IFSC DFBe

*

See FichaCatalograficaCorrigida2.pdf

This thesis is dedicated to all people who work hard against misinformation
and to developing science in Brazil.
Most importantly, it is dedicated to all, direct and indirect, victims of this
pandemic.

A countless number of people pass through our lives. Most of them are with us
just for brief moments, while some become essential and irreplaceable
constants. Yet, all of them represents a piece of our totality, in one way or
another. I am the sum of several crucial parts, and for that, I would like to
thank and acknowledge you all:

I am very grateful to my loving family: my parents, Lilian and Jefferson, my
brother, Arthur, my grandmother, Beatriz, and my younger family members,
Minnie (RIP), Bella and Mia. You represent my foundations and the most special
constant behind me and my achievements. Thank you for your unconditional
support and love throughout my life, especially during the last years. The
path I chose is uncertain and tortuous, but I am very fortunate for having you
by my side. In fact, there is no André without you.

I also would like to thank my advisor, Prof. Dr. Frederico Brito, for sharing
his knowledge, providing support during the execution of this project and,
most importantly, all the trust and patience. I have grown in many ways during
this time, and I am sure I have become a better person and, without a doubt, a
better scientist after his guidance.

Many great friends also joined me during this trajectory, both on and offline:
"I am glad you are here with me, at the end of all things…". A special thanks
to my "roommates" Gabriela and Lais for sharing much more than a simple
office; to Cleverson and all the members of the "Jovens quânticos" for all the
passionate scientific discussions; to Rodrigo and Pedro for existing and
making life more bearable; to Dimitrios and Madeira for the several coffees,
advice and funny moments; to Diogo, Jessica, Guilherme, Raul, Edmilson and
Cesar for welcoming me when I moved to IFSC; to Adriane for all our talks and
mutual support; to Letícia for all the help and assistance; to Paula,
Fernando, Maria Luiza and Artur for sharing a story and being a second loving
family for many years; to my hometown friends, José, Rodrigo and Ikeda for
also being important constants during my life. Last but not least, I am
grateful to Thalyta for all her patience, care and shared memories. Thank you
for being supportive and staying by my side during these arduous times.

Finally, I would like to acknowledge all the staff from IFSC, especially
Ricardo, for supporting me since day one. This study was financed in part by
the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil
(CAPES) – Finance Code 001, and I am very greatfull for the financial support.

“The unexamined life is not worth living.”
Socrates

[Abstract]

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During the last decades, there have been many theoretical and experimental
advances both in the extension of thermodynamics to comprise microscopic
systems out-of-equilibrium and in the understanding of quantum mechanics.
Along with the state-of-the-art capability of controlling fragile quantum
systems in a wide variety of physical platforms, this context has paved the
way for the current strategic efforts to develop a thermodynamic theory of
quantum systems. In this sense, the research field coined as quantum
thermodynamics (QT) already plays a key role in the design and development of
future quantum-based technologies. More specifically, QT aims both to apply
the usual thermodynamic concepts and notions to describe arbitrary non-
equilibrium quantum systems and to understand the emergence of classical
thermodynamic behaviour from the underlying fundamentally quantum dynamics.
However, despite all current progress, there is still no consolidated
formalism for a general thermodynamic description of fully autonomous quantum
objects. Besides, the lack of consensus on some central aspects, such as the
definitions of quantum counterparts of thermodynamic quantities, is
particularly notorious. In this thesis, we focus on the energetic analysis
within autonomous quantum systems. To this aim, we propose a novel and general
formalism for a dynamic description of the energy exchanges between
interacting subsystems. From the Schmidt decomposition approach, we identify
effective Hamiltonians as the representative operators for characterizing the
local internal energies, whose expectation values satisfy the usual
thermodynamic notion of energy additivity. In contrast to the currently used
methodologies, such procedure treats the subsystems with equal footing and do
not rely on any sort of approximations and additional hypotheses, e.g., semi-
classical description, weak-coupling regime, strict energy conservation and
Markovian dynamics. In short, our proposal contributes to the development of
QT by providing a new formalism that does not suffer from the usual
restrictive shortcomings and establishes a new and exact route for defining
other general thermodynamic quantities to the quantum regime.

Keywords: Quantum thermodynamics. Open quantum systems.

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Durante as últimas décadas, houve muitos avanços teóricos e experimentais
tanto na extensão da termodinâmica para abranger sistemas microscópicos fora
de equilíbrio quanto na compreensão da mecânica quântica. Somada a capacidade
de última geração de controlar sistemas quânticos frágeis em uma ampla
variedade de plataformas físicas, esse contexto abriu caminho para os atuais
esforços estratégicos para desenvolver uma teoria termodinâmica de sistemas
quânticos. Nesse sentido, o campo de pesquisa cunhado como termodinâmica
quântica (TQ) já desempenha um papel fundamental no projeto e desenvolvimento
de futuras tecnologias baseadas em fenômenos quânticos. Mais especificamente,
a TQ visa tanto aplicar os conceitos e as noções termodinâmicas usuais para
descrever sistemas quânticos arbitrários fora do equilíbrio quanto entender o
surgimento do comportamento termodinâmico clássico a partir da dinâmica
fundamentalmente quântica subjacente. No entanto, apesar de todo o progresso
atual, ainda não existe um formalismo consolidado para uma descrição
termodinâmica geral de objetos quânticos totalmente autônomos. Além disso, é
particularmente notória a falta de consenso em relação a alguns aspectos
centrais, como as definições de análogos quânticos de grandezas
termodinâmicas. Nesta tese, focamos na análise energética em sistemas
quânticos autônomos. Para isso, propomos um novo formalismo geral para uma
descrição dinâmica das trocas energéticas entre subsistemas interagentes. A
partir da abordagem da decomposição de Schmidt, identificamos Hamiltonianos
efetivos como os operadores representativos para caracterização das energias
internas locais, cujos valores esperados satisfazem a noção termodinâmica
usual da aditividade de energia. Ao contrário das metodologias atualmente
utilizadas, tal procedimento trata os subsistemas em pé de igualdade e não
depende de nenhum tipo de aproximações e hipóteses adicionais, por exemplo,
descrição semiclássica, regime de acoplamento fraco, conservação de energia
estrita e dinâmica Markoviana. Em suma, nossa proposta contribui para o
desenvolvimento da TQ fornecendo um novo formalismo que não sofre das
restrições usuais e estabelece uma nova e exata rota para definir outras
grandezas termodinâmicas gerais para o regime quântico.

Palavras-chave: Termodinâmica quântica. Sistemas quânticos abertos.

lof.0lof.0List of FiguresList of Figureslof.0

###### List of Figures

  1. 1 (a) Purity dynamics for two interacting qubits. While the whole system maintains pure for all t, both qubits oscillate continuously between pure and mixed states with equal purity; (b) Dynamical behaviour of the von Neumann entropies. On the one hand, given $\hat{\rho}^{(0)}(t)\equiv|\Psi(t)\rangle\langle\Psi(t)|$, the whole system’s von Neumann entropy maintains null throughout the unitary dynamics. On the other hand, the equal qubits’ von Neumann entropies illustrate the oscillation between the degree of entanglement within the bipartition. For the computation of all the plots, it was assumed $a^{(1)}=(5)^{-1/2},b^{(1)}=2(5)^{-1/2}$, $a^{(2)}=(10)^{-1/2}$ and $b^{(2)}=3(10)^{-1/2}$ for the initial states, $\hbar\equiv 1$, $\omega_{1}=1$, $\omega_{2}=5$ and $g=1.3$.
  2. (a) Purity
  3. (b) von Neumann entropy
  4. 2 Dynamical behaviour of the expectation values of the local bare Hamiltonians. On the one hand, since $[\hat{H}^{(2)},\hat{H}_{int}]=0$, the populations of $\hat{\rho}^{(2)}$ in the $\hat{H}^{(2)}$ basis are time-invariant and, therefore, $\frac{d}{dt}\langle\hat{H}^{(2)}\rangle(t)=0$ for all $t$. On the other, $\frac{d}{dt}\langle\hat{H}^{(1)}\rangle(t)\neq 0$ and $\langle\hat{H}^{(1)}\rangle(t)$ continuously oscillates in time.
  5. 3 According to Alicki’s form of work and heat fluxes (see Eqs. (142,143)): (a) Qubit $(1)$ does not perform work, and all energy exchange with the interaction term is interpreted as heat, i.e., $\frac{d}{dt}W_{1}^{(1,2)}(t)=0$ and $\frac{d}{dt}Q_{1}^{(1)}(t)=-\frac{d}{dt}\langle\hat{H}_{int}\rangle(t)$; (b) The energy of qubit $(2)$ is neither submitted to work-like nor heat-like energy changes.
  6. (a) Alicki’s proposal for qubit $(1)$
  7. (b) Alicki’s proposal for qubit $(2)$
  8. 4 According to Eqs. (152, 153): (a) The energetics between qubit $(1)$ and the interaction term has separated contribution from both work and heat, i.e., $\frac{d}{dt}\langle\hat{H}^{(1)}\rangle(t)=\frac{d}{dt}W_{3}^{(1)}(t)+\frac{d}{dt}Q_{3}^{(1)}(t)=-\frac{d}{dt}\langle\hat{H}_{int}\rangle(t)$; (b) Qubit $(2)$ neither performs work nor exchanges heat. Note this happens despite its dynamics and the clear oscillation of its von Neumann entropy $S_{vN}[\hat{\rho}^{(2)}(t)]$ (see Figure 1(b)).
  9. (a) [214, 215] proposal for qubit $(1)$
  10. (b) [214, 215] proposal for qubit $(2)$
  11. 5 Dynamical behaviour of the expectation values of the local effective Hamiltonians and its relationship with the whole’s internal energy. (a) Interplay between $\langle\tilde{H}^{(1)}(t)\rangle$ and $\langle\tilde{H}^{(2)}(t)\rangle$. If $\tilde{H}^{(1,2)}(t)$ are interpreted as the representative operators for quantifying local internal energies, then all energy lost from qubit $(1)$ is obtained by qubit $(2)$, and vice versa; (b) The sum of the expectation values of the local effective Hamiltonians is equal to the whole’s internal energy. This plot illustrates the energy additivity.
  12. (a) Local effective Hamiltonians
  13. (b) Whole’s internal energy
  14. 6 Dynamical behaviour of the spectrum of the bare and local effective Hamiltonians, for both qubits. The bare eigenvalues $\\{b_{\pm}^{(1,2)}\\}$ are constant, while the local effective ones $\\{\tilde{\epsilon}_{\pm}^{(1,2)}(t)\\}$ clearly change in time. (a) Qubit $(1)$; (b) Qubit $(2)$.
  15. (a) Spectrum - Qubit $(1)$
  16. (b) Spectrum - Qubit $(2)$
  17. 7 Dynamical behaviour of the spectral gaps of the bare and local effective Hamiltonians, for both qubits. The changes in the spectrum modulate the local gaps $\tilde{\omega}_{1,2}(t)=\tilde{\epsilon}_{+}^{(1,2)}(t)-\tilde{\epsilon}_{-}^{(1,2)}(t)$ between the effective energy levels while $\hbar\omega_{1,2}=b_{+}^{(1,2)}-b_{-}^{(1,2)}$ is maintained fixed. (a) Qubit $(1)$; (b) Qubit $(2)$.
  18. (a) Spectral gaps - Qubit $(1)$
  19. (b) Spectral gaps - Qubit $(2)$
  20. 8 Work and heat fluxes according to Eqs. (156, 157) for qubit $(1)$. (a) Work flux; (b) Heat flux.
  21. (a) Alicki’s form of work flux for qubit $(1)$
  22. (b) Alicki’s form of heat flux for qubit $(1)$
  23. 9 Work and heat fluxes according to Eqs. (156, 157) for qubit $(2)$. (a) Work flux; (b) Heat flux.
  24. (a) Alicki’s form of work flux for qubit $(2)$
  25. (b) Alicki’s form of heat flux for qubit $(2)$
  26. 10 Net energy fluxes according to Eqs. (156, 157). (a) The total work flux performed and the total heat flux transferred throughout the dynamics; (b) All net energetic exchanges involved during the whole system dynamics sum up to zero, Eq. (155).
  27. (a) Net work and net heat fluxes
  28. (b) Net energetic exchange
  29. 11 Energy fluxes according to Eqs. (163, 164). (a) Qubit $(1)$; (b) Qubit $(2)$.
  30. (a) Work and heat fluxes for qubit $(1)$
  31. (b) Work and heat fluxes for qubit $(2)$
  32. 12 Net energy fluxes according to Eqs. (163, 164). (a) The total work flux performed and the total heat flux transferred throughout the dynamics; (b) All net energetic exchanges involved during the whole system dynamics sum up to zero, Eq. (155).
  33. (a) Net work and net heat fluxes
  34. (b) Net energetic exchange

*

toc.0toc.0ContentsContentstoc.0

###### Contents

  1. 0 Introduction
  2. 1 Contextualization
    1. 1 Thermodynamics
      1. 1 Classical equilibrium thermodynamics
        1. The laws of thermodynamics
      2. 2 Stochastic thermodynamics
        1. Fluctuations theorems
    2. 2 Quantum mechanics
      1. 1 Mathematical description of quantum systems
        1. Operators and observables
        2. Projective measurements
        3. Composite systems
        4. Density operator
      2. 2 Quantum dynamics
        1. Isolated and closed system dynamics
        2. Open systems dynamics
    3. 3 Quantum thermodynamics
  3. 2 Schmidt decomposition approach to quantum thermodynamics
    1. 1 The setting
    2. 2 Schmidt basis dynamics and local effective Hamiltonians
    3. 3 Local states dynamics
      1. 1 Unitary dynamics
        1. Semi-classical external drive
    4. 4 Internal energy and additivity
      1. 1 The whole and the parts internal energies
        1. Local effective internal energy
    5. 5 Local phase gauge
      1. 1 Frame change
      2. 2 Recap and gauge fixing proposal
    6. 6 Thermodynamics
      1. 1 Remarks on thermal states
      2. 2 Remarks on quantum thermodynamic entropy
        1. General context
        2. Issues of the von Neumann entropy
      3. 3 Remarks on quantum work and heat
        1. General context
        2. Road to effective work and heat
    7. 7 Proof of principle
    8. 8 Generalization for mixed states
      1. 1 Schmidt basis dynamics and local effective Hamiltonians
      2. 2 Local states dynamics
      3. 3 Local effective internal energy
        1. Pure state level
        2. Ensemble level
        3. Brief remarks
    9. 9 Discussion and summary
  4. 3 Conclusion and outlook

*

##  Chapter 0 Introduction

Quantum thermodynamics is a rapidly growing and promising field of research,
accumulating efforts from several distinct perspectives. Its flourishing is
inextricably linked to recent technological advances and current independent
theoretical progress in the fields of nonequilibrium thermodynamics and
quantum mechanics. On the one hand, state-of-the-art experiments allow the
precise control of small, and possibly quantum, systems in several platforms.
On the other, yet the laws of thermodynamics have been successfully verified
to microscopic scales when considering ensemble realizations of the
experiment, and studies on quantum information elucidated the technological
potential of harnessing genuine quantum phenomena. However, the task of
developing a consistent and somehow meaningful thermodynamic theory of quantum
systems proved to be non-trivial and far from being straightforward. In this
sense, the most current approaches are restricted to specific scenarios of
approximative descriptions and semi-classical treatments, i.e., it is commonly
assumed externally driven systems (by means of classical agents), weak-
coupling regime and Markovian dynamics. Along these lines, there is no
unifying formalism for characterizing the thermodynamics within general
autonomous quantum systems, and many fundamental open questions remain
unanswered. Remarkably, it is unclear how to define quantum counterparts of
the most basic thermodynamic quantities. This context highlights the
importance of further research, especially on conceptual aspects of the
theory. The work in question falls within this scope111The main discussions
and results introduced in this thesis were presented in [1] after its
defense..

This thesis is organized as follows: Chapter (1) contextualizes the field of
quantum thermodynamics. It initially introduces the basic concepts and
formalism of thermodynamics and quantum mechanics separately. Then it provides
a brief overview of quantum thermodynamics and discusses its current status
and open problems; Chapter (2) contains the main sections of the thesis. It
develops and proposes a novel framework for analyzing the energetics within
autonomous pure bipartite quantum systems. This proposal is exact, symmetrical
and based on a well-established mathematical tool: the Schmidt decomposition
procedure. More specifically, it introduces time-dependent local effective
Hamiltonians interpreted as the representative operators for characterizing
the physical local internal energies. This identification allows the
description of the subsystems effective dynamics within general interacting
subsystems, regardless of their particular properties and coupling regimes,
and also recovers the usual thermodynamic property of energy additivity.
Besides, it briefly discusses the current approaches and difficulties for
defining general quantum versions of core thermodynamic quantities. After
that, the introduced formalism is extended for mixed states; Finally, the
conclusions of this thesis are addressed in Chapter (3). It summarizes its
main messages and provides a perspective for future work considering the
general context of quantum thermodynamics.

##  Chapter 1 Contextualization

This chapter aims to contextualize the field of research in which this work is
inserted, known as quantum thermodynamics. To this end, a brief detour is
necessary to introduce its main ingredients, namely thermodynamics and quantum
mechanics. Of course, this is not an extensive review of both areas. Instead,
these notes intend to provide an overview of the context and main motivations
for the development of a thermodynamic theory of quantum systems.

In this sense, Section (1) presents the basic concepts of classical
equilibrium thermodynamics and the recently established research area of
stochastic thermodynamics. Then, Section (2) discusses the formalism of
quantum mechanics. Finally, Section (3) formally introduces quantum
thermodynamics and its current status.

###  1 Thermodynamics

Thermodynamics is, undoubtedly, one of the most successful physical theories
in the history of science. It is no overstating to say that the foundations of
modern society were significantly influenced by the technological innovations
brought by it. Its range of predictions and applicability is surprisingly far-
reaching, especially if considering its restrictions and the circumstances of
its developments. In this sense, all branches of modern science and
engineering function accordingly to the laws of thermodynamics. Nevertheless,
even after centuries of discussions, its results and statements continue to
inspire and instigate novel research and surprising connections.

Despite earlier developments, the theory of thermodynamics was born during the
17th century and flourished as an influential branch of physics during the
18th and 19th centuries111For a historical perspective on the development of
thermodynamics, see. [2]. Since its early days, thermodynamics was developed
as a phenomenological discipline and approached semi-empirically. It initially
progressed through scientists’ efforts to understand better the behaviour of
gases and the "abhors" nature of the vacuum222"Natura abhorret vacuum". Then,
it proceeded as a fundamental way to characterize different aspects of energy
exchange through work and heat, along with the notions of irreversibility and
entropy. As often happens in science, these investigations were partially
motivated by the technological advancements of the time: the development of
new instruments, such as vacuum pumps, the thermometer and steam engines,
provided the necessary means to perform novel and better measurements and push
even further the progress of the theory. Surprisingly, this context also
triggered the unprecedented technological transformation that led western
society toward the Industrial Revolution and, eventually, the thriving of
modern machinery. In fact, the progress of thermodynamics was driven
considerably by the attempts to understand and develop efficient machines. In
this sense, despite the pragmatical and sober motivation, it is remarkable how
these studies guided scientists toward the establishment of general and
fundamental laws of nature.

Seeing from the privilege of modern science, the phenomenological approach to
thermodynamics could hardly have been different. The microscopic behaviour of
matter remained hidden from scientists’ eyes for a long time, and the tools of
analytical mechanics, along with its mechanistic view of nature, were being
developed in parallel to thermodynamics. In this sense, its history is also
intrinsically connected to the concomitant developments of chemistry and
classical mechanics. Along these lines, at that time, the empirical
observation that mass333Strictly speaking, mass does not conserve due to the
mass-energy equivalence. However, in the present thermodynamic context, this
difference is negligible., momentum and energy were conserved quantities in
isolated systems was still under scrutiny and far from consensus. In fact,
even these individual notions - including "isolated systems" \- were not
entirely established. Heat, for instance, was not directly associated with the
mechanical motions of particles or the "vis viva", the earlier version of
kinetic energy, until the efforts of Joule, when he established the
relationship between energy, work and heat. Instead, the orthodox explanation
was given by the so-called caloric theory. Interestingly, this was also the
context for the development of heat engines and the theoretical studies
concerning their performances, which set the stage for the works of Sadi
Carnot (1796 - 1832) - and later Rudolf Clausius (1822 - 1888) \- on what we
know today as the second law of thermodynamics and the concept of entropy.

After that, names like James Clerk Maxwell (1831 - 1879), Ludwig Boltzmann
(1844 - 1906) and Josiah Willard Gibbs (1839 - 1903) were responsible for
providing the microscopic and mechanical basis of phenomenological equilibrium
thermodynamics. Such progress is depicted by the kinetic theory of gases and
the following foundation of statistical mechanics. Most importantly, these
developments represent a crucial paradigm shift concerning the understanding
of the macroscopic nature, i.e., the introduction of probabilistic reasoning
and thermodynamic ensembles provided the necessary tools for - at least
partially - bridging the gap between the underlying microscopic motion and the
observed macroscopic behaviour. More recently, during the 20th century,
physicists, such as Lars Onsager (1903 - 1976) and Ilya Prigogine (1917 -
2003), began to extrapolate thermodynamics to the uncharted territory of non-
equilibrium regimes. After all, most natural processes occur under these
conditions. Along these lines, the analysis of systems close to equilibrium,
known as the linear regime, provided the formalism for investigating several
physical phenomena and represented the starting point of the characterization
of non-equilibrium states.

As mentioned earlier, thermodynamics has still been an active field of
research. During the last decades, one can find great efforts to extend the
theory and its results to even broader scenarios, namely microscopic and far-
from-equilibrium settings. In such regimes, thermal fluctuations play a
critical role in the dynamics of the desired physical system, and the usual
thermodynamic quantities are updated to random variables. In this sense,
stochastic thermodynamics and its fluctuation theorems have been providing
seminal results concerning non-equilibrium processes and irreversibility.

####  1 Classical equilibrium thermodynamics

Let us briefly present and review the classical equilibrium thermodynamics and
its main statements. Of course, there are several traditional textbooks
written on the subject. For a comprehensive discussion of thermodynamics see.
[3, 4, 5, 6, 7]

Classical thermodynamics restricts itself to a specific, though meaningful,
physical regime and scope. Essentially, it was developed to describe the
behaviour of large scale properties of classical macroscopic objects and their
general relationships. Since it is a phenomenological description, it does not
require any microscopic consideration concerning the structure of matter.
Instead, it provides a macroscopic and fully general characterization in terms
of a few measurable gross variables $\\{X_{j}\\}$, known as thermodynamic
coordinates, such as volume ($V$), pressure ($P$) and temperature ($T$). In
this sense, once obtained the relevant coordinates for describing a system,
one has specified its physical state and, along with it, has access to all
thermodynamic properties that are functions of it, referred to as state
functions, i.e., $F(\\{X_{j}\\})$. Such a procedure is only possible because
classical thermodynamics revolves around the concept of states in
thermodynamic equilibrium. These states are characterized by their static, or
time-invariant, nature, such that all possible thermodynamic coordinates
remain fixed444One can also refer to equilibrium states relative to specific
macroscopic coordinates, such as mechanical, chemical and thermal.. In
contrast, non-equilibrium states might not even have specified values for
them. It is an empirical fact that many physical systems naturally
(spontaneously) evolve in time from non-equilibrium states towards these
constant states. It is worth mentioning that everything that happens during
this dynamic process is outside the scope of classical equilibrium
thermodynamics.

Along these lines, it is also possible to define a succession of changes
connecting different equilibrium states, which establishes a thermodynamic
process. If the initial and final states are the same, the whole process is
known as a thermodynamic cycle, which represents the building blocks of any
heat engine. Visually, thermodynamic processes and cycles are usually depicted
by trajectories in the space of equilibrium states defined in terms of the
thermodynamic coordinates $\\{X_{j}\\}$. Of course, any change presupposes
some degree of variation of a macroscopic variable that, in general, would
perturb the system and kick it out of equilibrium until it reaches a novel one
in terms of the new setting. Nevertheless, as long as the changes are kept
infinitesimal, the system is weakly perturbed and maintained close to
equilibrium. Such a convenient condition characterizes a so-called quasi-
static process. Notice that any ideal thermodynamic process would require an
infinite amount of time to be performed. However, in many realistic scenarios,
this theoretical abstraction represents a satisfactory approximation.
Interestingly, the rate of processes is not addressed by classical equilibrium
thermodynamics, which highlights that the theory is not a truly dynamic one,
at least not in the usual sense.

##### The laws of thermodynamics

The physical object under scrutiny, or the thermodynamics system, is usually
classified according to the ways it interacts with its exterior, especially
concerning the possibility of exchanging matter or energy with its
surroundings. In this sense, isolated systems, as the name suggests, do not
interact with other objects and, therefore, exchange neither things. Closed
systems only exchange energy, while open systems exchange both energy and
matter. Naturally, these interactions are accompanied by thermodynamic
processes, both from the system and its surroundings. The possible state
transformations are described, and constrained, by a set of empirical and
mathematical statements known as the laws of thermodynamics. Let us now
briefly introduce them.

###### Zeroth law of thermodynamics

As mentioned earlier, thermodynamic systems eventually reach equilibrium
states. It implies that if two independent systems are allowed to interact,
the single entity constituted by both individuals will also equilibrate. Thus,
it is an experimental fact that interacting bodies that are specified by
distinct temperatures will reach thermal equilibrium. Along these lines, if
systems $A$ and $B$ are individually in thermal equilibrium with system $C$,
then $A$ and $B$ are also in equilibrium. Such transitivity property is known
as the zeroth law of thermodynamics.

###### First law of thermodynamics

Essentially, the first law of thermodynamics refers to the conservation of
energy within thermodynamic systems. However, instead of being written in
terms of the sum of all the kinetic and potential contributions of its
microscopic constituents, the internal energy relative to state $k$, $U_{k}$,
is a state function specified by the relevant macroscopic variables
$\\{X_{j}^{k}\\}$, i.e.,

$U_{k}\equiv U(\\{X_{j}^{k}\\})+U_{0},$ (1)

where $U_{0}$ is the energy of a reference state. Notice that it means that
the internal energy change $U_{b}-U_{a}$ from a state $a$ to $b$ is
independent of the thermodynamic process, $\gamma$, connecting them, such that

$\int_{\gamma}\,dU=\int_{\gamma^{\prime}}\,dU=U_{b}-U_{a},$ (2)

where $dU$ is an exact differential, and

$\oint\,dU=0$ (3)

for any cyclic transformation.

It is clear that isolated system, by construction, maintains its internal
energy fixed since any energetic change requires the system to be submitted to
a thermodynamic process. Thus, let us now consider closed systems.
Thermodynamics, and the first law specifically, not only state energy
conservation but also splits its possible changes into two different
"flavours": heat, $Q$, is the type of energy exchanged once systems of
different temperatures are interacting or, as commonly said, put into thermal
contact; work, $W$, in contrast, is the energy transferred (not
stochastically) by the external change of the thermodynamic coordinates. Along
these lines, in the simplest scenario consisting of two interacting bodies
reaching thermal equilibrium, the energetic exchange of the system under
consideration is completely due to heat, or

$\Delta U=Q.$ (4)

However, for a thermally insulated system, one can perform work and modify the
system’s state simply by externally controlling its macroscopic variables,
such that

$\Delta U=W.$ (5)

This kind of procedure where no heat is involved, is characterized as an
adiabatic process. Nevertheless, considering general interactions, both
contributions might be present for arbitrary processes. Thus, for an
infinitesimal energy change, $dU$, the first law of thermodynamics is stated
as

$dU\equiv\delta Q+\delta W,$ (6)

where $\delta W$ and $\delta Q$ are the infinitesimal work performed on the
system and the heat transferred into it, respectively. In contrast with
internal energy, both individual quantities are not state functions. Instead,
they are intrinsic to thermodynamic processes. Such path dependency is
mathematically represented by the inexact differentials $\delta W$ and $\delta
Q$. Thus, given two possible trajectories, $\gamma$ and $\gamma^{\prime}$,
connecting different states $a$ and $b$, we have

$\displaystyle W_{\gamma}$ $\displaystyle\equiv\int_{\gamma}\,\delta
W\neq\int_{\gamma^{\prime}}\,\delta W\equiv W_{\gamma^{\prime}},$ (7)
$\displaystyle Q_{\gamma}$ $\displaystyle\equiv\int_{\gamma}\,\delta
Q\neq\int_{\gamma^{\prime}}\,\delta Q\equiv Q_{\gamma^{\prime}},$ (8)

even though

$U_{b}-U_{a}=W_{\gamma}+Q_{\gamma}=W_{\gamma^{\prime}}+Q_{\gamma^{\prime}}.$
(9)

It is worth mentioning that, for open systems, one just must consider an
additional term $dU_{matter}$ for the energy flow due to matter exchange.

###### Second law of thermodynamics

Interestingly, there are several distinct, though ultimately equivalent, ways
for formulating the second law of thermodynamics. In this sense, according to
Lord Kelvin (William Thomson, 1824 - 1907) [3]:

> "It is impossible by means of inanimate material agency to derive mechanical
> effect from any portion of matter by cooling it below the temperature of the
> coldest of the surrounding objects."

On the same matter, Max Planck (1858 - 1947) wrote:

> "It is impossible to construct an engine which, working in a complete cycle,
> will produce no effect other than the raising of a weight and the cooling of
> a heat reservoir."

However, both statements above are often combined into the so-called Kelvin-
Planck statement of the second law:

> "It is impossible to construct an engine that, operating in a cycle, will
> produce no effect other than the extraction of heat from a reservoir and the
> performance of an equivalent amount of work."

Also, in the words of Clausius:

> "It is impossible to construct a device that, operating in a cycle, will
> produce no effect other than the transference of heat from a cooler to a
> hotter body."

Finally, Carnot’s theorem states:

> "No heat engine operating between two heat reservoirs can be more efficient
> than a reversible heat engine operating between the same two reservoirs."

Thus, according to these words, the second law of thermodynamics is a
statement of what is impossible to achieve in the attempt of interchanging
heat and work. However, these statements transcend the context of heat engines
and refrigerators and have surprisingly far-reaching and less down-to-earth
consequences. Essentially, the second law captures a fundamental feature of
natural phenomena not covered by the first law, i.e., there are classes of
processes not prohibitive from the perspective of the conservation of energy
that still does not happen in nature. In this sense, the second law is an
independent and complementary statement that also portrays a fundamental
natural restriction. Moreover, it constrains even further the set of possible
thermodynamic processes.

At the core of the second law also lies the concepts of reversibility and
irreversibility. Suppose a thermodynamic system that interacts with its
surroundings. Naturally, both work and heat may be exchanged during any change
of the macroscopic variables. Nevertheless, a reversible process is, by
definition, a process that, once realized, the system and its surroundings
must be returned to their previous states, without any other change, by
restoring the settings to their initial conditions. However, this is only
achievable - a priori - by requiring processes executed by slow infinitesimal
transformations (quasi-statically)555In fact, every reversible process is
necessarily quasi-static, while the converse is not true. and in the absence
of dissipative effects, such as friction. Along these lines, it is clear that
reversible processes are theoretical idealizations and, consequently, every
natural process occurs in a finite time and is irreversible to some extent.

Along with (ir)reversibility, entropy is a cornerstone of the second law. By
introducing it, Clausius provided the means for a consistent mathematical
formulation of the previous statements. Interestingly, the notion of entropy
transcended thermodynamics and soon achieved the status of one of the most
fundamental quantities of modern physics. As a direct extension of the work of
Carnot on heat engines and cyclic thermodynamic processes [8], Clausius showed
what we know today as Clausius’s theorem [9],

$\oint\,\frac{\delta Q}{T}=0,$ (10)

where $T$ is the temperature at which the heat exchange took place666This
equality is obtained by noticing that any thermodynamic cycle can be seen as a
composition of infinitesimal Carnot cycles and summing up all the ratios
$\frac{Q_{i}}{T_{i}}$.. Eq. (10) is valid for arbitrary cyclic and reversible
processes and also hints at the existence of a state function specified by the
macroscopic variables $\\{X_{j}\\}$, i.e., $\frac{\delta Q}{T}$ which depends
only on the thermodynamic coordinates and its integral between two different
states $a$ and $b$ is independent of the path (similar to Eq. (3)). Thus,
Clausius proposed a new quantity $S$ and named it entropy777Entropy means
transformation, from the Greek word $\tau\rho\pi\eta$. [7], such that

$dS\equiv\frac{\delta Q}{T}$ (11)

and, therefore, the entropic change is

$\Delta S\equiv S_{b}-S_{a}=\int_{a}^{b}\frac{\delta Q}{T}.$ (12)

Notice that, despite $\delta Q$ being a path-dependent quantity, the ratio
$\frac{\delta Q}{T}$ is an exact differential, $dS$. It is imperative to
highlight both the close relationship between entropy and heat and the
importance of the reversibility hypotheses. Let us suppose a system of
interest interacting with a reservoir at temperature $T$. During any
reversible process the heat exchanged, $Q$, between the system and reservoir
happens at the same temperature, in such a way that their local entropic
changes are the opposite, i.e.,

$\Delta S_{system}=-\Delta S_{reservoir}.$ (13)

The equality above is true for every reversible process and sketches an even
more general result: The sum of the entropies of the system of interest and
all its surroundings, also known as the universe888In the context of
thermodynamics, the word universe does not have a cosmological sense., is
invariant over reversible processes, i.e., given $S_{universe}\equiv
S_{system}+S_{surroundings}$, we have

$\Delta S_{universe}=0.$ (14)

Of course, as mentioned earlier, reversible processes do not represent the
typical behaviour in nature. Since entropy is a state function, given a system
at equilibrium, any possible cyclic process, reversible or not, would still
satisfy $\oint dS_{system}=0$. However, if any irreversibility is taken into
account, such as friction and other dissipative effects, Eq. (10) should be
updated to the following inequality

$\oint\,\frac{\delta Q}{T}\leq 0,$ (15)

which also implies that

$dS_{system}\geq\frac{\delta Q}{T}.$ (16)

Besides, in these cases, the entropic change from the system of interest is
not completely compensated by the changes that occur in the surroundings. In
fact, in general, one has for the second law of thermodynamics

$\Delta S_{universe}\geq 0.$ (17)

Clearly, these expressions automatically include the previous ones: if no
irreversible processes take place, then the equalities are recovered. Eq. (17)
above is the most common mathematical statement of the second law: the entropy
of the universe always increases or remains constant. It is also directly
related to the notion of the arrow of time, i.e., since every natural process
inexorably increases the entropy, one can distinguish the past from the
future.

Note that the entropic change of an irreversible process is higher than a
reversible one with the same exchanged heat, $Q$, and temperature $T$. More
modern approaches often rewrite the inequality from Eq.(16) into the following
equality:

$dS_{system}=d\varPhi+d\Sigma,$ (18)

where $d\varPhi\equiv\frac{\delta Q}{T}$ is recognized as the entropy change
due to the external energy flow999If the system is open, there is also an
energetic contribution due to matter flow., known as entropy flux, and
$d\Sigma$, or the entropy production, is the change due to the irreversible
processes. Along these lines, it is clear that entropy $dS_{system}$ and
$d\varPhi$ might acquire positive or negative values, while entropy production
is always a positive quantity,

$d\Sigma\geq 0.$ (19)

Interestingly, while the first law states that the energy of an isolated
system is conserved, the second implies that its entropy is not and, more
importantly, can only increase during natural processes. Of course, both
fundamental quantities are intrinsically connected, and many applications of
thermodynamics intend to characterize their changes.

###### Third law of thermodynamic

Finally, the third law of thermodynamics is mathematically stated as

$T\rightarrow 0_{+}\,\Rightarrow\,S\rightarrow S_{0},$ (20)

i.e., when the temperature $T$ approaches the limiting absolute zero, the
entropy approaches a constant value, $S_{0}$, independent of all macroscopic
variables.

###### Brief remarks

In the context of thermodynamics, physical properties are commonly separated
into two distinct categories: On the one hand, intensive quantities are
independent of the mass or the size and number of constituents of the system;
On the other, extensive ones are those proportional to the system’s mass,
number of constituents, or size. Along these lines, let us consider a
thermodynamic system in equilibrium described by the macroscopic variables
temperature $T$, volume $V$ and pressure $P$. Naturally, one can also
prescribe an internal energy $U$ and entropy $S$. Then, suppose one can split
the whole system into two equal smaller subsystems. Both partitions will
maintain their previous temperature $T$ and pressure $P$, while their new
volume, internal energy and entropy will be divided by two ($V/2$, $U/2$ and
$S/2$). Thus, it is clear that $T$ and $P$ are intensive quantities, while
$V$, $U$ and $S$ are extensive ones. In particular, notice that extensivity
also means that these properties are additive, i.e., the whole energy and
entropy, for instance, are the sum of the subsystem’s energy and entropy,
respectively.

Finally, it is worth emphasizing - once again - that the laws of
thermodynamics were developed and established for macroscopic systems in
equilibrium. It was not necessary any considerations concerning the
microscopic behaviour of matter. Surprisingly, however, it does not mean that
these laws are not valid in other regimes.

####  2 Stochastic thermodynamics

The microscopic foundations of classical equilibrium thermodynamics were
provided by the advent of statistical mechanics. It also brought a new
probabilistic perspective to the understanding of natural phenomena at the
molecular level. These efforts allowed the field to shift from purely
phenomenological reasoning to more solid ground. Later, linear response theory
contributed to the first expeditions outside the equilibrium context. More
recently, however, the scenarios are even more extreme: the current high
degree of control of systems and devices below the microscopic scale urged a
thermodynamic description of small systems far from equilibrium. In such
circumstances, essentially, one is interested in characterizing the energetics
of finite objects at finite times. Biological systems at the cellular level,
molecular machines, and colloidal particles trapped by optical tweezers, for
instance, are paradigmatic examples of physical objects operating under these
conditions. In contrast with macroscopic scenarios, the energy within small
systems is of the order of $k_{B}T$ and its fluctuations are not negligible.
In fact, deviations from the average behaviour are typical features of such
regimes and carry valuable thermodynamic information.

In general, these discussions belong to the recent field of stochastic
thermodynamics [10, 11, 12, 13, 14, 15], whose developments has been
successfully helping to bridge the gap between the well known macroscopic laws
and the small-scale behaviour of matter. In this framework, thermodynamic
systems are both driven by external classical forces and constantly influenced
by the fluctuations induced by their environments. Along these lines,
thermodynamic quantities, such as work, heat and entropy production, are
understood as fluctuating entities specified at the phase space trajectory
level and characterized by stochastic dynamical processes, often depicted by
Markovian master equations. Despite the shared probabilistic spirit,
stochastic thermodynamics exceeds the scope of equilibrium statistical
mechanics and the linear response regime, i.e., it allows the description of
general microscopic scenarios and the consideration of arbitrary non-
equilibrium processes, in such a way that the connection with the well
established macroscopic laws and conclusions are directly obtained once
considered the statistical ensemble level. Furthermore, this perspective
provided novel insights on the relationship between irreversibility and the
second law of thermodynamics. Interestingly, although predicting the
positivity of the average of entropy production, stochastic thermodynamics
highlighted the existence of trajectories with negative entropy production and
provided the mathematical formalism for quantifying their probabilities of
occurrence.

##### Fluctuations theorems

Along with a more refined view of thermodynamics, the so-called fluctuation
theorems (FTs) represent the most influential results of the field. Initially
discovered during the ’90s [16, 17, 18, 19, 20, 21, 22, 23, 24] and later
experimentally verified in several setups [25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 15], the FTs were instrumental in triggering further studies on non-
equilibrium systems and fostering the foundation of stochastic thermodynamics.
Essentially, the FTs are general mathematical statements concerning the
probability distributions of the relevant thermodynamic quantities over an
ensemble of identically prepared systems. They are usually expressed according
to their integral (IFTs) or detailed (DFTs) forms, given, respectively, by
[12]

$\langle e^{-\zeta}\rangle=\int d\zeta\,P(\zeta)e^{-\zeta}=1$ (21)

and

$\frac{P(\zeta)}{P(-\zeta)}=e^{-\zeta},$ (22)

where $\zeta$ is a functional of the stochastic trajectory, e.g., fluctuating
work and heat, and $P(\zeta)$ is its respective distribution. From Eq. (21),
one can easily use Jensen’s inequality101010For the exponential function,
Jensen’s inequality becomes $e^{-\langle\zeta\rangle}\geq\langle
e^{-\zeta}\rangle$. and unveil the hidden inequality

$\langle\zeta\rangle\geq 0.$ (23)

For instance, suppose a system initially prepared in thermal equilibrium at
temperature $T$ and then decoupled with its surroundings. If the system’s
Hamiltonian is parametrized as $H(\lambda)$, where $\lambda$ is an externally
controlled parameter, work $w$ can be performed by simply changing it from
$\lambda_{i}$ to $\lambda_{f}$ since $w\equiv\int\frac{\partial
H(\lambda)}{\partial\lambda}d\lambda$. By defining the dissipated work along
with a specific work protocol $\lambda_{i}\rightarrow\lambda_{f}$ as
$w_{diss}\equiv w-\Delta F_{fi}$, where $\Delta F\equiv F_{f}-F_{i}$ is the
Helmholtz free energy difference for the equilibrium states associated with
$H(\lambda_{i})$ and $H(\lambda_{f})$, one can show the following IFT

$\langle e^{-\beta w_{diss}}\rangle=1\>\Longleftrightarrow\>\langle e^{-\beta
w}\rangle=e^{-\beta\Delta F}$ (24)

and, therefore,

$\langle w_{diss}\rangle\geq 0\>\Longleftrightarrow\>\langle
w\rangle\geq\Delta F_{fi},$ (25)

where $\beta^{-1}\equiv k_{B}T$ with $k_{B}$ the Boltzmann constant and the
ensemble averages $\langle.\rangle$ are computed after performing the same
protocol several - ideally infinite - times. Interestingly, these expressions
make no mention of the protocol’s execution speed or its specific details. In
fact, it just depends on the final and initial values $\lambda_{f}$ and
$\lambda_{i}$. Eq. (24) is the emblematic - and surprising - Jarzynski
equality [19] relating arbitrary non-equilibrium processes,
$\lambda_{i}\rightarrow\lambda_{f}$, with genuine equilibrium quantities
$\Delta F_{fi}$, while the Eq. (25) implied by Eq. (24) is one of the well-
known forms of the second law of thermodynamics. Notice, however, that
$\langle w\rangle\geq\Delta F_{fi}$ is even more revealing than the classical
thermodynamic one since it is the average value of work that is higher than
the difference of free energy, which also means that for individual elements
of the ensemble such inequality may be violated. Of course, this conclusion
does not represent a contradiction with the classical result: for macroscopic
systems and their energy scales, the fluctuations are not easily observed, in
a way that the work performed by the execution of the same protocol are
essentially equal and, consequently, their distributions are mathematically
depicted by delta functions. Additionally, if $P_{\Lambda}(w)$ is the
probability distribution of performing some work $w$ along the trajectory
defined by the protocol $\Lambda:\lambda_{i}\rightarrow\lambda_{f}$ and
$P_{\tilde{\Lambda}}(w)$ is the distribution related to the trajectory
obtained by the time-reversed protocol,
$\tilde{\Lambda}:\lambda_{f}\rightarrow\lambda_{i}$, under the assumption of
Markovian dynamics and initial equilibrium states for both protocols, one can
show the following DFTs-like expression

$\frac{P_{\Lambda}(w)}{P_{\tilde{\Lambda}}(-w)}=e^{-\beta
w_{diss}}=e^{-\beta(w-\Delta F_{fi})},$ (26)

also known as the Crooks theorem. [23] Interestingly, Eq. (26) means the
probability of observing trajectories whose work spent along with $\Lambda$ is
given by $w$ is exponentially more likely than the probability of observing
work $-w$ along with the reverse protocol $\tilde{\Lambda}$. In fact, their
probabilities are equal iff the work performed is equal to the free energy
difference, $w=\Delta F_{fi}\Longleftrightarrow
P_{\Lambda}(w)=P_{\tilde{\Lambda}}(-w)$. Moreover, it is easy to see that Eq.
(26) directly implies the Jarzynski equality, Eq. (24)111111$\int
dw\,P_{\tilde{\Lambda}}(-w)e^{-\beta w}=e^{-\beta\Delta F}\int
dw\,P_{\Lambda}(w)\Rightarrow\langle e^{-\beta w}\rangle=e^{-\beta\Delta F}$..

Finally, by properly identifying entropy production, $\Sigma$, along with
stochastic trajectories, one can also show the following IFT for arbitrary
protocols and initial conditions [12, 13]

$\langle e^{-\frac{\Sigma}{k_{B}}}\rangle=1.$ (27)

Again, it also implies what can be seen as a more refined version of the
second law of thermodynamics depicted by Eq. (19) concerning the average value
of entropy production, such that

$\langle\Sigma\rangle\geq 0.$ (28)

In short, these theorems are simple equalities concerning the distributions of
stochastic thermodynamic quantities. Surprisingly, they are valid even for
general processes far from equilibrium. In this sense, despite their
simplicity, they represented generalizations of the classical macroscopic
thermodynamic results and provided novel insights on the underlying
statistical nature of the laws of thermodynamics.

###  2 Quantum mechanics

Quantum mechanics (QM) is one of the main pillars of modern science, not only
because of its remarkable accuracy and outstanding success in predicting
physical phenomena but also because it represents, along with Einstein’s
theory of general relativity and Maxwell’s of electrodynamics, our best
attempt to understand the very fundamental aspects of nature. Since its early
days, at the beginning of the 20th century, it became clear that the
development of QM would require a novel and courageous change in the
traditional way of thinking. After all, physics - as was commonly believed at
the time - was not complete, and the failure of classical theories to explain
the contemporary experimental observations was undeniable evidence of this
defeat. Hence, despite centuries of immense accomplishment, physicists were
forced to adapt in the face of empirical confrontation. Along these lines, the
so-called ultraviolet catastrophe is a paradigmatic example of how nature’s
behaviour contradicted classical intuition and, in contrast, radical new
hypotheses were suddenly necessary. This interesting episode is commonly seen
as the birth of the quantum theory, and Max Planck (1858 - 1947) is regarded
as one of its founding fathers. Latter, seminal works by Albert Einstein (1879
- 1955), Niels Bohr (1885 - 1962), Louis de Broglie (1892 - 1987), Erwin
Schrödinger (1887 - 1961), Werner Heisenberg (1901 - 1976) and many others
helped develop and establish the basis and mathematical structure of the
theory.

During the first decades of the last century, QM flourished both from a
theoretical perspective and experimental validation. The revolutionary ideas
of energy quantization, particle-wave duality and the intrinsic probabilistic
behaviour at the atomic scale, although highly non-intuitive, seemed to be a
superior way to describe how nature works. This early progress was of
fundamental importance to pave the way for the later development of quantum
field theory and the conception of modern technologies, such as the
groundbreaking developments of the transistor, the laser and, more recently,
magnetic resonance machines. However, despite this great success, QM remained
challenging conceptually. On the one hand, it was unclear how to physically
interpret some fundamental aspects of the theory, such as the wave function
and non-local correlations. On the other, there was a debate to understand if
it was indeed a complete description of reality. Interestingly, even after one
century of development and crucial progress, some of these foundational
questions remain open, which, in a sense, preserve these old debates echoing
throughout modern research. This, however, did not prevent the field from
expanding and developing. Applications of QM to specific questions gave rise
to new and exciting research fields. The birth of quantum information and
computation in the ’80s and their first progress during the ’90s, for
instance, are recent and fascinating ramifications of the quantum theory in
the context of information processing.

Nowadays, QM maintains its influence as one of the most fundamental theories
of modern physics and keeps changing the world consistently in several aspects
ranging from the production of new materials to the development of novel - and
unique - technologies. Along these lines, this section will be dedicated to
briefly introducing - or reviewing - the elementary aspects of the theory and
its particular mathematical formalism. Of course, it does not aim for an
extensive presentation of the subject and its most modern tools. A
comprehensive discussion of QM can be found in the excellent textbooks that
also guided the following introduction. [36, 37, 38, 39, 40]

####  1 Mathematical description of quantum systems

Quantum theory, essentially, is mathematically written and stated with the
vocabulary provided by linear algebra. In fact, the whole formalism of QM, and
its basic elements, can be described by vectors and linear transformations. In
this sense, complex vector spaces with inner product, known as Hilbert spaces,
are the most fundamental underlying structures setting the stage for
describing quantum systems, whose physical states are simply represented by
vectors within these abstract entities.

Along these lines, for every quantum system, there is an associated Hilbert
space $\mathcal{H}$ with the appropriate dimensionality $d:=dim(\mathcal{H})$
\- possibly infinity - such that its vector state is fully characterized and
labelled by, according to Dirac’s notation, the so-called ket
$|\Psi\rangle\in\mathcal{H}$. From now on, let us assume only finite-
dimensional Hilbert spaces. These vectors, or kets, encode all possible
information concerning the physical system in question and are the quantum
analogous to the usual notion of classical states in the phase space, given by
$(\overrightarrow{\boldsymbol{p}},\overrightarrow{\boldsymbol{q}})$121212Of
course, the analogy only corresponds to their similar descriptive roles inside
their particular theoretical frameworks.. Just like any regular vector, every
ket $|\Psi\rangle$ can be represented in terms of a basis, i.e., given a set
$\\{|b_{j}\rangle,j=1,...,d\\}$ of $d$ linearly independent kets spanning
$\mathcal{H}$, one can describe the state as the following superposition
(linear combination)

$|\Psi\rangle=\sum_{j=1}^{d}\alpha_{j}|b_{j}\rangle,$ (29)

where the coefficients $\\{\alpha_{j},j=1,...,d\\}$ are complex numbers.
Besides, it is worth mentioning that Hilbert spaces inherit all properties and
basic operations defined for arbitrary vector spaces. Along with
$\mathcal{H}$, we should also define the so-called dual vector space
$\mathcal{H}^{*}$, whose for every possible element $|\Psi\rangle$ of
$\mathcal{H}$, there is a single component $\langle\Psi|$ \- known as bra \-
from $\mathcal{H}^{*}$, such that

$|\Psi\rangle=\sum_{j=1}^{d}\alpha_{j}|b_{j}\rangle\longleftrightarrow\langle\Psi|=\sum_{j=1}^{d}\alpha_{j}^{*}\langle
b_{j}|,$ (30)

where $\alpha_{j}^{*}$ denotes the complex conjugate of $\alpha_{j}$. Now we
have all the necessary ingredients to properly define the inner product
between two elements $|\psi\rangle$ and $|\phi\rangle$ of $\mathcal{H}$ as
simply as

$\langle\psi|\phi\rangle,$ (31)

where $\langle\psi|\phi\rangle=(\langle\phi|\psi\rangle)^{*}$ and
$\langle\psi|\psi\rangle\geq 0$. Thus, two non-null kets are orthogonal,
relative to Eq. (31), iff $\langle\psi|\phi\rangle=0$. Also, it allows the
identification of the norm of a ket $|\psi\rangle$ as
$|||\psi\rangle||\equiv\sqrt{\langle\psi|\psi\rangle}$. In this sense, every
possible basis $\\{|b_{j}\rangle,j=1,...,d\\}$ of $\mathcal{H}$ should contain
orthogonal and normalized (unit norm) elements, i.e., they should satisfy the
orthonormality condition stated by $\langle b_{j}|b_{k}\rangle=\delta_{jk}$,
where $\delta_{jk}$ is the usual Kronecker delta.

In QM, the system’s state is encoded only in the "direction" of
$|\psi\rangle$. In fact, any ket from the set
$\\{c|\Psi\rangle\in\mathcal{H}|c\in\boldsymbol{C}_{\neq 0}\\}$ represents the
same physical state. For this reason, and latter interpretative convenience,
it is commonly assumed normalized kets for describing quantum systems,
$\langle\Psi|\Psi\rangle=1$, which also implies that, given Eq. (29),

$\sum_{j=1}^{d}\alpha_{j}^{*}\alpha_{j}=\sum_{j=1}^{d}|\alpha_{j}|^{2}=1.$
(32)

Interestingly, one could add a complex phase $c=e^{i\theta}$ with unit modulus
such that $|\Psi\rangle\rightarrow e^{i\theta}|\Psi\rangle$, in such a way
that $\langle\Psi|\Psi\rangle=1$ is still satisfied. Nevertheless, this phase,
often called as global phase, does not affect any physical prediction obtained
from $|\Psi\rangle$.

##### Operators and observables

In the following sections, we denote operators by $\hat{(.)}$. In this
context, operators refer to linear transformations
$\hat{T}:\mathcal{H}\rightarrow\mathcal{H}^{\prime}$ mapping one element
$|\Psi\rangle$ from a particular Hilbert space $\mathcal{H}$ to another
$|\Phi\rangle$ from $\mathcal{H}^{\prime}$, such that
$\hat{T}|\Psi\rangle=|\Phi\rangle$ and

$\hat{T}\left(\sum_{j=1}^{d}\alpha_{j}|b_{j}\rangle\right)=\sum_{j=1}^{d}\alpha_{j}(\hat{T}|b_{j}\rangle).$
(33)

Nevertheless, most of the time, we are interested in operators $\hat{O}$
acting within a specific Hilbert space, such that
$\hat{O}:\mathcal{H}\rightarrow\mathcal{H}$. From now on, the set of linear
operators satisfying this is denominated by $\mathcal{L}(\mathcal{H})$.
Without any loss of generality, let us restrain our discussion for such cases.

As shown earlier, kets can be represented in terms of any basis
$\\{|b_{j}\rangle,j=1,...,d\\}$ of $\mathcal{H}$. Similarly, operators might
be depicted by $d\times d$ matrices. In this sense, any operator
$\hat{O}\in\mathcal{L}(\mathcal{H})$ is fully characterized by the complex
matrix elements $o_{kj}\equiv\langle b_{k}|\hat{O}|b_{j}\rangle$ and written
in terms of the outer products defined by $|b_{k}\rangle\langle b_{j}|$, such
that

$\hat{O}=\sum_{k=1}^{d}\sum_{j=1}^{d}o_{kj}|b_{k}\rangle\langle b_{j}|.$ (34)

It is worth mentioning that any orthonormal basis of $\mathcal{H}$ is a valid
and equivalent basis of description, in the sense that all possible
representations are directly connected via similarity transformations. This
fact gives us the freedom to perform calculations in the most suitable one.
Thus, given Eqs. (29, 34), we have - in general -
$\hat{O}|\Psi\rangle=\sum_{k=1}^{d}\sum_{j=1}^{d}\alpha_{j}o_{kj}|b_{k}\rangle$.
In particular, the so-called identity operator
$\hat{1}\in\mathcal{L}(\mathcal{H})$ that maps any ket to itself, such that
$\hat{1}|\Psi\rangle=|\Psi\rangle$, is clearly obtained iff
$o_{kj}=\delta_{kj}$ in the equation above. Hence,

$\hat{1}=\sum_{j=1}^{d}|b_{j}\rangle\langle b_{j}|$ (35)

This important equality is true for any possible basis, and it is known as the
completeness relation. Often, we are interested in the applications of
different operators in a given ket, such as
$\hat{O}_{2}\hat{O}_{1}|\Psi\rangle$. Since they are depicted by matrices, the
general non-commutative property is, naturally, inherited, i.e.,
$\hat{O}_{1}\hat{O}_{2}\neq\hat{O}_{2}\hat{O}_{1}$. Sometimes, however,
operators might commute and, for this reason, it is convenient to define the
commutator $[,]$ between two operators as

$[\hat{O}_{1},\hat{O}_{2}]\equiv\hat{O}_{1}\hat{O}_{2}-\hat{O}_{2}\hat{O}_{1}.$
(36)

Let us present some import classes of linear operators. The Hermitian
conjugate of $\hat{O}$ \- or simply adjoint - defined by $\hat{O}^{\dagger}$,
is the unique operator which guarantees that the bra
$(\langle\Psi|\hat{O}^{\dagger})$ is the dual vector of the ket
$(\hat{O}|\Psi\rangle)$. Operationally, the matrix $\hat{O}^{\dagger}$ is the
conjugate transpose of $\hat{O}$, which means that $\langle
b_{k}|\hat{O}|b_{j}\rangle=(\langle
b_{j}|\hat{O}^{\dagger}|b_{k}\rangle)^{*}$. Nevertheless, if

$\hat{O}=\hat{O}^{\dagger},$ (37)

then $\hat{O}$ is denominated as a Hermitian operator. Additionally, Unitary
operators $\hat{U}$ are those who satisfy the following equality

$\hat{U}\hat{U}^{\dagger}=\hat{U}^{\dagger}\hat{U}=\hat{1},$ (38)

which also implies that they preserve the inner product, such that
$\langle\psi|\phi\rangle=(\langle\psi|\hat{U}^{\dagger})(\hat{U}|\phi\rangle)$.
Both Hermitian and Unitary linear operators are examples of a broader class of
operators, known as Normal ones. A Normal operators $\hat{N}$, by definition,
commutes with its own Hermitian conjugate, i.e.,

$[\hat{N},\hat{N}^{\dagger}]=0$ (39)

or $\hat{N}\hat{N}^{\dagger}=\hat{N}^{\dagger}\hat{N}$. Along these lines,
this class of operators are accompanied by the spectral theorem, which states
that [39]: if $\hat{N}\in\mathcal{L}(\mathcal{H})$ is normal, then it is
unitarily diagonalizable, i.e., there is a basis
$\\{|N_{j}\rangle,j=1,...,d\\}$ of $\mathcal{H}$ such that

$\hat{N}=\sum_{j=1}^{d}n_{j}|N_{j}\rangle\langle N_{j}|,$ (40)

where $\\{n_{j},j=1,...,d\\}$ and $\\{|N_{j}\rangle,j=1,...,d\\}$ are the
respective eigenvalues and eigenvectors of $\hat{N}$, since
$\hat{N}|N_{j}\rangle=n_{j}|N_{j}\rangle$. The form of Eq. (40) above is
commonly referred to as the spectral decomposition of $\hat{N}$.
Interestingly, if two different normal operators $\hat{N}_{1}$ and
$\hat{N}_{2}$ have spectral decompositions according to the same basis, they
must commute, i.e., if $\hat{N}_{k}=\sum_{j=1}^{d}n_{k,j}|N_{j}\rangle\langle
N_{j}|$ for $k=1,2$ share the same eigenvectors, then it is guaranteed that
$[\hat{N}_{1},\hat{N}_{2}]=0$.

In QM, meaningful physical quantities, such as position, momentum and spin,
are known as observables and mathematically described by hermitian linear
operators and the machinery presented above. Along these lines, given an
observable $\hat{O}\in\mathcal{L}(\mathcal{H})$, its eigenvalues
$\\{o_{j}\\}_{j=1,...,d}$ are understood as the possible outcomes of an
eventual measurement performed on the quantum system. The hermiticity
condition, stated by Eq. (37), guarantee that these values are real
numbers131313Given Eq. (40), if $\hat{N}=\hat{N}^{\dagger}$, then
$\\{n_{j}\\}_{j}\in\mathbb{R}$..

##### Projective measurements

Suppose that $\hat{B}\in\mathcal{L}(\mathcal{H})$ is the observable we are
interested in measuring, with the following spectral decomposition

$\hat{B}=\sum_{j=1}^{d}b_{j}|b_{j}\rangle\langle b_{j}|.$ (41)

If $|\Psi\rangle$ is the system’s state, and since
$\\{|b_{j}\rangle,j=1,...,d\\}$ constitute a valid basis of $\mathcal{H}$, let
us write it in the following representation
$|\Psi\rangle=\sum_{j=1}^{d}\alpha_{j}|b_{j}\rangle$, where
$\alpha_{k}\equiv\langle b_{k}|\Psi\rangle$. Notice that, in general,
$|\Psi\rangle$ is in a superposition of the eigenvectors of $\hat{B}$. Once a
measurement is performed, two things happen: i) as a result of the process,
the measurement apparatus obtains one of the possible eigenvalues of
$\hat{B}$; ii) the system’s state immediately collapses into one of the
superposition elements. Such state update corresponds to the eigenvector
relative to the measurement outcome, i.e., if $b_{k}$ is measured, then

$|\Psi\rangle\rightarrow|\Psi^{\prime}\rangle\equiv|b_{k}\rangle.$ (42)

This whole event - known as a projective measurement - is seen as
fundamentally probabilistic, i.e., in the orthodox presentation of QM, all
information encoded by $|\Psi\rangle$ are the statistics of the outcomes,
which are characterized by the coefficients $\\{\alpha_{j},j=1,...,d\\}$
playing the role of probability amplitudes. In this sense, by defining the so-
called projectors $\hat{\Pi}_{k}\equiv|b_{k}\rangle\langle
b_{k}|\in\mathcal{L}(\mathcal{H})$, the probability of measuring $b_{k}$ and
the system collapsing to $|b_{k}\rangle$ is given by

$Prob(b_{k})\equiv\langle\Psi|\Pi_{k}|\Psi\rangle=|\alpha_{k}|^{2}\geq 0,$
(43)

while the collapse itself is mathematically written as

$|\Psi^{\prime}\rangle=\frac{\Pi_{k}|\Psi\rangle}{\sqrt{\langle\Psi|\Pi_{k}|\Psi\rangle}}.$
(44)

Notice that, it means that if any posterior measurement of $\hat{B}$ is
performed right after the first one, the system would still be found at the
same state $|\Psi^{\prime}\rangle=|b_{k}\rangle$ with $Prob(b_{k})=1$. Also,
as expected, the normalization $\langle\Psi|\Psi\rangle=1$ guarantees that

$\sum_{k=1}^{d}Prob(b_{k})=1.$ (45)

It is clear that the physical act of measuring is not being modelled at all,
and the treatment above is, essentially, phenomenological. In fact, Eq. (43)
is a fundamental postulate of QM and, therefore, it is not demonstrable from
first principles141414Still today, there are several discussions on this
matter, especially on foundational questions of QM. [41].

From the measurement statistics of $\hat{B}$ above, one can compute its
average value simply as $\sum_{k=1}^{d}b_{k}Prob(b_{k})$. Thus, it is easy to
see that the expectation value of any given observable
$\hat{O}\in\mathcal{L}(\mathcal{H})$ relative to the state $|\Psi\rangle$ can
be defined as

$\langle\hat{O}\rangle\equiv\langle\Psi|\hat{O}|\Psi\rangle.$ (46)

It is worth mentioning that, even though $|\Psi\rangle$ might represent a
single physical quantum system, in practice, such probabilities are only
accessible at an ensemble level analysis, in a way that it is implicitly
assumed several identical copies of the same physical ket. This identical copy
setting is known as a pure ensemble.

##### Composite systems

Let us now generalize the previous formalism to include the description of -
possibly - several quantum systems151515Notice that the majority of the
machinery presented here also works for a single quantum systems that has
several degrees of freedom, such as orbital motion, spin, etc.. In many cases,
the quantum system being described is composed of two or more, non necessarily
equal, subsystems. If $\mathcal{H}^{(k)}$ is the Hilbert space of the $k$th
element of a group of $N$ subsystems, the whole Hilbert space
$\mathcal{H}^{(0)}$ is constituted by the following tensor product
$\mathcal{H}^{(0)}=\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}...\otimes\mathcal{H}^{(N)}$.
Of course, each subsystem might have its own dimensionality
$d^{(k)}:=dim(\mathcal{H}^{(k)})$, such that $d^{(0)}=\Pi_{k=1}^{N}d^{(k)}$.
The representation of possible states in $\mathcal{H}^{(0)}$ is given by the
tensor product structure
$|\psi^{(1)}\rangle\otimes|\psi^{(2)}\rangle...\otimes|\psi^{(N)}\rangle=|\psi^{(1)}\psi^{(2)}...\psi^{(N)}\rangle$,
where $|\psi^{(k)}\rangle\in\mathcal{H}^{(k)}$ stands for an individual ket
belonging to the $k$th subsystem. It means that, in the particular case in
which every subsystem is prepared in the state $|\Psi^{(k)}\rangle$, the whole
system is described by
$|\Psi\rangle=|\Psi^{(1)}\rangle\otimes|\Psi^{(2)}\rangle...\otimes|\Psi^{(N)}\rangle$.

For simplicity, let us focus on bipartite systems, i.e., quantum systems
composed of two individual subsystems. Thus, let
$\mathcal{H}^{(0)}=\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$ and
$\\{|b_{j}^{(1,2)}\rangle,j=1,...,d^{(1,2)}\\}$ be a possible basis of
$\mathcal{H}^{(1,2)}$. Then any bipartite state can be written as

$\displaystyle|\Psi\rangle$
$\displaystyle=\sum_{j=1}^{d^{(1)}}\sum_{k=1}^{d^{(2)}}a_{jk}|b_{j}^{(1)}\rangle\otimes|b_{k}^{(2)}\rangle,$
(47)

where $a_{jk}\equiv\langle b_{j}^{(1)},b_{k}^{(2)}|\Psi\rangle$ is the
probability amplitude of finding each individual subsystem into the states
$|b_{j}^{(1)}\rangle$ and $|b_{k}^{(2)}\rangle$. Eq. (47) above is very
general. In some cases, if $a_{jk}=c_{j}d_{k}$, then the whole system can be
written as

$|\Psi\rangle=|\phi^{(1)}\rangle\otimes|\phi^{(2)}\rangle,$ (48)

where $|\phi^{(1)}\rangle=\sum_{j=1}^{d^{(1)}}c_{j}|b_{j}^{(1)}\rangle$ and
$|\phi^{(2)}\rangle=\sum_{k=1}^{d^{(2)}}d_{k}|b_{k}^{(2)}\rangle$, i.e., for
each subsystem, one can attribute an individual local state
$|\phi^{(1,2)}\rangle$ that it is independent of the other. Whenever the
previous form of Eq. (48) is obtained, the whole system is denominated
separable, and we have a product state. Otherwise, the subsystems are non-
separable and, therefore, called entangled. Entanglement is a genuine quantum
property that, essentially, means the different quantum systems have
fundamentally correlated states, regardless of their physical spatial
distance. Interestingly, it implies that the whole entangled system is not
understood as the simple composition of individual subsystems but an
indivisible entity instead.

In this context, local operators
$\hat{O}^{(k)}\in\mathcal{L}(\mathcal{H}^{(k)})$ are the ones that act on
individual subsystems and, therefore, specific Hilbert spaces. In a global
perspective, however, a local operator acting in the subsystem $(2)$ of
$|\psi^{(1)}\rangle\otimes|\psi^{(2)}\rangle$, for instance, translates to

$\left(\hat{1}^{(1)}\otimes\hat{O}^{(2)}\right)|\psi^{(1)}\rangle\otimes|\psi^{(2)}\rangle=|\psi^{(1)}\rangle\otimes\hat{O}^{(2)}|\psi^{(2)}\rangle,$
(49)

which means that $\hat{O}^{(2)}$ acts on $|\psi^{(2)}\rangle$, while
$|\psi^{(1)}\rangle$ remains unchanged due to the identity operator
$\hat{1}^{(1)}$. Besides, one could also be interested in applying the
operators $\hat{O}^{(1)}$ and $\hat{O}^{(2)}$ simultaneously. In this case,
the whole operator is written as $\hat{O}^{(1)}\otimes\hat{O}^{(2)}$ and

$\left(\hat{O}^{(1)}\otimes\hat{O}^{(2)}\right)|\psi^{(1)}\rangle\otimes|\psi^{(2)}\rangle=\hat{O}^{(1)}|\psi^{(1)}\rangle\otimes\hat{O}^{(2)}|\psi^{(2)}\rangle.$
(50)

Along these lines, bipartite states can be cast in a very convenient and
special form, given by the Schmidt decomposition. Since this procedure is the
basis of this work, let us discuss it in more detail.

###### The Schmidt decomposition

The Schmidt decomposition is a very useful theorem concerning the
representation form of pure bipartite quantum states (or any composite one
with a bipartite separation, actually). Its convenience has far-reaching
consequences and applications in quantum theory, especially in the context of
quantum information and quantum computation. Essentially, it claims that any
pure bipartite state can be written compactly as a single sum of correlated
orthonormal basis. In fact, as briefly sketched below, this is a direct
consequence of a more general result concerning matrix factorization, known as
Singular Value Decomposition (SVD for short). Despite its simplicity and
aesthetical appeal, the Schmidt decomposition facilitates both the
entanglement analysis and the computation of reduced states161616Formally
presented in Section (1). when dealing with such systems. It is also worth
mentioning that this tool is closely related to the concept of state
purification.

Consider a pure bipartite quantum state
$|\Psi\rangle\in\mathcal{H}^{(0)}=\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$,
such that $dim(\mathcal{H}^{(1,2)})=d^{(1,2)}$ and, without any loss of
generality, $d^{(1)}\leq d^{(2)}$. Given a pair of local orthonormal basis
$\\{|b_{j}^{(1,2)}\rangle,j=1,...,d^{(1,2)}\\}$ for $\mathcal{H}^{(1,2)}$, we
know that $|\Psi\rangle$ can be written according to Eq. (47). Notice that
this general expression contains a double sum
$\sum_{j=1}^{d^{(1)}}\sum_{k=1}^{d^{(2)}}$, each one relative to its
respective subsystem’s Hilbert space. Clearly, this represents a superposition
over all possible combination of basis elements, where $|a_{jk}|^{2}$ quantify
the probability associated with the ket
$|b_{j}^{(1)}\rangle\otimes|b_{k}^{(2)}\rangle$. Alternatively, the
coefficients $a_{jk}$ can be seen as the entries of a $d^{(1)}\times d^{(2)}$
matrix $\overleftrightarrow{A}$, such that
$a_{jk}=[\overleftrightarrow{A}]_{jk}$. The SVD guarantee the following matrix
factorization below

$\overleftrightarrow{A}=\overleftrightarrow{L}\overleftrightarrow{\Lambda}\overleftrightarrow{R}^{\dagger},$
(51)

where $\overleftrightarrow{L}$ and $\overleftrightarrow{R}$ are matrices with
orthogonal columns, i.e.,
$\overleftrightarrow{L}^{\dagger}\overleftrightarrow{L}=\overleftrightarrow{R}^{\dagger}\overleftrightarrow{R}=\overleftrightarrow{1}$,
and $\overleftrightarrow{\Lambda}$ is a diagonal positive semi-definite
matrix. Since we are assuming $d^{(1)}\leq d^{(2)}$, both
$\overleftrightarrow{L}$ and $\overleftrightarrow{\Lambda}$ are square
$d^{(1)}\times d^{(1)}$ matrices, while $\overleftrightarrow{R}$ is a
$d^{(2)}\times d^{(1)}$ one171717If $d^{(1)}\geq d^{(2)}$, their forms would
be altered: $\overleftrightarrow{R}$ would be a square matrix while
$\overleftrightarrow{L}$ would be rectangular.. The diagonal elements
$\\{\lambda_{j},j=1,...,d^{(1)}\\}$ of $\overleftrightarrow{\Lambda}$ are
called its singular values, and the number $n\leq d^{(1)}$ of non-zero
$\lambda_{j}$ is known as the Schmidt rank. Thus,

$a_{jk}=\sum_{\eta=1}^{n}L_{j\eta}\lambda_{\eta}R_{k\eta}^{*},$ (52)

and, therefore,

$|\Psi\rangle=\sum_{\eta=1}^{n}\lambda_{\eta}|\varphi_{\eta}^{(1)}\rangle\otimes|\varphi_{\eta}^{(2)}\rangle,$
(53)

where
$|\varphi_{\eta}^{(1)}\rangle\equiv\sum_{j=1}^{d^{(1)}}L_{j\eta}|b_{j}^{(1)}\rangle$
and
$|\varphi_{\eta}^{(2)}\rangle\equiv\sum_{k=1}^{d^{(2)}}R_{k\eta}^{*}|b_{k}^{(2)}\rangle$.
Additionally, the normality condition of $|\Psi\rangle$ and the
$\overleftrightarrow{L}(\overleftrightarrow{R})$ column’s orthogonality
guarantee that $\sum_{j=1}^{n}\lambda_{j}^{2}=1$ and
$\langle\varphi_{m}^{(1,2)}|\varphi_{n}^{(1,2)}\rangle=\delta_{mn}$,
respectively. Eq. (53) is the well known Schmidt decomposition of
$|\Psi\rangle$ in its full glory, where the local kets
$\\{|\varphi_{j}^{(1)}\rangle,j=1,...,d^{(1)}\\}$ and
$\\{|\varphi_{j}^{(2)}\rangle,j=1,...,d^{(2)}\\}$ form an orthonormal basis
for $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$, called Schmidt basis, and the
singular values $\lambda_{j}$ are also known as the Schmidt coefficients of
$|\Psi\rangle$181818Depending on the reference, sometimes the square of the
singular values are denominated as the Schmidt coefficients.. These
coefficients are unambiguously defined, while the Schmidt basis are unique up
to eventual degenerate coefficients and phases. Such form is general and far
from being intuitive, and, despite the obvious simplification compared with
Eq. (47), it has many interesting features that justify its use: notice that
the double sum is changed for a single one bounded by the Schmidt rank $n\leq
d^{(1)}$, which means that regardless the dimension $d^{(2)}$ only an
$n$-dimensional subspace of $\mathcal{H}^{(2)}$ is relevant for the whole
state description; besides, it becomes evident that in order to have entangled
bipartite systems, or non-separable
$|\Psi\rangle\neq|\varphi^{(1)}\rangle\otimes|\varphi^{(2)}\rangle$, it is
required at least two non-zero Schmidt coefficients, i.e., $|\Psi\rangle$ is
an entangled state iff $n>$1; plus, it gives all necessary information for
writing the reduced states (as discussed in Section (1)).

##### Density operator

The framework presented until now was built to describe pure states
$|\Psi\rangle\in\mathcal{H}$. As mentioned earlier, every physical ket encodes
the probabilities associated with measurements performed on a chosen basis.
Such statistics, however, is only observed at the ensemble level, i.e., after
the access of a collection of identical copies of $|\Psi\rangle$. Although
feasible, it is clear that this is a particular case of a more general
ensemble. In most realistic scenarios, one has access to a statistical mixture
of $N$ arbitrary states
$\\{P_{\eta},|\Psi_{\eta}\rangle\in\mathcal{H},\eta=1,...,N\\}$, where
$\\{P_{\eta}\\}\geq 0$ with $\sum_{\eta=1}^{N}P_{\eta}=1$ characterizes the
distribution of pure states $|\Psi_{\eta}\rangle$. The mathematical entity
representing quantum ensembles is known as density operators or density
matrices and written as

$\hat{\rho}\equiv\sum_{\eta=1}^{N}P_{\eta}|\Psi_{\eta}\rangle\langle\Psi_{\eta}|\in\mathcal{L}(\mathcal{H}).$
(54)

If $P_{j}=1$ and $P_{\eta}=0$ for all $\eta\neq j$, then the quantum state
$\hat{\rho}=|\Psi_{j}\rangle\langle\Psi_{j}|$ is called pure. Otherwise, it is
referred to as a mixed state. Eq. (54) is the most general representation of a
quantum system. In fact, the whole formalism of QM can be stated in terms of
density operators $\hat{\rho}$. Notice that, instead of being represented by
kets, states are depicted by operators according to this framework. Also, in
contrast with the probabilities mentioned in the context of pure ensembles,
the distribution of states $\\{P_{\eta}\\}$ is fully classical, in the sense
that it only captures our ignorance concerning the preparation \- or access -
of possible quantum states. It is worth mentioning the pure states within the
set $\\{|\Psi_{\eta}\rangle\\}$ are not necessarily orthogonal. Nevertheless,
since $\hat{\rho}$ is clearly hermitian, one can write it conforming to its
spectral decomposition, such that

$\hat{\rho}=\sum_{j=1}^{d}p_{j}|\varphi_{j}\rangle\langle\varphi_{j}|,$ (55)

where $\\{|\varphi_{j}\rangle,j=1,...,d\\}$ is a basis of $\mathcal{H}$ and
$\\{p_{j}\\}\geq 0$ with $\sum_{j}^{d}p_{j}=1$. Along these lines, it is clear
that density matrices correspond to a particular class of operators within
$\mathcal{L}(\mathcal{H})$, satisfying some properties to characterize
physical systems. The corresponding density matrices’ subset will be
represented by $\mathscr{D}(\mathcal{H})$. Mathematically, any
$\hat{\rho}\in\mathscr{D}(\mathcal{H})$ should meet the following conditions:

  1. 1.

Hermitian: $\hat{\rho}=\hat{\rho}^{\dagger}$;

  2. 2.

Unit trace: $tr\\{\hat{\rho}\\}=1$191919The trace $tr\\{\hat{O}\\}$ of any
operator $\hat{O}$ is simply the sum of its diagonal elements. Since this
quantity is independent of the representation choice, one can use any basis to
compute it.;

  3. 3.

Positive semi-definite: $\hat{\rho}\geq 0$ or $\\{p_{j}\\}\geq 0$.

While condition 1. implies real eigenvalues, condition 2. states the
normalization of $\hat{\rho}$ or $\sum_{j=1}^{d}p_{j}=1$ and condition 3.
guarantee positive probabilities, $p_{j}\geq 0$ for all $j$. Besides, every
density matrix $\hat{\rho}$ satisfies the following inequality

$tr\\{\hat{\rho}^{2}\\}\leq 1.$ (56)

Interestingly, by computing Eq. (56), one can easily verify if the state
$\hat{\rho}$ is pure or mixed. Notice that $tr\\{\hat{\rho}^{2}\\}=1$ iff
$\hat{\rho}=|\Psi\rangle\langle\Psi|$ is pure. Otherwise, it is mixed. For
this reason, Eq. (56) is commonly known as state purity.

###### Projective measurements

Concerning measurements and observables. Let us suppose, again, the observable
$\hat{B}=\sum_{j=1}^{d}b_{j}|b_{j}\rangle\langle b_{j}|$. If a projective
measurement is performed in an ensemble characterized by Eq. (55), the
probability of obtaining $b_{k}$ is

$Prob(b_{k})\equiv tr\\{\hat{\Pi}_{k}\hat{\rho}\\}=\sum_{j=1}^{d}p_{j}|\langle
b_{k}|\varphi_{j}\rangle|^{2}\geq 0,$ (57)

which is, essentially, a sum over the probability of observing $b_{k}$
conditioned by each possible element of the mixture
$\\{p_{j},|\varphi_{j}\rangle,j=1,...,d\\}$. After the measurement, as
expected, the new state is pure and given by

$\hat{\rho}^{\prime}\equiv\frac{\hat{\Pi}_{k}\hat{\rho}\hat{\Pi}_{k}^{\dagger}}{tr\\{\hat{\Pi}_{k}\hat{\rho}\\}}=|b_{k}\rangle\langle
b_{k}|.$ (58)

From the expressions above, the average value of these measurements,
$\langle\hat{B}\rangle$, can be easily computed by
$\sum_{k=1}^{d}b_{k}Prob(b_{k})$. It is straightforward to check that
$\langle\hat{B}\rangle=tr\\{\hat{B}\hat{\rho}\\}.$ This previous relation can
be generalized to any observable $\hat{O}\in\mathcal{L}(\mathcal{H})$, such
that its expectation value $\langle\hat{O}\rangle$ relative to the ensemble
$\hat{\rho}$ is defined as202020Due to the cyclic property of the trace, the
equality $tr\\{\hat{A}\hat{B}\\}=tr\\{\hat{B}\hat{A}\\}$ is satisfied for any
pair of operators $\hat{A}$ and $\hat{B}$.

$\langle\hat{O}\rangle\equiv
tr\\{\hat{O}\hat{\rho}\\}=tr\\{\hat{\rho}\hat{O}\\}.$ (59)

###### Reduced states

If the quantum system of interest is composed of different subsystems, the
density operator formalism provide a straightforward manner to describe them.
Often, one has access only to a part of a much broader quantum system, in a
way that it is impossible to obtain the whole state description. In this
sense, the reduced density operators are the representative ones for
describing local states.

Let us suppose a bipartite system, represented by
$\hat{\rho}^{(0)}\in\mathscr{D}(\mathcal{H}^{(0)})$ and
$\mathcal{H}^{(0)}=\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$. The reduced
density operators are defined by

$\displaystyle\hat{\rho}^{(1)}$ $\displaystyle\equiv
tr_{2}\\{\hat{\rho}^{(0)}\\}\in\mathscr{D}(\mathcal{H}^{(1)}),$ (60)
$\displaystyle\hat{\rho}^{(2)}$ $\displaystyle\equiv
tr_{1}\\{\hat{\rho}^{(0)}\\}\in\mathscr{D}(\mathcal{H}^{(2)}),$ (61)

where $tr_{k}\\{(.)\\}$ is the partial trace over subspace $k$. Of course,
these quantities are genuine density operators and, therefore, inherit all
desired properties. In this sense, the statistics of local observables agrees
with the expected behaviour, i.e., suppose one perform measurements of an
observable $\hat{O}^{(1)}$ in the subsystem $(1)$, then

$\langle\hat{O}^{(1)}\rangle\equiv
tr\left\\{\left(\hat{O}^{(1)}\otimes\hat{1}^{(2)}\right)\hat{\rho}^{(0)}\right\\}=tr_{1}\left\\{\hat{O}^{(1)}\hat{\rho}^{(1)}\right\\}.$
(62)

If the whole system is described by
$\hat{\rho}^{(0)}=\hat{\rho}^{(1)}\otimes\hat{\rho}^{(2)}$, the state is
separable, such that the reduced density matrices are clearly
$\hat{\rho}^{(1,2)}=tr_{2,1}\\{\hat{\rho}^{(0)}\\}$. In these cases, access to
the local descriptions is enough to characterize the whole system.
Nevertheless, if
$\hat{\rho}^{(0)}\neq\hat{\rho}^{(1)}\otimes\hat{\rho}^{(2)}$, then the
subsystems are somehow correlated, in a way that even knowing both individual
local states, $\hat{\rho}^{(1,2)}$, one still cannot infer the whole state
$\hat{\rho}^{(0)}$.

Let us now consider the particular case in which the whole bipartite system is
described by a pure state, such that
$\hat{\rho}^{(0)}=|\Psi\rangle\langle\Psi|$ and, without any loss of
generality, $d^{(1)}\leq d^{(2)}$. Thus, there is no classical uncertainty
concerning the ensemble. Interestingly, as will be shown below, this is not
necessarily true for the subsystems. As we saw, in general, any ket
$|\Psi\rangle$ can be written as Eq. (47). However, the Schmidt decomposition
presented in Eq. (53) provides a more convenient form. Along these lines, the
whole state can be cast in the following form

$\hat{\rho}^{(0)}=\sum_{\eta,\alpha=1}^{n}\lambda_{\eta}\lambda_{\alpha}|\varphi_{\eta}^{(1)}\rangle\langle\varphi_{\alpha}^{(1)}|\otimes|\varphi_{\eta}^{(2)}\rangle\langle\varphi_{\alpha}^{(2)}|,$
(63)

such that the local states can be immediately computed,

$\displaystyle\hat{\rho}^{(1)}$
$\displaystyle=\sum_{\eta=1}^{n}\lambda_{\eta}^{2}|\varphi_{\eta}^{(1)}\rangle\langle\varphi_{\eta}^{(1)}|,$
(64) $\displaystyle\hat{\rho}^{(2)}$
$\displaystyle=\sum_{\eta=1}^{n}\lambda_{\eta}^{2}|\varphi_{\eta}^{(2)}\rangle\langle\varphi_{\eta}^{(2)}|.$
(65)

Hence, as long the whole system is pure, its Schmidt decomposition gives all
necessary information for inferring the local spectral decompositions above,
i.e., the square of the Schmidt coefficients,
$\\{\lambda_{\eta}^{2}(t),\eta=1,...,n\\}$, characterize the distributions
across the pure states given by the Schmidt basis,
$\\{|\varphi_{\eta}^{(1,2)}(t)\rangle,\eta=1,...,n\\}$. Notice that, despite
having no uncertainty from a global perspective, the local subsystems are
represented by mixed states with identical probabilities. In particular, since
by hypotheses, $d^{(2)}\geq d^{(1)}$ the subsystem $(2)$ must necessarily
contain $(d^{(2)}-n)$ null eigenvalues. Interestingly, this is true for any
conceivable bipartite system, regardless of the dimensions in question. Such a
non-intuitive result is a direct consequence of entanglement: as mentioned
earlier in Section (1), for $|\Psi\rangle$ to be entangled it is required at
least two non-zero Schmidt coefficients, i.e., the Schmidt rank $n>1$. In
contrast, if $n=1$ such that $\lambda_{\eta}=\delta_{k\eta}$, then
$\hat{\rho}^{(1,2)}=|\varphi_{k}^{(1,2)}\rangle\langle\varphi_{k}^{(1,2)}|,$
and the whole system is clearly separable,

$|\Psi\rangle=|\varphi_{k}^{(1)}\rangle\otimes|\varphi_{k}^{(2)}\rangle\rightarrow\hat{\rho}^{(0)}=|\varphi_{k}^{(1)}\rangle\langle\varphi_{k}^{(1)}|\otimes|\varphi_{k}^{(2)}\rangle\langle\varphi_{k}^{(2)}|.$
(66)

It is worth mentioning that the reasoning above is also commonly explored the
other way around. Given a quantum system $(1)$ described by
$\hat{\rho}^{(1)}\in\mathscr{D}(\mathcal{H}^{(1)})$, by adding an auxiliary
system $(2)$, the former can be seen as a subsystem of a bigger pure one
depicted by $|\Psi\rangle$, i.e.,
$\hat{\rho}^{(1)}=tr_{2}\\{|\Psi\rangle\langle\Psi|\\}$. This procedure is
known as purification of $\hat{\rho}^{(1)}$.

####  2 Quantum dynamics

In the previous subsection, we presented the mathematical tools and formalism
for describing quantum systems. Let us now briefly discuss how to express
their time evolution considering the different possible contexts.

##### Isolated and closed system dynamics

The most simple dynamical scenario consists of isolated and closed quantum
systems. By definition, isolated systems do not interact - in any way - with
any other kind of object, classical or quantum. In these cases, the described
system is fully quantized, and no other external entity is included -
implicitly or explicitly \- in the picture, i.e., the system in question
represents the totality of elements in the context under scrutiny.
Consequently, an isolated quantum system evolves autonomously in time,
uninfluenced by anything else outside its own existence. Of course, the whole
system itself may be constituted by individual interacting parts. In contrast,
closed quantum systems represent less restrictive scenarios. It is allowed the
presence of external interaction with classical agents, which is often
justified by semi-classical reasoning. In these cases, the considered
influences are implicit, and the totality of elements is not fully quantized.
In general, such situations portray quantum systems evolving in time according
to externally controlled (deterministic) fields driving its dynamics.

Mathematically, both cases are differentiated by the eventual time-dependency
of their time-translation generator. On the one hand, isolated systems have
fixed Hamiltonians. On the other hand, the external interaction considered in
closed systems induces time-dependent ones. Interestingly, despite the
fundamental - and subtle - differences, their dynamic behaviour is described
by the same functional form.

Thus, let us proceed considering a closed quantum system at time $t$
represented by $|\Psi(t)\rangle\in\mathcal{H}$. In general, given the initial
state $|\Psi(t_{0})\rangle$, we are interested in describing $|\Psi(t)\rangle$
at any subsequent time $t\geq t_{0}$. Formally, this task is performed by the
so-called time-evolution operator
$\hat{\mathcal{U}}(t,t_{0})\in\mathcal{L}(\mathcal{H})$, such that

$|\Psi(t)\rangle=\hat{\mathcal{U}}(t,t_{0})|\Psi(t_{0})\rangle.$ (67)

Essentially, $\hat{\mathcal{U}}(t,t_{0})$ is the linear operator that maps any
initial physical ket to a time-evolved one. Under the general requirements of
$\underset{t\rightarrow t_{0}}{lim}\,\hat{\mathcal{U}}(t,t_{0})=\hat{1}$, the
state normalization $\langle\Psi(t)|\Psi(t)\rangle=1$ for all $t$ and map
composition
$\hat{\mathcal{U}}(t_{2},t_{0})=\hat{\mathcal{U}}(t_{2},t_{1})\hat{\mathcal{U}}(t_{1},t_{0})$,
one can show that $\hat{\mathcal{U}}(t,t_{0})$ must be unitary,
$\hat{\mathcal{U}}(t,t_{0})\hat{\mathcal{U}}^{\dagger}(t,t_{0})=\hat{1}$, and
satisfies the following differential equation

$i\hbar\frac{d}{dt}\hat{\mathcal{U}}(t,t_{0})=\hat{H}(t)\hat{\mathcal{U}}(t,t_{0}),$
(68)

where $\hbar$ is the Planck’s constant, and
$\hat{H}(t)=\hat{H}^{\dagger}(t)\in\mathcal{L}(\mathcal{H})$ is the - possibly
- time-dependent Hamiltonian and the time-translation generator of this
dynamics. The general solution of Eq. (68) above is given by

$\hat{\mathcal{U}}(t,t_{0})=\overleftarrow{\mathcal{T}}e^{-\frac{i}{\hbar}\int_{t_{0}}^{t}ds\,\hat{H}(s)}$
(69)

where $\overleftarrow{\mathcal{T}}$ is the chronological time-ordering
operator. Of course, as mentioned earlier, if the system is isolated then
$\hat{H}$ is constant, and the time-evolution operator is considerably
simpler:

$\hat{\mathcal{U}}(t,t_{0})=e^{-\frac{i}{\hbar}\hat{H}(t-t_{0})}.$ (70)

Finally, combining Eqs. (67, 68) one obtains the usual Schrödinger equation
for describing the state’s dynamics,

$i\hbar\frac{d}{dt}|\Psi(t)\rangle=\hat{H}(t)|\Psi(t)\rangle.$ (71)

Alternatively, if the system of interest is initially described by a mixed
state
$\hat{\rho}(t_{0})=\sum_{j=1}^{d}p_{j}|\varphi_{j}(t_{0})\rangle\langle\varphi_{j}(t_{0})|\in\mathscr{D}(\mathcal{H})$,
each element of the ensemble will certainly time evolve according to Eq. (71)
above, such that
$|\varphi_{j}(t)\rangle=\hat{\mathcal{U}}(t,t_{0})|\varphi_{j}(t_{0})\rangle$.
Thus, at any later time $t\geq t_{0}$, the density matrix is given by

$\hat{\rho}(t)\equiv\hat{\Phi}_{t,t_{0}}\hat{\rho}(t_{0})=\hat{\mathcal{U}}(t,t_{0})\hat{\rho}(t_{0})\hat{\mathcal{U}}^{\dagger}(t,t_{0}),$
(72)

where

$\hat{\Phi}_{t,t_{0}}:\mathscr{D}(\mathcal{H})\rightarrow\mathscr{D}(\mathcal{H})$
(73)

is the so-called (linear) dynamical map representing the unitary time
evolution212121Notice that $\hat{\Phi}_{t,t_{0}}$ acts on an operator instead
of a ket. For this reason, these linear operators are commonly known as
superoperators.. Along these lines, it is straightforward to obtain the
density matrix counterpart of Eq. (71) for the equation of motion, given by
the so-called Liouville-von Neumann equation,

$i\hbar\frac{d}{dt}\hat{\rho}(t)=[\hat{H}(t),\hat{\rho}(t)].$ (74)

It is important to highlight that the eigenvalues $\\{p_{j},j=1,...,d\\}$ of
$\hat{\rho}(t)$ remain invariant for this class of time evolution.

In short, the whole dynamics of both isolated and closed systems are fully
characterized once having access to initial conditions and the unitaries
$\hat{\mathcal{U}}(t,t_{0})$, written as functionals of the Hamiltonians,
mapping states at different instants of time.

##### Open systems dynamics

This class of dynamics is fundamentally distinct from the isolated and closed
ones. As the name suggests, open quantum systems consist of physical systems
that explicitly interact with other - possibly many - quantum objects.
Differently from the other cases, as a consequence of the interaction, the
dynamics of open systems are unavoidably coupled and correlated to their
environments, which directly affect their time evolutions and lead to the
phenomena of decoherence and dissipation. More specifically, their equation of
motion is no longer unitary nor described by Eq. (74) in such a way that a
more general description formalism is required. Along these lines, an
individual subsystem of a larger entity is - in general - treated as an open
quantum system. As discussed earlier in Section (1), subsystems are
characterized by reduced states. Therefore, essentially, we are interested in
finding their dynamical equations. Solving this task is the most fundamental
objective of the subfield known as open quantum systems and is of imperative
importance for understanding the complex behaviour of quantum matter and the
development of future quantum devices. It is worth mentioning that open
systems represent the standard approach for many realistic scenarios: the
intrinsic fragility of quantum systems means that they are easily disturbed by
external influences. Nevertheless, if the interactions are sufficiently weak,
the system might be treated as approximately isolated. Let us focus on the
general aspects of open system dynamics.

Suppose the system of interest is described, at any time $t$, by
$\hat{\rho}^{(1)}(t)\in\mathscr{D}(\mathcal{H}^{(1)})$ and corresponds to a
subsystem of a larger isolated quantum system depicted by
$\hat{\rho}^{(0)}\in\mathscr{D}(\mathcal{H}^{(0)})$, such that
$\mathcal{H}^{(0)}=\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$, where
$\mathcal{H}^{(2)}$ is the Hilbert space that encompasses the environment of
$(1)$. Notice this configuration establishes a bipartition between the
described system and "everything else". Since the total system is isolated -
by hypotheses - given an initial state $\hat{\rho}^{(0)}(t_{0})$, its time
evolution is guaranteed to be unitary, such that

$\hat{\rho}^{(0)}(t)=\hat{\mathcal{U}}^{(0)}(t,t_{0})\hat{\rho}^{(0)}(t_{0})\hat{\mathcal{U}}^{(0)\dagger}(t,t_{0}),$
(75)

where
$\hat{\mathcal{U}}^{(0)}(t,t_{0})=e^{-\frac{i}{\hbar}\hat{H}^{(0)}(t-t_{0})}\in\mathcal{L}(\mathcal{H}^{(0)})$
is its time evolution operator, and
$\hat{H}^{(0)}\in\mathcal{L}(\mathcal{H}^{(0)})$ is the Hamiltonian relative
to the whole bipartition. This Hamiltonian have the following structure

$\hat{H}^{(0)}:=\hat{H}^{(1)}\otimes\hat{1}^{(2)}+\hat{1}^{(1)}\otimes\hat{H}^{(2)}+\hat{H}_{int},$
(76)

where $\hat{H}^{(1,2)}\in\mathcal{L}(\mathcal{H}^{(1,2)})$ are the bare
Hamiltonians for each subsystem, and
$\hat{H}_{int}\in\mathcal{L}(\mathcal{H}^{(0)})$ is the term representing all
the physical interactions between $(1)$ and $(2)$. It is important to
highlight the fact the bare Hamiltonians are local operators restricted to
their individual subspaces while the interaction is clearly non-local, in the
sense that it acts on the total Hilbert space.

According to Section (1), at any time $t$, the local states are simply

$\hat{\rho}^{(1,2)}(t)=tr_{2,1}\\{\hat{\mathcal{U}}^{(0)}(t,t_{0})\hat{\rho}^{(0)}(t_{0})\hat{\mathcal{U}}^{(0)\dagger}(t,t_{0})\\}.$
(77)

Thus, using Eqs. (74, 76, 77), it is straightforward to show that their exact
dynamical equations are

$i\hbar\frac{d}{dt}\hat{\rho}^{(1,2)}(t)=tr_{2,1}\\{[\hat{H}^{(0)},\hat{\rho}^{(0)}]\\}=[\hat{H}^{(1,2)},\hat{\rho}^{(1,2)}(t)]+tr_{2,1}\\{[\hat{H}_{int},\hat{\rho}^{(0)}]\\},$
(78)

where the commutator $[\hat{H}^{(1,2)},\hat{\rho}^{(1,2)}(t)]$ resembles the
contribution that appears in Eq. (74) and
$tr_{2,1}\\{[\hat{H}_{int},\hat{\rho}^{(0)}]\\}$ is an extra component due to
the interaction term, $\hat{H}_{int}$. While the former is the unitary part of
the dynamics, the latter contains non-unitary contributions222222This operator
might contain terms of the following form $[(.),\hat{\rho}^{(1,2)}(t)]$ as
well. The remaining part, however, is non-unitary.. It is clear that the role
played by the interaction is of extreme relevance to the local dynamics. In
fact, one can easily check that its absence would imply that both subsystems
would be individually isolated and, therefore, evolving in time independently,
i.e.,
$\hat{H}^{(0)}=\hat{H}^{(1)}\otimes\hat{1}^{(2)}+\hat{1}^{(1)}\otimes\hat{H}^{(2)}$
and
$i\hbar\frac{d}{dt}\hat{\rho}^{(1,2)}(t)=[\hat{H}^{(1,2)},\hat{\rho}^{(1,2)}(t)]$.
Also, from a global perspective, the interaction is responsible for inducing
the formation of both classical and quantum correlations within the system in
a way that
$\hat{\rho}^{(0)}(t)\neq\hat{\rho}^{(1)}(t)\otimes\hat{\rho}^{(2)}(t)$ for
general situations. Of course, the resulting local dynamics of subsystem $(1)$
fundamentally depends on the nature of $(2)$ that, in principle, could be as
simple as a single qubit or a complex many-body system. Nevertheless, in many
scenarios involving open quantum systems, it is commonly assumed that $(2)$ is
a reservoir with an infinite number of degrees of freedom or a large heat
bath. These situations, as expected, might be very complicated to be analysed
by the exact expressions above or even treated numerically. For this reason,
the usual approaches and techniques rely on approximative methods and
different hypotheses aiming to simplify the description, which also restrains
the analysis for specific regimes of validity. Along these lines, initial
uncorrelated states, weak-coupling and Markovian dynamics represent the most
common assumptions leading to the orthodox microscopic derivations of the
phenomenological master equations.

As we can see, the interaction breaks the unitary mapping given by Eq. (72).
This relationship, however, might be generalized by defining the dynamical map
$\hat{\Lambda}_{t,t_{0}}^{(1)}:\mathscr{D}(\mathcal{H}^{(1)})\rightarrow\mathscr{D}(\mathcal{H}^{(1)})$,
such that

$\hat{\rho}^{(1)}(t)\equiv\hat{\Lambda}_{t,t_{0}}^{(1)}\hat{\rho}^{(1)}(t_{0})=tr_{2}\\{\hat{\mathcal{U}}^{(0)}(t,t_{0})\hat{\rho}^{(0)}(t_{0})\hat{\mathcal{U}}^{(0)\dagger}(t,t_{0})\\}.$
(79)

Any physical dynamical map of this kind should be completely positive and
trace-preserving, also known as a CPTP-map, i.e.,

  1. 1.

Completely positive: $\hat{\Lambda}_{t,t_{0}}^{(1)}\hat{\rho}^{(1)}(t_{0})\geq
0$;

  2. 2.

Trace-preserving:
$tr\\{\hat{\Lambda}_{t,t_{0}}^{(1)}\hat{\rho}^{(1)}(t_{0})\\}=tr\\{\hat{\rho}^{(1)}(t_{0})\\}=1$.

Similarly to the time evolution operator when considering physical kets, these
conditions are the most basic requirements that guarantee a linear mapping
between proper density matrices. Thus, the whole dynamics is fully
characterized by a set of CPTP dynamical maps,
$\\{\hat{\Lambda}_{t,t_{0}}^{(1)},t\geq t_{0}\\}$, describing the density
matrix throughout the time evolution. In some scenarios, the set of dynamical
maps satisfies the so-called semigroup property
$\hat{\Lambda}_{t_{2},t_{0}}^{(1)}=\hat{\Lambda}_{t_{2},t_{1}}^{(1)}\hat{\Lambda}_{t_{1},t_{0}}^{(1)}$
for $t_{2}\geq t_{1}\geq t_{0}$ which also implies the usual Lindblad-like
form for master equations.

For instance, let us consider initially uncorrelated systems, such that the
whole bipartite system is depicted by a product state
$\hat{\rho}^{(0)}(t_{0})=\hat{\rho}^{(1)}(t_{0})\otimes\hat{\rho}^{(2)}(t_{0})$.
If the spectral decomposition of subsystem $(2)$ at time $t_{0}$ is given by

$\hat{\rho}^{(2)}(t_{0})=\sum_{\alpha=1}^{d^{(2)}}p_{\alpha}(t_{0})|\varphi_{\alpha}^{(2)}(t_{0})\rangle\langle\varphi_{\alpha}^{(2)}(t_{0})|,$
(80)

then Eq. (79) becomes

$\hat{\rho}^{(1)}(t)=\hat{\Lambda}_{t,t_{0}}^{(1)}\hat{\rho}^{(1)}(t_{0})=\sum_{\alpha=1}^{d^{(2)}}\sum_{\beta=1}^{d^{(2)}}\hat{K}_{\beta\alpha}^{(1)}(t,t_{0})\hat{\rho}^{(1)}(t_{0})\hat{K}_{\beta\alpha}^{(1)\dagger}(t,t_{0}),$
(81)

where

$\hat{K}_{\beta\alpha}^{(1)}(t,t_{0})\equiv\sqrt{p_{\alpha}(t_{0})}\langle\varphi_{\beta}^{(2)}(t_{0})|\hat{\mathcal{U}}^{(0)}(t,t_{0})|\varphi_{\alpha}^{(2)}(t_{0})\rangle\in\mathcal{L}(\mathcal{H}^{(1)})$
(82)

and
$\sum_{\alpha,\beta=1}^{d^{(2)}}\hat{K}_{\beta\alpha}^{(1)\dagger}(t,t_{0})\hat{K}_{\beta\alpha}^{(1)}(t,t_{0})=\hat{1}^{(1)}.$
The form of Eq. (81) is known as the Kraus representation \- or operator sum
representation - and is the general structure of a dynamical map, where the
operators
$\\{\hat{K}_{\beta\alpha}^{(1)}(t,t_{0}),\beta,\alpha=1,...,d^{(2)}\\}$ are
the Kraus operators. In particular, if there is only one Kraus operator
$\hat{K}_{j}^{(1)}(t,t_{0})$, then the map simplifies to
$\hat{\Lambda}_{t,t_{0}}^{(1)}\hat{\rho}^{(1)}(t_{0})=\hat{K}_{j}^{(1)}(t,t_{0})\hat{\rho}^{(1)}(t_{0})\hat{K}_{j}^{(1)\dagger}(t,t_{0})$
and the time evolution is unitary, since
$\hat{K}_{j}^{(1)\dagger}(t,t_{0})\hat{K}_{j}^{(1)}(t,t_{0})=\hat{1}^{(1)}$.

In short, the dynamics of open systems are considerably more complicated than
the isolated and closed ones and require a more general approach for treating
them. As mentioned earlier, in realistic scenarios, any object is unavoidable
coupled to its external environment. Its interactions often lead to an
intricate non-unitary time evolution that is not easily described by analytic
means. In this sense, all the formalism and approximations developed within
the context of open quantum systems are useful tools for understanding the
behaviour of interacting systems.

###  3 Quantum thermodynamics

Given the previous introductions from Sections (1) and (2), it is clear that
thermodynamics and quantum mechanics are both successful and well established
physical theories developed and applied to different aspects of natural
phenomena. However, it is not a priori obvious to understand how they connect.
The research program coined as quantum thermodynamics (QT) is relatively young
and has been emerging as a meaningful scientific field during the last couple
of decades. Its name is self-explanatory232323Interestingly, this is not the
case for many things in science. since QT has the ambition to develop a
consistent theory that both extends and applies thermodynamics for general,
and possibly nonequilibrium, quantum systems and to describe the classical
thermodynamic behaviour from the underlying fundamental quantum dynamics. Both
major goals are clearly complementary and represent different valid
perspectives on the same subject, which partially explains the diverse
community forming around QT and the plethora of approaches and ideas coming
from distinct - but closely related - areas, especially from quantum
information theory and open quantum systems. For a general introduction to
this topic, see. [42, 43, 44, 45, 46] A more detailed discussion on QT can be
found in. [47, 48, 49]

Although the recent impressive booming of the field, it is interesting to
notice that, historically, thermodynamics has always been assisting quantum
mechanics. In fact, it was present at the birth of quantum theory [50, 51]
and, during the last century, supported some crucial technological advances,
in particular in the development of the lasers and masers. [52, 53, 54, 55]
However, despite being centenary theories, it is no surprise, nor coincidence,
that questions concerning the thermodynamics of quantum systems have been
rapidly coming to light during present days: On the one hand, since the early
90s seminal results on nonequilibrium thermodynamics [16, 17, 18, 19, 21], the
field of stochastic thermodynamics flourished and has been successfully
bridging the gap between the understanding of macroscopic and microscopic
thermodynamic processes [10, 12]; On the other hand, the current state-of-the-
art technology allows the precise fabrication, control and measurement of
truly quantum objects in a wide variety of platforms, ranging from solid-state
systems to trapped ions and optical setups. In this context, questions
concerning whether or not it is possible to expand the laws of thermodynamics
to even smaller - and possibly quantum - systems is both a logical step and a
current technological urgency. In fact, along with quantum computation and
quantum information, QT is an important player in the contemporary
technological revolution, the so-called quantum technologies 2.0. Its primary
purpose is to design quantum devices capable of intentionally harvesting
quantum properties, such as coherence and entanglement, to outperform their
classical counterparts and - hopefully - execute functions that no classical
machines could do, even in principle. At the core of these technologies, it is
necessary to understand the underlying interplay between work, heat and
information. On this wise, QT provides the most natural framework to deal with
the design and manipulation of quantum heat engines [53, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67], which is a fundamental step toward the
development of efficient and stable functioning quantum machines. However, it
is important to emphasize that, to this date, it is still not clear to what
extent such kinds of devices would be feasible in practical scenarios.
Physicists are optimistic, though. Currently, QT investigations are already
accessible and performed in the traditional experimental setups for quantum
mechanics studies, e.g., superconducting devices [68, 69, 70, 71, 72, 73, 74,
75, 76], nitrogen-vacancy (NV) centers [77, 78], nuclear magnetic resonance
(NMR) [79, 80, 81, 82], trapped ions [83, 84, 85, 86], ultracold atoms [87,
88, 89] and others. In this sense, much experimental research in QT is driven
by questions concerning cyclic processes, quantum heat engines and quantum
refrigerators. [84, 77, 80, 81, 85, 86, 90] For more details and further
references, see. [91] Additionally, from a theoretical perspective, QT has the
potential to shed some light on fundamental questions on the foundations of
quantum mechanics and statistical physics, especially those concerning the
emergence of macroscopic behaviour from first principles, e.g., the transition
from quantum to classical mechanics, irreversibility, thermalisation and the
measurement problem.

As briefly mentioned earlier, QT has been approached from many different
perspectives and attitudes, each one tackling particular problems with its own
set of tools. This diversity gave rise to smaller communities and several
research branches within QT. Along these lines, in addition to studies with
quantum heat engines, one can find investigations on quantum thermometry [92,
93, 94, 95, 96, 97, 98], quantum batteries [99, 100, 101, 102, 103],
stochastic quantum thermodynamics [104, 105, 106, 107, 108, 109] and
thermodynamics of quantum information [110, 111, 112, 113, 71, 114], just to
name a few. All these subareas have been reporting interesting and relevant
theoretical progress, although not in a cohesive manner yet. In particular,
pertinent developments are being achieved by resource theoretic notions
imported from quantum information theory and applied to QT. [115, 116, 117,
118, 119, 120] Resource theories are robust axiomatic and operational
mathematical descriptions of the so-called resources and free operations:
while the former refers to desired - and possibly scarce (just like any
resource) - physical properties, like entanglement and coherence, the latter
refers to the set of feasible and accessible operations. Interestingly, using
this framework, one can derive general results concerning possible state
transformation under a given set of constraints. At present, there is a
plethora of different resource theories, each one dealing with its particular
resources and set of operational restrictions. [121, 122] In the thermodynamic
context, nonequilibrium states are seen as resources, while free operations
include the addition of thermal baths and global unitaries preserving energy.
[123, 124] It is interesting to appreciate how similar such reasoning is
compared with the usual phenomenological approach to thermodynamics.

Alternatively, instead of knowing whether or not a given transformation is
possible, one might be interested in the process itself [125, 126], i.e., in
all the dynamical aspects within a time-dependent transformation, which
certainly includes both energetic and entropic changes. In this sense,
describing the dynamics of any desired property in a given protocol is of
fundamental importance for a complete quantum thermodynamic theory.
Furthermore, such knowledge is clearly necessary for technological
applications where is expected a high degree of control during the preparation
and manipulation of sophisticated and fragile quantum states. In fact, the
most promising - and interesting - applications will demand local descriptions
of interacting quantum systems, especially when dealing with quantum heat
engines and contexts like quantum control and quantum sensing. For these
reasons, the framework of open quantum systems [40] is a natural - and
unavoidable - candidate to approach quantum thermodynamics and has been
extensively used in several contexts. Along these lines, both the
understanding of open quantum systems and the tools used to describe them are
slowly pushing the usual orthodox thermodynamic scenario to a wide range of
situations.

Given the relatively young age of QT, there is no surprise to notice the
presence of several open problems. In fact, considering its stage, this is
expected. However, despite all current advances and efforts, the lack of
consensus on central aspects of the theory is particularly notorious. This
situation can be explained, at least partially, by the still unknown
thermodynamic role of quantum properties, i.e., once considered interacting
quantum systems, subtleties concerning entanglement, coherence, and the
interaction should be carefully scrutinized. Of course, classical
thermodynamics give us some expectations of how things should occur.
Nevertheless, such properties vanish at the appropriate classical limit: while
genuine quantum phenomena are absent in any classical setting, the interaction
is negligible for macroscopic systems. In this sense, (remarkably) there is
still no acceptable general definitions for the quantum counterparts of the
most basic thermodynamic quantities, which highlights the need for further
investigations at the foundational and conceptual levels.

Along these lines, the obtention of a general quantum thermodynamic entropy
remains elusive. Different alternatives can be found in the literature [127,
128, 129, 130, 131], but, unsurprisingly, most current approaches are based on
information-theoretic perspectives. [132] Despite its success for microscopic
classical systems [133] and particular \- orthodox - quantum scenarios [56,
134], it does not satisfy the expected properties for a proper generalization
of thermodynamic entropy and the second law. Unfortunately, this is also the
case for internal energy and the first law. There is no ambiguity in
identifying the internal energy of isolated quantum systems, i.e., this role
is unambiguously assigned to the expectation value of its Hamiltonian.
However, it is not clear how to proceed once considering arbitrary open
quantum systems. In such cases, the notion of physical local internal energy
is clouded by existing non-negligible interactions and correlations between
the system of interest and its surroundings. Naturally, any attempt
description of energy exchanges inherits such basic dubiety. In this sense,
classical thermodynamics states that energy flow is divided into two
complementary and fundamentally distinct categories, work and heat. Such
splitting for the quantum case also carries some intrinsic difficulties. As
mentioned earlier in Section (1), work and heat are defined along with
trajectories, which establish an extra conceptual barrier for directly
translating them to the quantum realm. [135] More importantly, it is uncertain
how to account for the energetic contributions of coherence and quantum
correlations into these elements. Still, most of the current strategies are
grounded on the classical - and pragmatical - reasoning that work is
associated with the energy transferred in a controllable and deterministic
fashion via the precise control of external parameters, and heat is linked to
randomly exchanged energy during a given process and entropic variation. This
perspective has been explored in different ways and with distinct frameworks.
On the one hand, many efforts were driven by the quest for the quantum
versions of FTs and measurement-based approaches. [104, 136, 105] On the other
hand, a fully quantum dynamic description is also sought from a less
operational point of view. [56, 125, 126] Interestingly, it should be
emphasized that despite all the above discussion, even the usual assumption of
splitting the energy flow solely in terms of work and heat can be debatable
for the quantum case. [137]

Additionally, another critical aspect concerns the enormous difficulty of
describing the dynamics of open quantum systems. As briefly mentioned in the
overview of Section (2), the time-evolution of reduced states may be extremely
convoluted and rarely solvable exactly. Besides, unless we are dealing with
simple physical systems, this type of description and analysis is a formidable
task, even for numerical methods. Thus, most procedures assume several
approximations for microscopically deriving more tractable dynamical
equations, which inevitably restrains its validity for specific conditions and
scenarios. In particular, it is often considered weak-coupling regimes and
Markovian dynamics leading to the usual Lindblad-like form of master
equations.242424In the literature, it is also commonly referred to as Gorini,
Kossakowski, Lindblad, and Sudarshan (GKLS) equation. [138, 139, 140, 141, 56,
126, 142] For more complex situations, when several interacting subsystems are
being described, alternative approaches also consider the so-called local
master equations by strategically neglecting some interaction terms, which
simplifies the analysis even further. [143, 144] Nevertheless, these
approximations can lead to thermodynamic inconsistency or unphysical
situations when not taken into account carefully. [145, 146, 147, 148, 149]
Alternatively, it was recently shown the possibility to derive
thermodynamically compatible master equations by employing additional
selective hypotheses. [150] However, it is clear that relying on
approximations and other restrictive assumptions also poses a critical
limitation to the development of a general thermodynamic description of
quantum systems, especially if one intends to characterize strongly-coupled
systems and further arbitrary contexts. In this sense, there are efforts to
extend the usual approach for broader scenarios. [151, 152, 153, 154, 155,
156, 157, 158, 109, 159, 160, 161, 162, 163, 164]

In addition to the approximate treatment commonly employed in QT, a semi-
classical description is also implicitly assumed, i.e., although not usually
highlighted, the addition of a classical external agent is a fundamental part
of the standard formalism, especially for closed quantum systems. Essentially,
this agent is responsible for controlling the dynamics of the system of
interest and inducing its Hamiltonian time-dependency, which yield the energy
exchange directly interpreted as work. It is also the relevant party for
measuring the system and, eventually, processing the accessible information.
From a physical point of view, this control is mediated by the use of external
fields interacting with the system in question in such a way that it maintains
its quantum properties. However, they are not explicitly included in the
physical description. Instead, they are regarded as classical fields whose
interaction induces an effective time-dependency in the system’s Hamiltonian.
Despite its practical relevance, this perspective of the so-called "coherent
control" [165] does not contemplate autonomous quantum systems and, therefore,
limits ourselves to the thermodynamic description of classically driven
devices, where both the quantum nature of the fields is unimportant relative
to the system of interest and the system itself does not affect the control’s
state. In contrast to a semi-classical approach, autonomous quantum systems
are isolated and do not have Hamiltonian time-dependency, which means that all
relevant parties - including the control agent - are quantized. Of course, it
also implies that the semi-classical description is a limit regime. Along with
this more general and fundamental picture, one is interested in understanding
the thermodynamics within isolated quantum systems, which would enable the
design and characterization of autonomous quantum machines. In this sense, any
attempt of developing a thermodynamic description of quantum systems that
assumes a time-dependent Hamiltonian a priori is not fully quantum and,
essentially, a phenomenological approach. Interestingly, this fundamental -
and conceptual - issue is not mentioned very often in the literature of QT.
For some discussions on autonomous quantum machines, see. [166, 167, 168, 169,
170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181] More recently,
[182] presented a formal treatment (although restricted to the usual
thermodynamic scenario) on this topic. It is worth mentioning that these
questions are also at the heart of quantum (optimal) control theory. See [183,
184, 185, 186] for more discussions on this matter.

Finally, the recent global interest in developing a quantum thermodynamic
theory is growing fast each year. The contemporary version of the industrial
revolution is spearheaded by current progress in understanding and developing
genuine quantum technologies for information processing, communication and
sensing. In this context, a fully matured framework of QT will certainly play
a leading role in the design and operation of functional quantum devices.
Nevertheless, despite being a promising field, considering its actual stage of
achievements and open questions, it is still in its infancy. More
specifically, additional investigations are needed at the conceptual level, as
several fundamental aspects are still under scrutiny. In short, QT is an
interesting, stimulating and young research field whose investigations and
potential scientific breakthroughs will, in one way or another, help to shape
future technologies and the understanding of quantum mechanics. Hopefully,
future historians of physics will view this chapter of science as we see past
developments today.

##  Chapter 2 Schmidt decomposition approach to quantum thermodynamics

In Section (3), we briefly introduced the current efforts and growing progress
on the development of a self-consistent thermodynamic theory of quantum
systems. Despite clear advancements toward this goal, the field of quantum
thermodynamics (QT) still has some fundamental issues to be addressed. In this
sense, most theoretical frameworks inherit some of the phenomenological spirit
of the classical theory and do not provide suitable tools to characterize and
understand thermodynamic processes within genuine autonomous quantum machines,
i.e., most modern approaches are built on the top of semi-classical
descriptions and approximative regimes, which clearly limit their range of
applicability.

In this chapter, we are interested to address and contribute to such more
foundational aspects of the theory. In the following pages, we are going to
present a novel approach to the thermodynamic analysis of autonomous quantum
systems. Our proposal is exact and based on the well-known procedure of the
Schmidt decomposition for bipartite systems111Or single systems where there is
coupling between diferent degrees of freedom, e.g., spin-orbit interaction..
Interestingly, despite being simple and providing a powerful statement, it is
still not explicitly explored in the context of QT. This framework will allow
us to describe the dynamics and energetics within generic interacting
subsystems in a symmetrical fashion, i.e., regardless of their individual
properties, details and dimension, they will be treated on equal footing. In
addition, it will not require any complementary hypotheses and approximations,
such as the commonly used ones concerning the Hamiltonian structure,
interaction regimes and type of dynamics, i.e., strict energy conservation,
weak-coupling, markovianity, etc. Formally, we will introduce time-dependent
local effective Hamiltonians that naturally embrace both their respective bare
ones and the contributions of the interaction term. These elements will be
identified as the representative operators for characterizing the subsystem’s
physical internal energies, in a way that will allow us to extend the usual
classical thermodynamic notion of energy additivity to general interacting
quantum systems222The following main discussions and results can be found in
[1], submitted after this thesis defense..

The outline of this chapter is the following: Section (1) formally introduces
and details the main setup of analysis, which consists of an isolated pure
bipartite quantum system. Also, it establishes the mathematical notation used
throughout the chapter. Then, Section (2) presents the foundations of our
formalism. More specifically, it discusses and describes the Schmidt basis
dynamics and identifies their time-translation generators as the so-called
local effective Hamiltonians. The following Section (3) briefly mentions how
the local dynamics are represented relative to these operators and the Schmidt
basis/coefficients. Then, in Section (4), the notion of local internal
energies is discussed in the context of QT. Also, it is argued that the
effective operators previously defined are suitable candidates for
characterizing the subsystem’s physical internal energy. Along these lines,
Section (5) introduces and discusses the consequences of the intrinsic
phase/frame gauge degree of freedom underlying the Schmidt decomposition and,
therefore, inherited by our formalism. More importantly, it is presented a
procedure for fixing it. Section (6) focuses on the analysis of current
approaches for defining quantum versions of thermodynamic quantities. After
that, Section (7) shows a proof of principle of the proposed formalism. Then,
Section (8) generalizes the previous results to mixed bipartite states. And,
finally, Section (9) briefly discusses these results and summarizes the
chapter.

###  1 The setting

As mentioned earlier, we consider a finite, isolated and nondegenerate pure
quantum system composed of two smaller interacting subsystems. Throughout this
thesis, the whole system and its global quantities will be labelled by $(0)$,
while the parts and their relative local properties will be identified by
$(1)$ and $(2)$. Let $\mathcal{H}^{(k)}$, with $k=0,1,2$, be their Hilbert
spaces with dimensions $d^{(k)}:=dim(\mathcal{H}^{(k)})$, such that
$d^{(0)}=(d^{(1)}+d^{(2)})$ and - without any loss of generality \-
$d^{(1)}\leq d^{(2)}$. Since we are interested in describing a fully quantum
autonomous object, the whole system Hamiltonian $\hat{H}^{(0)}$ generating its
dynamics is time-independent and given by

$\hat{H}^{(0)}:=\hat{H}^{(1)}\otimes\hat{1}^{(2)}+\hat{1}^{(1)}\otimes\hat{H}^{(2)}+\hat{H}_{int},$
(1)

where $\hat{1}^{(1,2)}\in\mathcal{L}(\mathcal{H}^{(1,2)})$ are the identity
operators, $\hat{H}^{(1,2)}\in\mathcal{L}(\mathcal{H}^{(1,2)})$ are the local
bare Hamiltonians of each subsystem and
$\hat{H}_{int}\in\mathcal{L}(\mathcal{H}^{(0)})$ is the term that encompasses
all the internal interactions between them. It is important to emphasize that
no additional hypothesis will be considered, especially concerning dynamical
features or the Hamiltonian/interaction structure, e.g., markovianity,
uncorrelated states, weak coupling, specific interaction Hamiltonian, etc. The
following description is general and exact.

At any time $t$ the whole pure system is described by a ket
$|\Psi(t)\rangle\in\mathcal{H}^{(0)}=\mathcal{H}^{(1)}\otimes\mathcal{H}^{(2)}$.
Besides, since it is isolated, its dynamics is governed by the usual
Schrödinger equation
$i\hbar\frac{d}{dt}|\Psi(t)\rangle=\hat{H}^{(0)}|\Psi(t)\rangle$; thereby, for
any initial state $|\Psi(t_{0})\rangle$ and $t\geq t_{0}$ we have

$|\Psi(t)\rangle=\hat{\mathcal{U}}(t,t_{0})|\Psi(t_{0})\rangle,$ (2)

where
$\hat{\mathcal{U}}(t,t_{0})=e^{-\frac{i}{\hbar}\hat{H}^{(0)}(t-t_{0})}\in\mathcal{L}(\mathcal{H}^{(0)})$
is the time-evolution operator of the whole. As usual, such bipartite state
could be written in any conceivable basis, e.g.,
$|\Psi(t)\rangle=\sum_{i=1}^{d^{(1)}}\sum_{j=1}^{d^{(2)}}\psi_{ij}(t)|b_{i}^{(1)},b_{j}^{(2)}\rangle$,
nevertheless the well known Schmidt decomposition guarantee the following
specific and convenient form (see (1))

$|\Psi(t)\rangle=\sum_{j=1}^{d^{(1)}}\lambda_{j}(t)|\varphi_{j}^{(1)}(t)\rangle\otimes|\varphi_{j}^{(2)}(t)\rangle,$
(3)

for every instant $t$, where $\\{\lambda_{j}(t)\geq 0;j=1,...,d^{(1)}\\}$ and
$\\{|\varphi_{j}^{(k)}(t)\rangle;j=1,...,d^{(1)}\\}$333Unless it is necessary,
from now on, any set will be represented simply as $\\{a_{j}\\}_{j}$ where $j$
is the index counting its elements, and the range is
implicit.$\in\mathcal{H}^{(k)}$ are the time-local Schmidt coefficients and
local Schmidt basis of subsystem $(k)$, respectively. The normalization
condition $\langle\Psi(t)|\Psi(t)\rangle=1$ implies that
$\sum_{j=1}^{d^{(1)}}\lambda_{j}^{2}(t)=1$, and the orthonormality of the
local basis elements assure that
$\langle\varphi_{m}^{(\alpha)}(t)|\varphi_{n}^{(\beta)}(t)\rangle=\delta_{\alpha\beta}\delta_{mn}$.
From now on, the Schmidt decomposition form showed in Eq. (3) will be the
standard description of $|\Psi(t)\rangle$.

As already mentioned in chapter (1), the representation above is compelling
and useful for a number of reasons: notice that despite a potential huge
discrepancy between $d^{(1)}$ and $d^{(2)}$ there is a single sum bounded by
the smallest dimension in question, by hypothesis $d^{(1)}$444Even though the
sum extends up to $d^{(1)}$ we may have null Schmidt coefficients thus, in
practice, the sum goes until the Schmidt rank, defined as the number of non-
zero coefficients. To keep track of these dimensions, we will maintain
$d^{(1)}$ in the sums.; apart from that, it is symmetrical in the sense that
for each ket $|\varphi_{j}^{(1)}(t)\rangle$ from subsystem $(1)$ there is a
related ket $|\varphi_{j}^{(2)}(t)\rangle$ from $(2)$; additionally, as we are
going to present below, it turns out that it gives all necessary information
for representing the subsystem’s local states; also, it is guaranteed that the
Schmidt coefficients are unambiguously defined, while the Schmidt basis are
unique up to eventual degenerate coefficients and a phase degree of freedom,
in the sense that Eq. (3) is invariant over simultaneous local phases changes
(this point will be discussed later in Section (5)); finally, it makes easy to
verify whether the subsystems are entangled or not, i.e., a product state will
be obtained iff there is a single non-zero Schmidt coefficient555Schmidt rank
equal to one. such that $\lambda_{j}(t)=1$, $\lambda_{m}(t)=0$ for all $m\neq
j$ and
$|\Psi(t)\rangle=|\varphi_{j}^{(1)}(t)\rangle\otimes|\varphi_{j}^{(2)}(t)\rangle$.
As a last remark, it is important to highlight the fact that the set
$\\{|\varphi_{j}^{(2)}(t)\rangle\\}_{j}$ of Schmidt basis is not formally
complete since we only have $d^{(1)}$ elements, still we are always allowed to
find the remaining $(d^{(2)}-d^{(1)})$ orthonormal kets to form a complete
basis of $\mathcal{H}^{(2)}$.

Let us now turn our attention to the parts: the individual state description