ𝑇 ∗ = 𝜂ˆcan,0 , then the GPT is valid globally. The vector representation

56

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

of the ideally optimal state to extract the maximum work is 𝜂ˆcan,0 . However, if𝜂ˆ0,𝑦 ≠ 0, then 𝜂ˆ𝑇 cannot be the ideally optimal state 𝜂ˆcan,0 for any 𝑇 in general,since the rotation is limited around the 𝑦-axis. Although any realizable final statescannot be the ideally optimal state, we can figure out what is the "closest" (minimum
divergence) state from the ideally optimal state in a set of realizable final states. The"closest" state is the experimentally optimal state to extract work and satisfies the
local orthogonal condition (see Figure 7).

!
)-./,1 (")"!#$ %&' [)*∗ ∥ )-./,1 "! ]"!#$ %&' [)* ∥ )-./,1 "! ]Tangent space at )*∗

)*∗

)*

"!#$ %&' [)* ∥ )*∗ ]Set of realizable final states˜
Figure 7: The GPT locally holds when the geodesic line connecting 𝜌𝑇 ∗ and 𝜌can,0 ( 𝛽)is orthogonal to the tangent space at 𝜌𝑇 ∗ . For any 𝜌𝑇 in the neighborhood of 𝜌𝑇 ∗ , 𝜌𝑇
is considered in the tangent space of 𝜌𝑇 ∗ .
The validity condition of the local GPT becomes the following local orthogonal condition between the tangent vector at 𝜌𝑇 ∗ and the geodesic line connecting
˜ and 𝜌𝑇 ∗ ,𝜌can,0 ( 𝛽)lim∗

Δ

𝑇→𝑇 𝑇 ∗ − 𝑇∝𝑑 𝜂ˆ𝑇 ∗ ,𝑧𝑑 𝜂ˆ𝑇 ∗𝑑 𝜂ˆ𝑇 ∗· ( 𝜂ˆcan,0 − 𝜂ˆ𝑇 ∗ ) =· 𝜂ˆcan,0 ∝
=0
∗
∗
𝑑𝑇
𝑑𝑇
𝑑𝑇 ∗

57

(3.87)

3

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY

where we used 𝑑 ( 𝜂ˆ𝑇 · 𝜂ˆ𝑇 )/𝑑𝑇 = 𝑑1/𝑑𝑇 = 0 and 𝜂ˆcan,0 = (0, 0, 1). 𝜂ˆ𝑇,𝑧 is written as1𝜂ˆ𝑇,𝑧 = Tr[𝜎𝑧 (𝑢𝑇 𝜂ˆ0 · 𝜎𝑢 −𝑇 )]2(3.88)

𝑢𝑇 = e−i𝜋𝜎𝑦 e−i(𝜔𝑇 𝜎𝑧 −𝜋𝜎𝑦 )/2 ei𝜋𝜎𝑦 /2 .

(3.89)

where

The calculation of 𝑑 𝜂ˆ𝑇,𝑧 /𝑑𝑇 is straightforward but tedious. We show only the final
result,n𝜂ˆ𝑇,𝑧 = 1 − cosp𝜔 2𝑇 2 + 𝜋 2

o𝜋𝜔 2𝑇 2 + 𝜋 2

−𝜔𝑇 𝜂ˆ0,𝑦 + 𝜋 𝜂ˆ0,𝑧 − 𝜂ˆ0,𝑧(3.90)

and
!(
!
√
√
𝑑 𝜂ˆ𝑇,𝑧2𝜋𝜔
𝜔𝑇𝜔 2𝑇 2 + 𝜋 2
𝜔 2𝑇 2 + 𝜋 2
= 2 2
cos
(−𝜔𝑇 𝜂ˆ0,𝑦 + 𝜋 𝜂ˆ0,𝑧 )
sin
√
2
𝑑𝑇
2
2
𝜔 𝑇 +𝜋𝜔 2𝑇 2 + 𝜋 2
!
√
)

2𝜔𝑇
𝜔 2𝑇 2 + 𝜋 2
+ sin
− 2 2
(3.91)−𝜔𝑇 𝜂ˆ0,𝑦 + 𝜋 𝜂ˆ0,𝑧 − 𝜂ˆ0,𝑦2
𝜔 𝑇 + 𝜋2
There are clearly two ways of solving 𝑑 𝜂ˆ𝑇 ∗ ,𝑧 /𝑑𝑇 ∗ = 0 that yield two sets ofp√solutions for 𝑇 ∗ . One way is when sin( 𝜔2𝑇 ∗ 2 + 𝜋 2 /2) = 0 so that 𝜔𝑇 ∗ = 4𝑛2 − 1𝜋
˜(𝑛 = 1, 2, ...). For these 𝑇 ∗ , the scaled KL divergence 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌𝑇 ∗ k 𝜌can,0 ( 𝛽)]takes the constant value since 𝜂ˆ𝑇 ∗ ,𝑧 = −𝜂ˆ0,𝑧 and the divergence only depends on 𝜂ˆ𝑇 ∗ ,𝑧
i.e.,




˜
˜
 ℏ𝜔

ℏ𝜔
𝛽ℏ𝜔
𝛽ℏ𝜔
˜ =
𝛽 𝐷 𝐾 𝐿 [𝜌𝑇 ∗ k 𝜌can,0 ( 𝛽)]tanh
1 − 𝜂ˆ𝑇 ∗ ,𝑧 =tanh
1 + 𝜂ˆ0,𝑧2
2
2
2(3.92)
˜−1This value also corresponds to the quasi-static process of 𝑇 → ∞(Ω → 0) as
illustrated in Figures 8 and 9.
Another set of solutions is obtained when the terms inside the curly brackets in58

RECONSIDERATION OF THE GENERALIZED SECOND LAW BASEDON INFORMATION GEOMETRY"! #$ %&' [)* ∥ ),-.,0 "! ]3

Ω = 4/6

˜ versus Ω(= 𝜋/𝑇)
Figure 8: The scaled KL divergence 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌𝑇 k 𝜌can,0 ( 𝛽)]for 𝜑 = 0 in 𝜂ˆ0 , |𝜂0 | = 1 and 𝜔 = ℏ = 1. The dashed line shows the level of thedivergence
for the quasi-static process, Ω = 0. The local minimums are realized for√2
𝑇 = 4𝑛 − 1𝜋/𝜔 (𝑛 = 1, 2, ...) and the local maximums are realized for the second
series of solutions.
Eq. (3.91) sum to zero. (In this case a simple expression for 𝑇 ∗ is not available.)˜ at these 𝑇 ∗ become local
The scaled KL divergences 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌𝑇 ∗ k 𝜌can,0 ( 𝛽)]maximum or local minimum according to the initial conditions as illustrated in
Figures 8 and 9. For 𝜑 = 0 in 𝜂ˆ0 (see Figure 8), the divergences of the second series
take local maximum. However, for 𝜑 = 𝜋/3 (see Figure 9), the divergence takes
the globally minimum, 0.024792... at Ω∗ = 𝜋/𝑇 ∗ = 0.358859... that is one of thesolutions of the second series. This interesting result tells us that we can extract
more work than the quasi-static process by choosing the optimal parameter Ω∗ in thecase of 𝜂ˆ0,𝑦 ≠ 0 [52]. The local GPT or the generalized projection theorem gives us
candidates of the realizable optimal state that is "closest" to the ideal optimal state.

59

CONCLUDING REMARKS'&() *+, [./ ∥ .123,5 '& ]4

Ω = #/%˜ versus Ω(= 𝜋/𝑇) for
Figure 9: The scaled KL divergence 𝛽˜−1 𝐷 𝐾 𝐿 [𝜌𝑇 k 𝜌can,0 ( 𝛽)]𝜑 = 𝜋/3 in 𝜂ˆ0 , |𝜂0 | = 1 and 𝜔 = ℏ = 1. The dashed line shows the level of thedivergence for the quasi-static process, Ω = 0. The value of the globally minimumis 0.024792... that is much lower than the level of the quasi-static process, Ω = 0.

4

Concluding remarks

In this thesis, we presented the generalized second law and its alternative informationgeometric foundation. The generalized second law for a transition between nonequilibrium states gives a perspective on some situations involving the interplay of energy
and information. We have presented the generalization for both thermally isolated
systems and systems in contact with a heat reservoir. As shown in Section 2, theformer case is more fundamental since a total isolated system may be divided into asystem of interest coupled to a larger system that acts as a reservoir.
In Section 3, we introduced "thermodynamic distance" as an energy-dimensional
KL divergence scaled by temperature. The change in "thermodynamic distance"
gives us the dissipative work. The GPT based on three "thermodynamic distances"
gives us a geometric interpretation of the maximum work formulation of the generalized second law. From the nonequilibrium initial state, the canonical state with the60

4

CONCLUDING REMARKS

effective temperature is the closest "point" in "thermodynamic distance." Since our
new concept of "thermodynamic distance" has an important role in nonequilibrium
statistical mechanics, it can also be called "thermodynamic divergence."
The most important result in this paper is that the geometric structure based on
the KL divergence scaled by a variable temperature is different from that based on thebare KL divergence. Scaling the divergence with variable temperature completelychanges the geometrical structure, such as how to take the coordinate systems andthe orthogonal conditions in the sense of the GPT. Conversely, scaled divergences
appear by taking different coordinates and convex functions. We would have failed
to recognize this result if we had only considered isothermal systems, as manyother studies have done. For isothermal processes with constant temperature T ,multiplying by temperature simply acts as a constant scale factor. It does not change
the geometrical structure. The validity condition of the GPT based on the bare
KL divergences is an isoenergetic condition. On the other hand, for a variabletemperature, the validity condition of the GPT based on the energy-scaled KL
divergences is an isentropic condition.
The geometrical interpretation of the generalized maximum work formulation
gives us a systematic method in figuring out a protocol to realize the optimal operation. In this thesis, we applied the geometrical interpretation of the generalized
maximum work formulation to a simple two-level quantum system. The optimal
cyclic operation to extract work from a nonequilibrium state was determined by minimalizing the scaled KL divergence between the final state and the final canonicalstate.
In Appendix A, we extended the information geometrical interpretation basedon the scaled KL divergence to the case of Tsallis statistics. In the framework of
Tsallis’s 𝑞-functions, Amari-Ohara’s 𝑞-divergence is a Bregman divergence, and theGPT holds. By scaling this 𝑞-divergence with temperature again, a geometricalstructure appears in which temperature and 𝑞-entropy are dual coordinates. From
61

4

CONCLUDING REMARKS

the GPT for the scaled 𝑞-divergence, the maximum 𝑞-work is obtained.
From section 3 onwards, we discussed only adiabatic processes in thermally isolated Hamiltonian systems in order to highlight the information-geometrical foundation of thermodynamics. We excluded any phenomenological assumptions suchas the existence of a heat reservoir when considering its relation to informationgeometry. However, we would still need high and low temperature heat reservoirs toconsider a heat engine [33]. Since the information geometry is flexibly applicablefor any parametrized probability distributions, the scaled KL divergence would play
an important role in the heat engine. One may consider scaling the KL divergenceby a quantity with power dimension, such as temperature divided by time. The GPTbased on this new scaled KL divergence would give us important knowledge for
optimizing the power. We will discuss this problem in the near future.

62

A

EXTENSION TO THE Q-GEOMETRY

A

Extension to the q-geometry

A.1

Tsallis’s and Amari-Ohara’s 𝑞-divergence

In 1988, Tsallis introduced his 𝑞-entropy [38, 40],𝑆˜𝑞 [𝜌] ≡1h𝜌 𝑞−1 − 1|𝜌i.
1−𝑞

(A.1)

The 𝑞-entropy satisfies axioms of Shannon–Khinchin of entropy except the additivity.
The 𝑞-entropy is rewritten as 1˜
𝑆 𝑞 [𝜌] = hlog𝑞
|𝜌i
𝜌

(A.2)

where we used the 𝑞-logarithmic form of Tsallis’s functions for convenience. The
𝑞-logarithmic function is defined as
log𝑞 (𝑥) ≡𝑥 1−𝑞 − 1
,
1−𝑞

(A.3)

and its inverse function, the 𝑞-exponential function, is defined as
1

exp𝑞 (𝑥) ≡ {1 + (1 − 𝑞)𝑥} 1−𝑞 .

(A.4)

In this Appendix, the subscript 𝑞 always means Tsallis’s functions [39, 41]. When𝑞 → 1, they become normal logarithmic and exponential functions.
Tsallis introduced his 𝑞-divergence as 
𝜌𝐵˜
𝐷 𝑞 [𝜌 𝐴 ||𝜌 𝐵 ] ≡ −hlog𝑞|𝜌 𝐴 i,𝜌𝐴(A.5)

which is known to be the same as 𝛼-divergence up to a constant factor where63

A

EXTENSION TO THE Q-GEOMETRY

𝛼 = 1 − 2𝑞 [42, 43]. Amari and Ohara, However, introduced their normalized 𝑞divergence later by the conformal transformation of Tsallis’s one, which led to thedual flat structure as a Bregman divergence. Amari-Ohara’s 𝑞-divergence is definedas1
𝐷˜ 𝑞 [𝜌 𝐴 k 𝜌 𝐵 ]ℎ 𝑞 (𝜌 𝐴 )


1
𝑞 1−𝑞
1 − h𝜌 𝐴 𝜌 𝐵 i=(1 − 𝑞)ℎ 𝑞 (𝜌 𝐴 )𝐷 𝑞 [𝜌 𝐴 k 𝜌 𝐵 ] =(A.6)

where ℎ 𝑞 (𝜌) = h1|𝜌 𝑞 i is a conformal factor. This divergence (A.6) is derived
naturally by considering the 𝑞-exponential family which consists of 𝑞-exponential
distributions in the form of𝜌 𝑞 (𝑥; θ) = exp𝑞 θ · H (𝑥) − 𝜓 𝑞 (θ) .

(A.7)

By taking θ as a coordinate and 𝜓 𝑞 (θ) as a convex function and calculating the
Bregman divergence
𝐷 [θ 𝐴 k θ𝐵 ] = 𝜓 𝑞 (θ 𝐴 ) − 𝜓 𝑞 (θ𝐵 ) − ∇𝜓 𝑞 (θ𝐵 ) · (θ 𝐴 − θ𝐵 ),(A.8)

we obtain Amari-Ohara’s 𝑞-divergence (A.6). Amari-Ohara’s 𝑞-entropy is also
adjusted by the conformal transformation as
𝑆 𝑞 [𝜌] ≡1 ˜𝑆 𝑞 [𝜌].
ℎ 𝑞 (𝜌)

64

(A.9)

A

A.2

EXTENSION TO THE Q-GEOMETRY

maximum 𝑞-work formulation

Using the above framework, we will consider the maximum 𝑞-work formulation.
The 𝑞-work is the work with respect to the escort distribution, 𝜌es , as𝑊𝑞 ≡ h𝐻𝑇 |𝜌es,𝑇 i − h𝐻0 |𝜌es,0 i.

(A.10)The escort density is a probability distribution. It is proportional to the density
raised to the power of 𝑞,
𝜌es (𝑥) ≡𝜌 𝑞 (𝑥)ℎ 𝑞 (𝜌)(A.11)

where ℎ 𝑞 (𝜌) serves as the normalization constant which does not depend on timein Hamiltonian dynamics. By using the escort distribution, Amari-Ohara’s 𝑞divergence and 𝑞-entropy are rewritten respectively as
𝐷 𝑞 [𝜌 𝐴 k 𝜌 𝐵 ] = hlog𝑞 (𝜌 𝐴 )|𝜌es,𝐴 i − hlog𝑞 (𝜌 𝐵 )|𝜌es,𝐴 i(A.12)

𝑆 𝑞 [𝜌] = −hlog𝑞 𝜌|𝜌es i.

(A.13)

andIt should be noted that all expectation values change to those with the escort distribution.
In order to calculate this 𝑞-work, we first check the correspondence between 𝑞canonical distribution and 𝑞-exponential distribution. The 𝑞-canonical distributionis𝜌 𝑞−can (𝛼) =1exp𝑞 (−𝛼𝐻 (𝑥)) ,𝑍 𝑞 (𝛼)65(A.14)

A

EXTENSION TO THE Q-GEOMETRY

which is calculated as follows:1
1{1 + (1 − 𝑞) (−𝛼𝐻 (𝑥))} 1−𝑞
𝑍 𝑞 (𝛼)1no 1−𝑞𝑞−1
𝑞−1= 𝑍 𝑞 (𝛼) + (1 − 𝑞) −𝛼𝑍 𝑞 (𝛼)𝐻 (𝑥)1(!) 1−𝑞
𝑞−1
𝑍
(𝛼)
−
1
𝑞
𝑞−1
= 1 + (1 − 𝑞) −𝛼𝑍 𝑞 (𝛼)𝐻 (𝑥) +
1−𝑞
!
𝑞−1
𝑍 𝑞 (𝛼) − 1
𝑞−1= exp𝑞 −𝛼𝑍 𝑞 (𝛼)𝐻 (𝑥) +1−𝑞

𝜌 𝑞−can (𝑥; 𝛼) == 𝜌 𝑞 (𝑥; θcan ( 𝛼))ˆ(A.15)

where in the last line we put following notations:
H (0) = −𝐻 (𝑥)𝑞−1(A.16)(𝛼)(A.17)

θcan ( 𝛼)ˆ = ( 𝛼,ˆ 0, · · · , 0)

(A.18)𝛼ˆ = 𝛼𝑍 𝑞𝑞−1

1 − 𝑍 𝑞 (𝛼)= − log𝑞 (𝑍 𝑞−1 (𝛼)).
𝜓 𝑞 (θcan ( 𝛼))ˆ =1−𝑞(A.19)

Another way of writing the 𝑞-canonical distribution,𝜌 𝑞−can (𝛼) = exp𝑞 𝛼(𝐹ˆ 𝑞 (𝛼) − 𝐻 (𝑥)) ,(A.20)

gives the 𝑞-free energy as
1−𝑞

𝑞−1

𝑍 𝑞 (𝛼) 𝑍 𝑞 (𝛼) − 1
11
ˆ = log𝑞 (𝑍 𝑞−1 (𝛼)) =𝐹𝑞 (𝛼) = − 𝜓 𝑞 (θcan ( 𝛼))
𝛼ˆ
𝛼ˆ
𝛼
1−𝑞
1−𝑞

1 1 − 𝑍 𝑞 (𝛼)1== − log𝑞 (𝑍 𝑞 (𝛼)).
𝛼
1−𝑞
𝛼

66

(A.21)

A

EXTENSION TO THE Q-GEOMETRY

We then obtain a 𝑞-work identity similar to Eq. (2.31),
𝑊𝑞 = Δ𝐹𝑞 (𝛼) +1
1
𝐷 𝑞 [𝜌𝑇 k 𝜌 𝑞−can,𝑇 (𝛼)] − 𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛼)].
𝛼ˆ𝛼ˆ(A.22)The proof of Eq. (A.22) is given by a straightforward calculation:
𝑊𝑞 = h𝐻𝑇 |𝜌es,𝑇 i − h𝐻0 |𝜌es,0 i= Δ𝐹𝑞 (𝛼) − 𝛼ˆ −1 𝛼ˆ 𝐹𝑞,𝑇 (𝛼) − 𝐻𝑇 |𝜌es,𝑇 + 𝛼ˆ −1 𝛼ˆ 𝐹𝑞,0 (𝛼) − 𝐻0 |𝜌es,0− 𝛼ˆ −1 𝑆 𝑞 [𝜌𝑇 ] + 𝛼ˆ −1 𝑆 𝑞 [𝜌0 ]= Δ𝐹𝑞 (𝛼) − 𝛼ˆ −1 log𝑞 𝜌 𝑞−can,𝑇 (𝛼) |𝜌es,𝑇 + 𝛼ˆ −1 hlog𝑞 (𝜌𝑇 )|𝜌es,𝑇 i+ 𝛼ˆ −1 log𝑞 𝜌 𝑞−can,0 (𝛼) |𝜌es,0 − 𝛼ˆ −1 hlog𝑞 (𝜌0 )|𝜌es,0 i= Δ𝐹𝑞 (𝛼) + 𝛼ˆ −1 𝐷 𝑞 [𝜌𝑇 k 𝜌 𝑞−can,𝑇 (𝛼)] − 𝛼ˆ −1 𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛼)](A.23)

where we used the conservation of 𝑞-entropy in Hamiltonian system.
The non-negativity of Amari-Ohara’s 𝑞-divergence gives the inequality
𝑊𝑞 ≥ Δ𝐹𝑞 (𝛼) −1
𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛼)]𝛼ˆ(A.24)

which holds for any 𝛼. If it is a cyclic operation, Δ𝐹𝑞 = 0, the maximum work is
1
𝑊𝑞 ≥ − 𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛼)] ≡ W𝑞,𝐿𝐵 (𝛼).
𝛼ˆA.3

(A.25)The scaled 𝑞-divergence

Let us now consider cyclic operations for simplicity. Of course, the argument for
non-cyclic operation is essentially the same. We want to find the maximum valueof the right-hand side of Eq. (A.25) to get the maximum work that can be achieved,

67

A

EXTENSION TO THE Q-GEOMETRY

just as we did in Section 2 and 3. Condition for the derivative to be zero, i.e.,𝜕
W𝑞,𝐿𝐵 (𝛼) = 0,𝜕𝛼(A.26)

gives the iso-𝑞-entropic condition
𝑆 𝑞 [𝜌 𝑞−can,0 (𝛽eff )] = 𝑆 𝑞 [𝜌0 ](A.27)

where 𝛽eff is the effective inverse temperature.
We, however, derive this iso-𝑞-entropic condition by using the concept of scaled
𝑞-divergence and its GPT. We introduce the scaled 𝑞-divergence between 𝜌 and𝜌 𝑞−can (𝛼) as 𝛼ˆ −1 𝐷 𝑞 [𝜌 k 𝜌 𝑞−can (𝛼)]. Note that the scaling variable is 𝛼,ˆ not 𝛼. The
minimization of this scaled 𝑞-divergence means the maximization of the 𝑞-work in
Eq. (A.25).
The GPT based on scaled 𝑞-divergences holds as the following theorem:
Theorem 3. If the iso-𝑞-entropic condition is satisfied, i.e.,𝑆 𝑞 [𝜌 𝑞−can,0 (𝛽eff )] = 𝑆 𝑞 [𝜌0 ],then the GPT holds,𝛼ˆ −1 𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛼)]−1= 𝛽ˆeff
𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛽eff )] + 𝛼ˆ −1 𝐷 𝑞 [𝜌 𝑞−can,0 (𝛽eff ) k 𝜌 𝑞−can,0 (𝛼)].

(A.28)68

A

EXTENSION TO THE Q-GEOMETRY

Proof. The derivation is straightforward,𝛼ˆ −1 𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛼)]−1− 𝛽ˆeff
𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛽eff )] − 𝛼ˆ −1 𝐷 𝑞 [𝜌 𝑞−can,0 (𝛽eff ) k 𝜌 𝑞−can,0 (𝛼)]−1= −𝛼ˆ −1 𝑆 𝑞 [𝜌0 ] − 𝐹𝑞,0 (𝛼) + h𝐻0 |𝜌es,0 i + 𝛽ˆeff
𝑆 𝑞 [𝜌0 ] + 𝐹𝑞,0 (𝛽eff ) − h𝐻0 |𝜌es,0 i+ 𝛼ˆ −1 𝑆 𝑞 [𝜌 𝑞−can,0 (𝛽eff )] + 𝐹𝑞,0 (𝛼) − h𝐻0 |𝜌es,𝑞−can,0 (𝛽eff )i−1= −𝛼ˆ −1 𝑆 𝑞 [𝜌0 ] + 𝛽ˆeff
𝑆 𝑞 [𝜌0 ] + 𝐹𝑞,0 (𝛽eff ) + 𝛼ˆ −1 𝑆 𝑞 [𝜌 𝑞−can,0 (𝛽eff )] − h𝐻0 |𝜌es,𝑞−can,0 (𝛽eff )i.

(A.29)
Substituting the following relation into Eq. (A.29),−1
𝐹𝑞,0 (𝛽eff ) − h𝐻0 |𝜌es,𝑞−can,0 (𝛽eff )i = − 𝛽ˆeff
𝑆 𝑞 [𝜌 𝑞−can,0 (𝛽eff )],(A.30)

we obtain𝛼ˆ −1 𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛼)]−1− 𝛽ˆeff
𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛽eff )] − 𝛼ˆ −1 𝐷 𝑞 [𝜌 𝑞−can,0 (𝛽eff ) k 𝜌 𝑞−can,0 (𝛼)]−1−1= −𝛼ˆ −1 𝑆 𝑞 [𝜌0 ] + 𝛽ˆeff
𝑆 𝑞 [𝜌0 ] + 𝛼ˆ −1 𝑆 𝑞 [𝜌 𝑞−can,0 (𝛽eff )] − 𝛽ˆeff
𝑆 𝑞 [𝜌 𝑞−can,0 (𝛽eff )]−1= 𝛽ˆeff− 𝛼ˆ −1 𝑆 𝑞 [𝜌0 ] − 𝑆 𝑞 [𝜌 𝑞−can,0 (𝛽eff )]= 0,(A.31)

where we used the isentropic condition (A.27) in the last line.



From the non-negativity of 𝑞-divergence, Eq. (A.28) gives the following inequality−1𝛼ˆ −1 𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛼)] ≥ 𝛽ˆeff
𝐷 𝑞 [𝜌0 k 𝜌 𝑞−can,0 (𝛽eff )](A.32)

which is valid for any 𝛼. The right-hand-side of Eq. (A.32) provides the greatest
lower bound of 𝑊𝑞 consistent with the iso-𝑞-entropic condition (A.27).
69

B

NON-NEGATIVITY OF THE KL DIVERGENCE

Theorem 3 shows that the theory in the case of 𝑞-divergence is completely
parallel to that of the KL divergence which is discussed in Section 3. This is because
the natural Bregman divergence based on the 𝑞-exponential distribution is AmariOhara’s 𝑞-divergence. As for the geometric structure, we also find that the scaled
𝑞-divergence leads to the different geometric structure from bare 𝑞-divergence.
While the bare 𝑞-divergence is a Bregman divergence with 𝜓 𝑞 = −𝛼𝐹ˆ 𝑞 (𝛼) as aconvex function, the scaled 𝑞-divergence is a Bregman divergence with −𝐹𝑞 (𝛼) as aconvex function. Its coordinate system is 𝛼ˆ −1 and its dual coordinate system is −𝑆 𝑞 .
This difference is shown in Figs.10 and 11. Figure 10 illustrates the GPT for bare𝑞-divergences
𝐷 𝑞 [𝜌 k 𝜌 𝑞−can (𝛼)] = 𝐷 𝑞 [𝜌 k 𝜌 𝑞−can (𝛽)] + 𝐷 𝑞 [𝜌 𝑞−can (𝛽) k 𝜌 𝑞−can (𝛼)](A.33)

which holds under the isoenergetic conditionh𝐻|𝜌es,𝑞−can (𝛽)i = h𝐻|𝜌i(A.34)On the other hand, Figure 11 illustrates the GPT (A.28) for scaled 𝑞-divergence.

B

Non-negativity of the KL divergence

We consider two probability distributions 𝜌(𝑥) and 𝜎(𝑥). The non-negativity of the
KL divergence 𝐷 𝐾 𝐿 [𝜌 k 𝜎] is easy to demonstrate using the elementary inequalityof log 𝑥 ≤ 𝑥 − 1 or − log 𝑥 ≥ 1 − 𝑥. We have
 
 
𝜎
𝜎
|𝜌
𝐷 𝐾 𝐿 [𝜌 k 𝜎] = − log |𝜌 ≥ 1 −𝜌𝜌= h1|𝜌i − h1|𝜎i = 0

70

(B.1)

B

NON-NEGATIVITY OF THE KL DIVERGENCE

Iso-𝑞-energetic
surface𝜌

𝐷! [𝜌 ∥ 𝜌!"#$% (𝛼)]𝐷! [𝜌 ∥ 𝜌!"#$% (𝛽)]𝜌!"#$% (𝛼)𝜌!"#$% (𝛽)Set of 𝑞-canonicaldistribution

𝐷! [𝜌!"#$% (𝛽) ∥ 𝜌!"#$% (𝛼)]Figure 10: The image of the GPT based on the 𝑞-divergences. Subscripts indicatingtime are omitted.
Iso-𝑞-entropic
surface𝜌!
𝛼' #* 𝐷" [𝜌! ∥ 𝜌"#$%&,! 𝛼 ]#*𝛽,())
𝐷" [𝜌! ∥ 𝜌"#$%&,! 𝛽()) ]𝜌"#$%&,! (𝛼)Set of 𝑞-canonicaldistribution

𝛼' #* 𝐷" [𝜌"#$%&,! 𝛽()) ∥ 𝜌"#$%&,! (𝛼)]𝜌"#$%&,! (𝛽()) )Figure 11: The image of the GPT based on the scaled 𝑞-divergences.
Note that equality occurs only when 𝜎(𝑥)/𝜌(𝑥) = 1 for all 𝑥, i.e., when the two
distributions are identical.
For the quantum case involving density matrices 𝜌 and 𝜎 we consider their

71

C

CONCAVITY OF W𝐿𝐵 WITH RESPECT TO THE TEMPERATURE

orthonormal decompositions: 𝜌 =Í

𝑖 𝑝 𝑖 |𝜙𝑖 ih𝜙𝑖 | and 𝜎 =Í𝑗 𝑞 𝑗 |𝜓 𝑗 ih𝜓 𝑗 |. Then wehave
𝐷 𝐾 𝐿 [𝜌 k 𝜎] = Tr [𝜌 log 𝜌] − Tr [𝜌 log 𝜎]ÕÕ Õ=𝑝𝑖 log 𝑝𝑖 −𝑝𝑖
𝑃𝑖 𝑗 log 𝑞 𝑗 ,
𝑖

𝑖

(A.4)

𝑗

Í
where 𝑃𝑖 𝑗 = h𝜙𝑖 |𝜓 𝑗 ih𝜓 𝑗 |𝜙𝑖 i = |h𝜙𝑖 |𝜓 𝑗 i| 2 . It is clear that 𝑃𝑖 𝑗 ≥ 0, 𝑖 𝑃𝑖 𝑗 = 1 andÍ
𝑗 𝑃𝑖 𝑗 = 1. (As a matrix 𝑃𝑖 𝑗 is doubly stochastic.)ÍNow, we let 𝑟𝑖 = 𝑗 𝑃𝑖 𝑗 𝑞 𝑗 . (The 𝑟𝑖 is regarded as a probability distribution.)
Since the logarithm is a concave function, the logarithm of 𝑞 𝑗 is less than or equal to
the tangential line at 𝑞 𝑗 = 𝑟𝑖 , log 𝑞 𝑗 ≤ log 𝑟𝑖 + (𝑞 𝑗 − 𝑟𝑖 )/𝑟𝑖 . By taking the expectation
Ívalue with respect to 𝑃𝑖 𝑗 we obtain the Jensen-like inequality, 𝑗 𝑃𝑖 𝑗 log 𝑞 𝑗 ≤ log 𝑟𝑖 .
This tells us that
𝐷 𝐾 𝐿 [𝜌 k 𝜎] ≥Õ𝑖𝑝𝑖 log

𝑝𝑖
.
𝑟𝑖

(A.5)

The right-hand-side here is proved to be greater than or equal to zero by the same
argument we used for the classical probability distribution above. Thus, 𝐷 𝐾 𝐿 [𝜌 k𝜎] ≥ 0.
Our discussion here largely follows that in Chapter 11 of Nielsen and Chuang[53]. For alternative proofs, one can consult Cover and Thomas [54].

C

Concavity of W𝐿𝐵 with respect to the temperature

To determine the effective inverse temperature we use the fact that W𝐿𝐵 (𝛼) has one
global maximum as a function of 𝛼. This is most easily seen from the concavityof W𝐿𝐵 as a function of the temperature, 𝛼−1 . To show this we use the facts
72

C

CONCAVITY OF W𝐿𝐵 WITH RESPECT TO THE TEMPERATURE(known from thermodynamics and also holding for a canonical distribution) that thederivative of the Helmholtz free energy with respect to temperature is minus theentropy and the derivative of the entropy with respect to temperature is positive.
We have,𝜕𝜕
W𝐿𝐵 (𝛼) =𝐹𝑇 (𝛼) + 𝑆[𝜌0 ]
−1
𝜕𝛼
𝜕𝛼−1

Then,

𝜕2
−1 2

= −𝑆[𝜌can,𝑇 (𝛼)] + 𝑆[𝜌0 ].

(C.1)

𝜕
𝑆[𝜌can,𝑇 (𝛼)].
𝜕𝛼−1

(C.2)

W𝐿𝐵 (𝛼) = −(𝜕𝛼 )The derivative of the entropy with respect to temperature is proportional to the heat
capacity or equivalently to the variance of the energy, which is positive. Thus wehave

𝜕2(𝜕𝛼−1 )

2W𝐿𝐵 (𝛼) < 0,(C.3)

which means that W𝐿𝐵 (𝛼) is a concave function of 𝛼−1 .
Suppose 𝜕𝛼−1 W𝐿𝐵 (𝛼) = 0 at 𝛼−1 = 𝛽˜−1 . The concavity of W𝐿𝐵 (𝛼) tells us that𝜕𝛼−1 W𝐿𝐵 (𝛼) is a strictly monotonic function (of 𝛼−1 ) so that 𝜕𝛼−1 W𝐿𝐵 (𝛼) cannot
be zero for any other value of 𝛼−1 than 𝛽˜−1 . Since 𝛼−1 and 𝛼 are in one-to-one
correspondence, W𝐿𝐵 (𝛼) has then only one global maximum as a function of 𝛼;˜
the one at 𝛼 = 𝛽.

73

DTHE GENERALIZED PYTHAGOREAN THEOREM BASED ON THE KL
DIVERGENCED

The generalized Pythagorean theorem basedon the KL divergence

The importance of the GPT based on the scaled KL divergence is shown in Sections 3
and 4. Here, we will describe the GPT based on the bare (non-scaled) KL divergencefor comparison. The geometric structure based on the bare KL divergence is different
from the geometric structure based on the scaled KL divergence. We also derive
the well-known principle of maximum entropy. While its proof usually just uses the
non-negativity of the KL divergence [54,55], it is also be derived from the GPT [17].
In this Appendix, for brevity, we suppress the subscripts representing time.
Using bare KL divergences gives us the following theorem;
Theorem 4. Under the isoenergetic condition, i.e.,h𝐻|𝜌i = h𝐻|𝜌can (𝛽)i,(D.1)

the GPT based on the KL divergence holds,
𝐷 𝐾 𝐿 [𝜌 k 𝜌can (𝛼)] = 𝐷 𝐾 𝐿 [𝜌 k 𝜌can (𝛽)] + 𝐷 𝐾 𝐿 [𝜌can (𝛽) k 𝜌can (𝛼)].

(D.2)

Proof. This theorem is easy to confirm as,
𝐷 𝐾 𝐿 [𝜌 k 𝜌can (𝛼)] − 𝐷 𝐾 𝐿 [𝜌 k 𝜌can (𝛽)] − 𝐷 𝐾 𝐿 [𝜌can (𝛽) k 𝜌can (𝛼)]= −𝑆[𝜌] − 𝛼(𝐹 (𝛼) − h𝐻|𝜌i) − {−𝑆[𝜌] − 𝛽(𝐹 (𝛽) − h𝐻|𝜌i)}− {𝛽(𝐹 (𝛽) − h𝐻|𝜌can (𝛽)i) − 𝛼(𝐹 (𝛼) − h𝐻|𝜌can (𝛽)i)}= 𝛼h𝐻|𝜌i − 𝛽h𝐻|𝜌i − 𝛼h𝐻|𝜌can (𝛽)i + 𝛽h𝐻|𝜌can (𝛽)i= (𝛽 − 𝛼) (h𝐻|𝜌can (𝛽)i − h𝐻|𝜌i) = 0

74

(D.3)

D

THE GENERALIZED PYTHAGOREAN THEOREM BASED ON THE KL
DIVERGENCE

where we used the isoenergetic condition of (D.1) in the last line.

The geometric image of the GPT based on KL divergences is illustrated in
Figure 12. The (geodesic) line connecting 𝜌 and 𝜌can (𝛽) on the isoenergetic surface
is orthogonal to the parametric line of canonical distributions in terms of the KL
divergence.
Isoenergetic
surface𝜌

𝐷$% [𝜌 ∥ 𝜌!"# (𝛼)]𝐷$% [𝜌 ∥ 𝜌!"# (𝛽)]𝜌!"# (𝛼)𝜌!"# (𝛽)Set of canonicaldistribution

𝐷$% [𝜌!"# (𝛽) ∥ 𝜌!"# (𝛼)]Figure 12: The image of the GPT based on the KL divergences.
By substituting the following expressions of the KL divergences into the GPT,
Eq. (D.2),
𝐷 𝐾 𝐿 [𝜌 k 𝜌can (𝛼)] = −𝑆[𝜌] − 𝛼(𝐹 (𝛼) − h𝐻|𝜌i)
𝐷 𝐾 𝐿 [𝜌can (𝛽) k 𝜌can (𝛼)] = −𝑆[𝜌can (𝛽)] − 𝛼(𝐹 (𝛼) − h𝐻|𝜌can (𝛽)i),

(D.4)
(D.5)

Eq. (D.2) is rewritten as−𝑆[𝜌] − 𝛼(𝐹 (𝛼) − h𝐻|𝜌i)= 𝐷 𝐾 𝐿 [𝜌 k 𝜌can (𝛽)] − 𝑆[𝜌can (𝛽)] − 𝛼(𝐹 (𝛼) − h𝐻|𝜌can (𝛽)i).

75

(D.6)

D

THE GENERALIZED PYTHAGOREAN THEOREM BASED ON THE KL
DIVERGENCE

From the isoenergetic condition Eq. (D.1),−𝑆[𝜌] = 𝐷 𝐾 𝐿 [𝜌 k 𝜌can (𝛽)] − 𝑆[𝜌can (𝛽)].

(D.7)

Using the non-negativity of the KL divergence in the right-hand-side, we obtain
𝑆[𝜌] ≤ 𝑆[𝜌can (𝛽)].

(D.8)

This inequality holds for any probability distribution 𝜌 satisfying the isoenergetic
condition (D.1). The canonical distribution has the maximum entropy among isoenergetic distributions. This property is the well-known principle of maximum entropy.

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REFERENCES[48] When the initial state is a canonical state, the effective temperature is thetemperature of the canonical state from the isentropic condition. The divergencein the right-hand-side of Eq. (2.37) vanishes so that no work is extractable
for any cyclic operation. This argument is applicable to an isothermal system
obtained by dividing a thermally isolated total system into the system of interest
and a reservoir with a canonical distribution as discussed in section 2.3. We
note that the microcanonical state is usually regarded as the equilibrium state
in a thermally isolated system. As is well known though, the expectation valueof any physical observable for a canonical state is approximated by that of thecorresponding microcanonical state in a many-body thermodynamic system inthe sense of the law of large numbers.
[49] Batalhao T et al. 2014 Phys. Rev. Lett. 113 140601[50] Messiah A 2014 Quantum mechanics: Two volumes bound as one. (New York:
Dover)[51] The temperature has already been fixed as the effective temperature so thescale factor is common for three divergences in Eq. (3.81). Thus, the validity
condition of the GPT of the scaled KL divergences is the same as that of thebare KL divergences here.
[52] For 𝜑 = 0 the state is the ground state after the sudden change at 𝑡 = 0.
The former set of operations including the quasi-static operation keeps theground state. The maximum work is realized for the former set of operations.
For 𝜑 = 𝜋/3, the state has higher energy than the ground state after thesudden change at 𝑡 = 0. The former set of operations including the quasi-static
operation keep this higher energy. On the other hand some quick operationsin the latter set induce transitions to make the energy of the state lower. As a80

REFERENCES

result, we can extract more work than the quasi-static process by the optimal
operation in the latter set.
[53] Nielsen M and Chuang I 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)[54] Cover T and Thomas J 2006 Elements of Information Theory (Hoboken: WileyInterscience)[55] Pauli W 2000 Pauli lectures on physics. vol. 4: Statistical mechanics (MITpress, Cambridge 1973 and reprinted by Dover in 2000)

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