Quantum Thermodynamics: control and non-Markovianity

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DIPARTIMENTO DI FISICA

Corso di Laurea Magistrale in Fisica Teorica

Tesi di laurea 25/06/2018

Quantum Thermodynamics:

control and non-Markovianity

Candidato: Paolo Abiuso Matricola 509474 Relatore: Vittorio Giovannetti Anno Accademico 2017–2018

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Ringraziamenti

Questo lavoro non sarebbe esistito senza la supervisione del mio relatore, che colgo l’occasione di ringraziare per il supporto alla tesi e, soprattutto, per avermi introdotto e dato l’opportunità di specializzarmi nel mondo dell’informazione quantistica. Ringrazio anche Andrea Mari e Vasco Cavina per gli utili consigli e discussioni.

Grazie alla mia famiglia, Mamma, Papà e Pierluigi, per avere assecondato da sempre le mie inclinazioni e per essere sempre rimasti un porto sicuro a cui fare riferimento nei momenti di difficoltà.

Grazie ad Erika per sopportarmi dolcemente da anni, per essermi vicina anche quando i chilometri ci separano.

Grazie a tutte le persone fantastiche che ho conosciuto e con cui ho passato questi cinque anni: grazie per il tempo insieme, ognuno di voi mi ha insegnato qualcosa.

Paolo

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Introduction

"If physical theories were people, thermodynamics would be the village witch." [29]

Thermodynamics is perhaps one of the oldest and most resilient theories of physics: its paradigms resisted in the centuries, while interfacing with all the major schemes, such as classical mechanics, fluid dynamics, field theory and general relativity. As Einstein put it: "A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts". Besides the theoretical side, Thermodynamics has been highly successful, impacting strongly on the natural sciences and enabling the development of technologies that have changed our lives, from fridges to jet planes. Until recently, it was applied to large systems described by the laws of classical physics.

With the strong understanding of Thermodynamics from statistical physics, followed by the Quantum Mechanical revolution in the 20th century which replaced at a funda-mental level the previous comprehension of the microscopic laws, one last big encounter, between the quantum world and thermodynamics, was inevitable. Moreover, with modern technologies miniaturizing down to the nanoscale and into the quantum regime, testing the applicability of thermodynamics in this new realm has become an exciting scientific challenge. Quantum Thermodynamics [26] was born and rapidly grew in the last decades. Fuelled by high experimental control of quantum systems and engineering at micro-scopic scales, one of the central goals of physicists is to push the limits of conventional thermodynamics, and the extension of standard models and cycles to include quantum ef-fects and small ensemble sizes. Powerful numerical methods, and the development of novel theoretical tools, for instance in non-equilibrium thermodynamics and quantum informa-tion theory, brought big enhancements and understanding of the matter. Beyond the drive to clarify fundamental physical issues, these models may also turn out to be relevant from a more practical point of view: it is expected that industrial need for miniaturisation of technologies to the nanoscale will benefit from the understanding of quantum thermody-namic processes. In both biology, for example, and nanotechnology, where the benefits from a cooling at the atomic scales are clear, refrigerators models [41,60] based on quan-tum thermal machines could find actual application. Moreover, proposals for experimental realisations of quantum engines were made considering various physical platforms, and

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many were actually realised (cf. Section3.5).

Thermodynamics is, intrinsically, a theory of non-isolated systems, due to the interac-tion of some working fluid with thermal baths and work storage systems. The descripinterac-tion of open systems, i.e. systems in touch with external environments, needs, especially in cases where the number of degrees of freedom of the surroundings is big, an effective description on the local d.o.f.s by means of some approximation or assumption: in general the exact solution of the equation of motion of the entire "universe" is, more than unfeasible, even useless, if the relevant information is lost in the massive amount of data it involves.

Probably the most important class of approximations on the dynamics of open sys-tems is what goes under the name of Markovian dynamics. Andrei Andreyevich Markov (1856-1922) was a Russian mathematician who worked on stochastic theory, and analysed the properties of processes describing the unfolding of probabilistic variables undergoing memoryless evolution, that is evolution that can be forecast by the only knowledge of the present state of the system, without knowing the full past history. From the physical point of view, this is associated to systems interacting with large unperturbed environments that "spread away the information" contained in the system.

Many recent works (see e.g. [6,9,43,44,51,52,62,70]), have started to investigate how the breaking of the Markov approximations in quantum dynamics can affect control and performance of quantum systems, motivated both by the necessity to overcome the approx-imation on very small systems, and by the speculation of non-Markovianity possibly being an actual resource in practical tasks. We try to contribute here to the ongoing scientific debate.

This Master’s Degree thesis is located in the field of Quantum Thermodynamics and its links with Information Theory [29].

The starting purpose of the work is to study how memory effects (non-Markovian dy-namics, or n-M) can affect the thermodynamic performances of a quantum system. In-cidentally we work out an optimal control theory for a class of simple thermal machines performing Carnot cycles.

The mathematical treatment is based on tools from Open Quantum System Dynamics (OQS) [8, 46]: evolution is described, under the most general physical assumptions, by means of CPT maps (Completely Positive, Trace-preserving, linear maps) acting on the density matrix of the system under consideration. The definition of Markovian process was born historically in a classical context, and has not an immediate, canonical counterpart in Quantum Mechanics. In the area of OQS there has been an intensive study of the issue during these years, which has raised different definitions of quantum Markovianity [54,7]. In this work we briefly review the main definitions, their relation and the connection with information/correlations backflow from the environment to the system. We stand by the most standard approach which identifies the Markovian character of a quantum process with its CP-divisibility [53], that means described by a map Λt,0that satisfies Λt,0 = Λt,s◦ Λs,0 (t ≥ s ≥ 0)where all Λs are CPT. Under this assumption a standard form can be derived for the Master equation generating the evolution, that is the

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Sudarshan-Lindblad form (GKSL) [30,40].

Based especially on OQS, our study of Quantum Thermodynamics will assume as a postulate the existence of thermal baths [27], and will focus on the analysis of the perfor-mance of thermal engines. In this framework thermalisation effects induced by the inter-action with a thermal reservoir will be modelled with Master Equations having the Gibbs state ωβ = e−βH/Tr[e−βH] as the only stationary state. To simulate sensible restraints on the system, it will be also assumed external control on dynamical parameters such as the coupling constant to the thermal baths, as well as the local Hamiltonian of the sys-tem. The internal energy of the system will be finally Tr[ρH], adopting the canonical OQS work+heat definition which identifies the former with dW = Tr[ρdH] and and the latter with dQ = Tr[dρH]. This definition is well justified by discrete formulations of quan-tum Thermodynamics [3], or in the standard thermal Markovian maps setting, by means of monotonicity properties of the relative entropy S(ρ||ωβ), or simply by the identification of work residing in the Hamiltonian external control parameters variation, and the heat given by the "disordered" change in energy responsible for the variation of entropy of the system. Once defined the basic thermodynamic quantities and potentials, we review the main quantum thermodynamic cycles, i.e. the Quantum Carnot Cycle and the Quantum Otto Cycle. The typical example is given on the "smallest" possible engine, that is a 2-level system (qubit) in contact with 2 thermal reservoirs with temperatures TH and TC.

In the above setting we develop an optimal control theory for simple quantum engines such as qubits in a wide class of dynamical models. Thermodynamic cycles are sensitive to the "experimentalist" control, and it is in general difficult - involving numerical/varia-tional methods - to characterise protocols that maximise efficiency, power, or other figures of merit for a given engine and a set of constraints on the control parameters. For this kind of study it is necessary to depart from the idealizations of quasistatic processes and analyse Finite Time Thermodynamics (FTT). One important tool in this study is given by a perturbative technique shown in [12], the Slow-Driving approximation (S-D). The idea be-hind this technique is the following: if a system is governed by a dynamical equation which varies very slowly in time, the evolution will follow approximately the (time-dependent) stationary state of the dynamics. More formally, the starting point is a Markovian Master equation of the form ˙ρ = Ltρ; in the limit of constant Lt = Lthe solution will be, after an exponentially small amount of time, the stationary state that satisfies Ltρ(0) = 0. In the limit of slow variation of Ltthe solution will then be ρ(0)_{(t) + δρ}where δρ is a small variation which can be studied perturbatively, the perturbation parameter being ε = τR/τ (here τ is the typical timescale for the variation of the dynamics generator τ ∼ ˙Lt/Lt, while τRis the relaxation time associated to the Lindblad generator). The S-D machinery turns out to be a well suited platform for the study of FTT, where the typical example is a thermalising Markovian map, possibly time dependent (e.g. due to slow external param-eters control on the Hamiltonian). The perturbative solution is very useful for it neglects initial conditions (which is what one desires over a number of cycles for a thermodynamic engine), and it is relatively simple to compute, hence we employ it to detect the optimal working points of quantum thermal machines: corrections from the quasistatic case can

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be found recursively, allowing a perturbative study of thermodynamic quantities (energy exchanges, efficiencies etc.). For the paradigmatic case of a quantum Carnot cycle we find that it is possible to pinpoint three main features which determine, in the S-D scheme, the thermodynamic performance:

• the shape of the protocol on the control parameters ~λ(t), • the speed of the protocol (for fixed shape),

• a model dependent S-D correction amplitude, namely how large the perturbative cor-rection to the steady state solution is.

The first contribution is in general a functional of the actual control parameters modified externally over a fixed time length. The S-D technique allows us to perform variational cal-culus to obtain the optimal shape for the control. The second feature is simply given by the possibility of changing, for a fixed shape, the velocity of the controls, by mapping t → αt; optimizing the power output over this property alone we recover efficiencies at maximum power typical of FTT, such as the Curzon-Ahlborn efficiency [16]. The last contribution encodes, for a given protocol "how distant" the Slow-Driving corrections are from the qua-sistatic case. It is a model dependent amplitude which we then exploit as the main figure of merit in the study e.g. of non-Markovian models. Our results for the Slow-Driving approx-imation help to generalise the optimal control theory for qubit Thermodynamics presented e.g. in [11]. In the article in fact the authors provide an exact optimal protocol for a spe-cific choice of a simple dissipator; the solution (quasi-Otto cycles) is however compatible with the Slow-Driving approximation, and at the same time results of this thesis are noted to be robust on the optimal trajectories. Given the fact that the Slow-Driving form of the dynamics relies less on the explicit form of the dissipators and dynamical equations, this gives our treatise a more general feature.

In the last part of the thesis we define a precise setup for the study of non-Markovian influence on a quantum thermodynamic system: we imagine to have an engine S in con-tact with the thermal baths, allowing however some degrees of freedom of the reservoirs (namely, a system C for the cold bath and H for the hot one) to be taken into account ex-plicitly and interact with S by an Hamiltonian HI. This means the dynamics in the bigger frame (system HSC ) is a standard Markovian dynamics, while the local reduced dynamics on S will in general be non-Markovian due to the memory effects and oscillations between S and H C . This particular setup choice is made in order to avoid problems and/or non-physicalities in the solution of the dynamics, that can happen using other techniques/ap-proximations for the study of non-Markovian processes. We have instead full control on the ignition of memory effects, whose origin can always be retrieved in the bigger picture, and we avoid apparent violations of the 2nd Law.

In the simplest model we have in mind H , S and C are qubits which interact via an exchange Hamiltonian between S and C (and S and H). When coupled to the cold reser-voirs, on both S and C the action of the bath is given by GKSL thermalising dissipators, and the controller has the ability to change the energy gap of the local Hamiltonian of S

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(C is thought as part of the reservoir); the situation is specular when coupled to the hot bath. We study this model in the Slow-Driving approximation; in particular we look for the expression of the S-D amplitude as a function of the setup parameters, to study how the rates defining the GKSL operators influence it.

We find that in a vast region of parameters space the introduction of S H and S C cou-pling can have improving effects on the FTT performance. We also examine the exact so-lution of an Otto cycle, confirming that the presence of the n-M enhances the power output. The main structure of this manuscript is tripartite; a background part is followed by the two main lines of original research:

Part I deals with introductory topics necessary for the understanding of the subsequent work, mainly Open Quantum Systems theory, Markov processes and their generali-sation to Quantum Mechanics, and introductory Quantum Thermal processes. Part II introduces arguments of Finite-Time-Thermodynamics, the low-dissipation regime,

and the strongly related Slow-Driving technique to solve the dynamics open quantum systems. We develop here an optimal control theory for the thermodynamic control of a qubit.

Part III revolves around the study of the dynamics and the Thermodynamics of the simple n-M model which allows to study thermodynamic efficiency and power of Carnot cycles and Otto cycles, and monitor the performance as non-Markovianity arises.

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Notations and conventions

Here we list the main notation choices for this manuscript. Units1

The Planck constant, whose value is ~ = 1.054571800(13) · 10−34_{J · s, is set to 1.}

The Boltzmann constant, whose value is kB = 1.38064852(79) · 10−23J/K, is set to 1.

Acronyms

QT Quantum Thermodynamics

CPT(P) Completely Positive Trace-Preserving (linear map) EMP Efficiency at Maximum Power

FTT Finite Time Thermodynamics n-M non-Markovian

OQS Open Quantum Systems dynamics POVM Positive-Operator Valued Measure S-D Slow-Driving

d.o.f. degree of freedom

Physical systems and their Hilbert space

S, E the name of a system is in mathcal style.

H_S, H_E its Hilbert space is identified by H with a standard uppercase subscript. HS, HE operators acting on the respective Hilbert space.

ρs, ρe, ρse we use lowercase subscripts for the density matrices.

σ+_s, σ_ez, 1s we use lowercase subscripts also to identify the domain spaces of standard operators.

L(H) the set of linear operators acting on H.

S(H) ⊂ L(H) the set of states: positive, trace-one linear operators acting on H. L(L(H)) the set of linear super-operators acting on L(H).

|ρ| the absolute value of a matrix ρ, |ρ| = pρρ†.

k ρ k1 the trace-norm of ρ, given by Tr[pρρ†].

k ρ k2 the Hilbert-norm of ρ, given by pTr[ρρ†].

σx, σy, σz, σ± the Pauli operators σx=0 1 1 0 , σy ₌0 −i i 0 , σz ₌1 0 0 1 , σ±₌ σx± iσy 2

1_{Taken from}_{The NIST reference on Constants, Units, and Uncertainty}_{based on CODATA 2014}

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8 Thermodynamics

ωβor ωβ(H) the canonical thermal state given by e −βH Tr[e−βH_] =

e−βH Z(β,H).

Z(β, H) the partition function Z(β, H) = Tr[e−βH] U the mean internal energy of a system Tr[ρH].

dU = dQ + dW the 1st law (note the sign of W , it is the work performed on the system). S the Von Neumann entropy −Tr[ρ ln ρ].

F the free energy F = U − T S.

ηC = 1 −_THTC the efficiency of an ideal Carnot cycle.

ηO the efficiency of an Otto cycle.

ηCA = 1 −

q

TC

TH the Curzon-Ahlborn efficiency.

η∗ the efficiency at maximum power (EMP).

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Part I

Background

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The first part of this manuscript is intended to be an introduction to the concepts and definitions useful to understand the whole work, as well as the motivations of it.

Thermodynamics is, par excellence, a theory involving non-isolated systems. This is why, whichever the description of it, whichever underlying physical theory, it must take into account the interaction and evolution induced by external degrees of freedom on a system. When dealing with quantum mechanics, this is the task of Open Quantum Systems theory. A wide class of standard OQS evolutions is called Markovian.

Markovianity is a property of the dynamics associated with the absence of persistent influence of a system over its environment. As a consequence, the environment induces an effective dynamics which has no memory of the previous history of the system. The formal definition and characterisation of Markovianity and memory effects is part of an ongoing debate at the present time. However the realisation, occurred during the 20th century, of Thermodynamics being closely related with Information Theory, as well as recent studies considering non-Markovianity as a possible resource in e.g. Quantum Communication and Quantum Control, incited the scientific community to investigate the interplay between these "memory effects" and thermal machines.

We are going to familiarize with these topics, in the following order:

Chapter 1 is a recap of notions of general Quantum Mechanics formalism and linear algebra. However, not all the presented topics are, in general, covered by standard Quantum Mechanics courses. Open Quantum Systems theory in his essentials is introduced, as well as some useful technicalities, selected for our purposes.

Chapter 2 deals with the concept of Markovianity and its abstract definition, from the mathe-matical and quantitative point of view.

Chapter 3 presents a brief set up of the Quantum Thermodynamics framework we will move in, as well as introducing the thermodynamic cycles we will analyse in detail thereafter. Readers experienced in this sector can skip the whole background, or just part of it, such as Chapter1. One suggested segment, useful to bear in mind for the understanding of the subsequent PartsII-III, is the description of the Quantum Carnot Cycle and Quantum Otto Cycle given in Section3.4.

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Chapter 1

Open quantum systems dynamics

The evolution of a closed quantum mechanical system is generated by its Hamiltonian. Nevertheless closed systems, even in classical mechanics, are an idealisation; any realistic system is influenced non-negligibly by its surroundings. This is especially true in the case study of Thermodynamics, where environmental systems with many degrees of freedom - namely the thermal reservoirs - are put in contact with the "engine" during its operations and determine its steady state. It is therefore manifest the need to investigate evolutions of density matrices beyond the unitary scheme.

The study of the evolution of quantum mechanical systems in contact with their environment is the task of Open Quantum Systems Dynamics theory, which is, as we will see, the natural platform for the study of Quantum Thermodynamics.

In this chapter we introduce briefly the basics of OQS, before proceeding with a special focus on properties of Markovian (divisible) dynamics which have close relations with Thermodynamics. The best references in this context are the textbooks [46] and [8], which we will avoid to cite throughout the Chapter.

1.1 Quantum states

Quantum Mechanics postulates the state of a system is described by some normalised vector be-longing to an Hilbert space

|ψi ∈ H , hψ|ψi = 1 , (1.1) where hφ|ψi is the Hilbert-Schmidt product of the space.

The joint state of two systems is described by means of a tensor product of the relative Hilbert spaces,

|ψi ⊗ |ϕi ∈ H1⊗ H2 (1.2)

which is of course still a normalised vector living in the total Hilbert space. Note that not all vec-tors belonging to H1⊗ H2can be written in the form (1.2). States in this form are called separable,

otherwise they are said to be entangled, sharing non classical correlation between the system 1 and 2.

The kind of vectors above are called, in the language of Quantum Mechanics, pure states, that is completely specified states whose statistical properties derive only from the measurement postu-lates (cf. Section1.4). It is however necessary, in many cases, to allow a more general statistical

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Chapter 1. Open quantum systems dynamics 14 description of a system, including classical uncertainty given e.g. from an ensemble of possible pure states with assigned probabilities

{p_i, |ψii} . (1.3)

In this case any mean value of an observable Θ is given by X i pihψi| Θ |ψii =Tr[ρΘ] , (1.4) where ρ =X i pi|ψiihψi| (1.5)

is called density matrix and represents the totality of information we have about the system. As it easy to see from its definition, the most general density matrix is a positive, trace-one operator

ρ ≥ 0 Tr[ρ] = 1 . (1.6)

It is immediate, from the definition, to see that pure states correspond to rank-1 density matrices |ψihψ|which are the only ones that fill the following inequality

Tr[ρ2_{] ≤ 1 .} _(1.7)

Schroedinger equation is translated in the density matrix formalism as

i~ ˙ρ = [H, ρ] . (1.8)

An example of density matrix is given by the thermal Gibbs state ωβ(H) =

e−βH

Tr[e−βH_] , (1.9)

which is, for β ≥ 0, simply the representation of the Canonical Ensemble of statistical mechanics characterised by a temperature T = (βkB)−1.

We will denote by S(H) the set of density matrices acting on H. In case of a multipartite system -suppose for simplicity a bipartite H = H1⊗ H2, as a natural generalization of (1.2) it is possible to

identify as separable states all the density matrices which can be written in the form ρ_separable=X j pjρ(j)1 ⊗ ρ (j) 2 ρ (j) 1 ∈ S(H1) ρ(j)2 ∈ S(H2) , (1.10)

for some set of probabilities indexed by j. States which cannot be expressed in this form are called entangled.

When looking at a local observable of a multipartite system, e.g.

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15 Chapter 1. Open quantum systems dynamics it is easy to see how its value1

hΘi_ρ=Tr[(O1⊗ 12)ρ] =Tr1[O1Tr2[ρ]] (1.13)

is determined by the local density matrix

ρ1=Tr2[ρ] ∈ S(H1) , (1.14)

which is called reduced state on H1. Note that ρ1 is indistinguishable from ρ if we have access only

to local observables, and this is indeed a very natural mechanism for the appearance of mixed, non-pure states in Quantum Mechanics: in general, even if ρ12is pure, ρ1 =Tr2[ρ12]is not.

Example 1

Consider the pure state (called maximally entangled) |Ψi = √1 d d X n=1 |ni|ni ∈ H1⊗ H2, (1.15)

where H1∼= H2are isomorphic and |ni is a basis. Its reduced state on H1is

ρ1 =Tr2 1 d X n,m |ni|nihm|hm| = 11 d , (1.16)

which is the completely mixed state, corresponding to a Gibbs state (1.9) with infinite temperature. Example 2: Bloch vector

The 2×2 density matrix of a 2-level system (qubit) can be characterised in the Pauli basis σx=0 1 1 0 σy =0 −i i 0 σz=1 0 0 −1 , (1.17) as 1 + ~a · ~σ 2 , (1.18)

where ~a is called Bloch vector and represents an isomorphism between the states ρ ∈ S(H) of such a simple Hilbert space and the sphere of radius 1 in R3. In fact, while the trace-one condition is

guaranteed by the normalisation of Eq. (1.18), the positivity constraint requires |~a| ≤ 1, being the eigenvalues 1

2± |~a|

2 . The same form of the eigenvalues makes us realise that the states on the surface

of the Bloch sphere are pure states.

1_{By Tr}

1and Tr2we mean the partial trace on the subsystems 1 and 2 respectively, i.e.

Tr1[A12] =

X

n

2hn| A12|ni2 (1.12)

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Chapter 1. Open quantum systems dynamics 16

1.2 Entropies

Entropies play a crucial role in Information Theory; their goal is to quantify the disorder, or the amount of informationstored in the state of a system, or in the correlations between systems. Shannon. The most famous and historical one is Shannon entropy [59]:

given a discrete probability distribution {pi|i = 0, . . . , d − 1}, it is defined as

SShannon({pi}) = −

X

piln pi. (1.19)

Shannon entropy has the property of being positive and limited

ln d ≥ SShannon({pi}) ≥ 0 . (1.20)

Moreover, it is null only if the state represented by the distribution is totally determined i.e. SShannon({pi}) = 0 ⇔ pi = δi¯j (1.21)

for some 0 ≤ ¯j ≤ d − 1. On the opposite front, maximal entropy corresponds to the totally undetermined state

SShannon({pi}) = ln d ⇔ pi=

1

d ∀i , (1.22)

that is, a flat distribution.

von Neumann. The quantum generalisation of Shannon entropy is given by von Neumann en-tropy, S_{von N eumann}, or simply S: given a state ρ, it is defined as

S(ρ) =Tr[−ρ ln ρ]. (1.23) and it correspond to the Shannon entropy of the eigenvalues of ρ. That is,

ρ =X

i

λi|iihi| ⇒ S(ρ) = SShannon({λi}) , (1.24)

hence they share most of their properties. Moreover, S(ρ) = S(UρU†₎is invariant under unitary

operations (the eigenvalues being the same). For example, if the dimension of the Hilbert space is dim(H) = D, then for ρ ∈ H, 0 ≤ S(ρ) ≤ ln D, with the upper bound saturated in case ρ = 1

D,

while S is null only if ρ is pure.

Shannon entropy and von Neumann are both concave functionals (due to Jensen’s inequality applied to the function f(x) = −x ln x), that is

S X i piρi ≥ X i piS(ρi) {pi≥ 0, X i pi= 1} . (1.25)

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17 Chapter 1. Open quantum systems dynamics Relative entropy. The relative entropy for a pair of states ρ and σ is defined as

S(ρ k σ) =Tr[ρ ln ρ − ρ ln σ] . (1.26) Relative entropy is positive and unbounded in general,

S(ρ k σ) ≥ 0 (Klein inequality) , (1.27) S(ρ k σ) = 0 ⇔ ρ = σ , (1.28) and shares convexity properties and invariance under unitary transformations

S(ρ k σ) = S(U ρU†k U σU†) , (1.29) S λρ1+ (1 − λ)ρ2 k λσ1+ (1 − λ)σ2 ≤ λS(ρ1 k σ1) + (1 − λ)S(ρ2k σ2) . (1.30)

Moreover, given ρ, σ ∈ S(H), and a partial trace such that

{σ0 ≡Trpart[σ] , ρ0 ≡Trpart[ρ]} ∈ S(H0) (H0 ⊂ H) , (1.31)

then

S(ρ0 k σ0) ≤ S(ρ k σ) . (1.32) The properties above (decrease under partial trace, mixing etc.) led to the interpretation of the relative entropy as a sort of "asymmetric distance" between states, which quantifies how close they are in an information theory context.

Quantum mutual information. When looking at a multipartite system, it can be interesting to quantify the amount of information stored in the correlations between subsystems. Suppose ρ ∈ S(H)with H = H1⊗ H2, ρ1 = Tr2[ρ], and ρ2 = Tr1[ρ]. The mutual information between

system 1 and 2 is defined as

I(1 : 2) = S(ρ1) + S(ρ2) − S(ρ) = S(ρ k ρ1⊗ ρ2) , (1.33)

hence it is a positive measure of the correlations shared by the subsystems, and it is null only in case they are completely independent,

I(1 : 2) = 0 ⇔ ρ = ρ1⊗ ρ2 . (1.34)

1.3 From closed quantum dynamics to CPT maps

Suppose we are given a system S , living on a Hilbert space HS, and an environment E , on HE. The

total Hamiltonian can be written as

Htot= HS⊗ 1e+ 1s⊗ HE + HI. (1.35)

The study of open quantum systems starts from the simple observation that the whole S E system can be considered as closed, hence standard Schroedinger equation holds:

˙

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Chapter 1. Open quantum systems dynamics 18

ρse(t) = U (t)ρse(0)U†(t) , U (t) = e−iHtott. (1.37)

Suppose now to start from an initially monitored system S , so that ρse(0) = ρs⊗ ρe. In this case

we can write the state of S at a later time as

ρs(t) =TrE[U (t)ρs(0) ⊗ ρe(0)U†(t)] =

X

k

Mk(t)ρs(0)M_k†(t) , (1.38)

where the operators Mk, acting on Hs, are called Kraus operators and are given, using the

environ-ment eigenstate basis ρe(0) =Piλi|iiehi|, by

Mk(t) = ehj| U (t) |ii_e

p

λi k = (j, i) , (1.39)

and have the property descending from unitarity X

k

M_k†(t)Mk(t) = 1 ∀t. (1.40)

Any evolution given in the form (1.38) from a set of operators which have the property (1.40) is said to be in the Kraus form, and it is the most general representation of a physical evolution.

Equation (1.38) defines linear map Λ, that describes the evolution of ρ (we now omit the subscripts

for convenience):

Λt,0[ρ(0)] = ρ(t) (1.41)

is said to be Completely Positive and Trace-Preserving (CPTP or CPT). This means the map: CPTP does not modify the trace of ρ, Tr[Λ[ρ]] = Tr[ρ].

CPTP sends positive operators into positive operators ρ ≥ 0 ⇒ Λ[ρ] ≥ 0.

CPTP sends positive operators into positive operators also when combined with the identity map on higher dimensional Hilbert spaces, that is 1A⊗ ΛS is still a positive map for any Hilbert

space HA.

These requirements are the minimal conditions to ensure the image of a density matrix will still be a valid density matrix, hence must be respected by any physical evolution. In particular the third condition (Complete Positivity) guarantees that the maps can always be combined with the identity map - that is, doing nothing - on an auxiliary system (which is therefore left untouched by the evolution).

It is easy to verify how any evolution given in the Kraus form is CPTP. The interesting result here is that also the counterpart is true: any CPTP map admits a representation of the form (1.38) [46]. That is,

for any CPTP maps, it exist a physical representation of it,

which means it has a Kraus form, which can always be seen as the result of unitary evolutions and partial traces2_.

2_{Given the Kraus set {M}

k, k = 0, ..., m − 1}, it is sufficient to choose an m-dimensional environment and a unitary

Usuch that

ρe(0) = |0ieh0| U |ψis⊗ |0ie =

X

k

(Mk|ψi)s⊗ |kie (1.42)

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19 Chapter 1. Open quantum systems dynamics

1.3.1 Choi-Jamiolkowski state.

One possible proof of the previous statement relies on a famous isomorphism, given by the Choi-Jamiolkowski matrix [14,35]: consider the maximally entangled state

|Ψi = √1 d

d

X

n=1

|ni |ni |Ψi ∈ H ⊗ H , (1.43) where dim[H] = d and |ni is a basis. The Choi-Jamiolkowksi matrix representing a CPT map Λ acting on H is given by

ρC−J = [Λ ⊗ 1](|ΨihΨ|) . (1.44)

Choi’s theorem asserts that Λt,s is completely positive if and only if ρC−J ≥ 0, hence being an

actual state.

The non-trivial part of Choi’s theorem (ρC−J ≥ 0 ⇒ ΛCPT), which we avoid to fully report here,

consists in finding the Kraus representation from these last two equations.

Note that Choi’s theorem implies that, if 1A⊗ Λis a positive super-operator, and dim[A] ≥ dim[H],

then the positivity property is extended to any other finite dimensional Hilbert space, i.e. 1A⊗ Λ ≥

0 ∀A.

1.4 POVMs and Helstrom theorem

The most general description of a measurement in quantum mechanics, is given by the POVM for-malism (Positive-Operator Valued Measurement) [8,46]. A POVM is given by a set of positive oper-ators Eiwith the property

X

i

Ei= 1 , Ei ≥ 0 . (1.48)

A POVM represents a measure whose i-th outcome has a probability

Pi =Tr[Eiρ] . (1.49)

Clearly this is a generalization of the von Neumann projective measurement, where the POVM is given by the projectors on the eigenstates of the observable Ei = Πi.

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Chapter 1. Open quantum systems dynamics 20 the maximum distinguishability between two quantum states is related to the trace-norm k . . . k1:

given two states ρ1 (with probability p1) and ρ2 (with probability p2), and a measurement E1, E2

aiming to distinguish them, the minimum probability of error is [32] min

{E1,E2}∈POVMsPerr=

1 2 −

k p1ρ1− p2ρ2 k1

2 . (1.50)

Note that under a unitary dynamics k p1U ρ1U†− p2U ρ2U†k1=k U (p1ρ1− p2ρ2)U†k1=k p1ρ1−

p2ρ2 k1 the distinctness of two states stays constant, while from the lemma showed below [57],

under a CPT map it can only decrease.

Lemma: A trace-preserving linear map Φ is positive if and only if for any Hermitian operator H acting on H,

k Φ[H] k1≤k H k1 . (1.51)

Proof [⇒]

Suppose Φ is positive, and note that if H ≥ 0 then k H k1= Tr[H]. If H is not positive, it can

always be decomposed as H = H+_{− H}−_{with both H}+_{, H}−_{≥ 0. Hence we have}

k Φ[H] k1=k Φ[H+] − Φ[H−] k1≤k Φ[H+] k1+ k Φ[H−] k1=k H+k1 + k H−k1=k H k1

(1.52) [⇐]

Suppose now Φ is trace-preserving and the inequality (1.51) applies. Then for H ≥ 0

k H k₁=Tr[H] = Tr[Φ[H]] ≤k Φ[H] k1≤k H k1 . (1.53)

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Chapter 2

Markovianity vs non-Markovianity

The concepts of Markovian processes and Markovian dynamics were born in a classical context. The physical intuition underlying a Markovian dynamics is based on phenomenological assump-tions and/or approximaassump-tions involving the interaction between a "small" system S with a "large" environment E . Typically, a system with a finite number of degrees of freedom interacts locally with a small subset of the d.o.f.s of its environment, who is assumed to have a high density of en-ergy states, allowing for very small relaxation time-scale, much smaller than the system evolution time-scale. That is, the environment very quickly scatters away - to "distant" d.o.f.s - the fluctua-tions associated to its interaction with S , which is weak enough to allow an effective description of the resultant dynamics as if the state of E is unperturbed. These considerations translate in precise approximations that go under the name of Born-Markov approximations [8] and are used to derive master equations in different settings.

Turning now to the more mathematical point of view: in probability theory, given a system X de-scribed stochastic variable x evolving in time with a discrete process, its evolution is said to be Markovian if the state of the system at any time depends only on the first previous configuration, that is

pi(x|Xi−1, Xi−2, ...) = pi(x|Xi−1). (2.1)

A Markov process (or Markov chain [8]) can thus be described by transition matrices (possibly time-dependent) that link the probabilities at adiacent times, i.e.

pi(x) =

X

y

M_x,y(i−1)pi−1(y). (2.2)

The elements of the matrix M are the transition rates of the process and have to fullfill some re-quirements in order to consistently link probabilities vectors

M_x,yi ≥ 0 , X

x

M_x,yi = 1. (2.3) In a more compact form we can write ~pi = Mi−1~pi−1, which in the continuum limit leads to a

Markovian Master equation(MME) ˙ ~ pt= Lt~pt Lt∼ lim τ →0 Mt+τ,t− 1 τ . (2.4) 21

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Chapter 2. Markovianity vs non-Markovianity 22 The generalization of the formal definition of Markovian dynamics in the quantum regime is not straightforward (due for example to measure-feedback on the system) and it is still debated today [7,54]. However the most followed definitions rely on divisibility properties of the dynamics, which we present in the next section.

2.1 Divisibility, Master equations, G-K-S-L

The CPT maps that represent the time-evolution of a system from its initial state ρ(0)

Λt,0[ρ(0)] = ρ(t) (2.5)

form a one-dimensional continuous set of CPT linear super-operators parametrised by {t}, with Λ0,0 = 1. It is possible to define1a class of maps Λt,s(with two "temporal parameters" t, s) that act

as a connecting transformation between Λs,0and Λt,0, i.e.

Λt,0[ρ] = Λt,s◦ Λs,0[ρ] Λs,s = 1 ∀s t ≥ s ≥ 0 . (2.7)

In the above setting, the resultant dynamics is said to be divisible, if Λt,sis still a physical map [7,54],

or, more precisely:

• the evolution is said CP-divisible if Λt,sis CPT ∀{t ≥ s ≥ 0}.

• while simply P-divisible if Λt,sis only positive (PT) ∀{t ≥ s ≥ 0}.

If the evolution of a system is divisible it is possible to find a Master equation given by ˙

ρ(t) = lim

ε→0

(Λt+ε,t− 1)[ρ(t)]

ε = Lt[ρ(t)]. (2.8) The super-operator L is called Gorini-Kossakowski-Sudarshan-Lindblad operator, or Lindbladian, inheriting the name of the physicists who first found its most general form [30,40], that is

L_t[ρ] = −i[H, ρ] + D[ρ] = −i[H, ρ] +X k AkρA†k− 1 2{A † kAk, ρ} . (2.9)

Note the operators H and Akare in the most general case time-dependent. The non-Hamiltonian

term D of Eq. (2.9) is sometimes referred as dissipator.

If the underlying theory is time-translational invariant, Λt,s is only function of (t − s) ≥ 0, thus

defining a one parameter class of maps

Λt,s= Λt−s,0 (2.10) satisfying Λt,0◦ Λs,0= Λt+s,0. (2.11) 1_{If Λ} t,0is invertible, or injective, Λt,s= Λt,0◦ Λ−1s,0. (2.6)

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23 Chapter 2. Markovianity vs non-Markovianity This structure is known as quantum dynamical semigroup. It’s easy to see how in this case the generator L_t= lim ε→0 (Λt+ε,t− 1) ε = limε→0 (Λε,0− 1) ε = L (2.12)

becomes time independent.

We see now why the divisibility property has been chosen as the main method for quantum Marko-vianity: it allows for a natural generalisation of Eq. (2.4), that is local dynamical equations in time. Moreover, from property (1.51) we find that under a divisible dynamics the Helstrom matrix norm is continuously decreasing, hence the distinguishability of states. This is associated to a memory loss of the initial conditions of the dynamics: states "get closer" during their evolution, forgetting their initial conditions

k Λ_t0_,0[ρ] − Λ_t0_,0[σ] k₁≤k Λ_t,0[ρ] − Λ_t,0[σ] k₁ ⇔ t0 ≥ t . (2.13)

It is also possible to prove [46] that the relative entropy S(ρ k σ) decreases under the action of CPT maps, this implying

S Λt0_,0[ρ] k Λ_t0_,0[σ] ≤ S Λ_t,0[ρ] k Λ_t,0[σ] ⇔ t0≥ t , (2.14)

and as a corollary the mutual information between subsystems also decreases monotonously in time under the action of local CP-divisible maps Λ = Λ1⊗ Λ2.

This loss of memory and correlations is well interpreted under the physical intuition of Markovianity: the semigroup-divisibility properties of the dynamics can be associated to the environment dispers-ing all the information relative to the "previous history" {ρ(s), t ≥ s ≥ 0}, while the evolution is dictated, at any time, only by the instantaneous value of ρ(t). These previous considerations opened the road to the quantum generalisation of the definition of Markovian dynamics. However, although generally agreeing on the main divisibility-driven approach, different groups proposed slightly different definitions of Quantum Markovianity [7,54]. In this manuscript we assert that

an OQS dynamics is Markovian if and only if it is CP-divisible, following the path drawn in [53].

2.1.1 Examples of MME

As examples, we report here some well-known Markovian Master Equations, derived from standard physical models.

Quantum optical master equation [8]. Consider an atom or a molecule S, which interacts with a quantised radiation field. The atom is represented by some finite dimensional state ρ, while the free quantised radiation field will be represented by the Hamiltonian HB =P~_kPλ=1,2~ωkb†λ(~k)bλ(~k),

where the bosonic operators b†

λ(~k)are the creation operators of the electromagnetic mode~k with

po-larisation λ, ωkthe associated frequency. A dipole interaction is assumed HI = − ~D · ~E between the

field and the system S, (where ~Eis the standard electric field ~E = iP

~ k P λ=1,2 q 2π~ωk V ~eλ(κ) bλ(~k)−

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Chapter 2. Markovianity vs non-Markovianity 24

b†_λ(~k), ~eλ(~k)the polarisation vectors). An MME can be derived under the Born-Markov

approxima-tion, assuming the bosonic bath is in the Gibbs state, namely, in the interaction picture (neglecting a Lamb-Shift contribution), ˙ ρs(t) = X ω>0 4ω3 3~c3 1 + N (ω) ~ A(ω)ρsA~†(ω) − 1 2 _~ A†(ω) ~A(ω), ρs + +X ω>0 4ω3 3~c3N (ω) ~ A†(ω)ρsA(ω) −~ 1 2 _~ A(ω) ~A†(ω), ρs (2.15) where the operators ~Aare defined by the decomposition of the dipole operator of S (Π being the projectors onto the eigenspaces of the local Hamiltonian H0of S ) ~A(ω) ≡Pε0_−ε=~ωΠ(ε) ~DΠ(ε0),

while N(ω) is the mean thermal bosonic occupation number N = 1

eβ~ω_{− 1} (2.16)

at temperature 1/β and energy ω.

Equation (2.15) can be applied on to different system, such as an harmonic oscillator H0 = ~ωa†a

(in which case the operators A ∝ a), or a qubit H0 = E₂σsz, in which case it can be written as

˙ ρ = Γ(1 + N (E)) σ−ρσ+−1 2{σ +_σ−_{, ρ} + ΓN (E) σ}+_ρσ−₋1 2{σ − σ+, ρ} . (2.17) This is an explicit example of a thermalising map obtained from a large thermal environment: it is easy to check indeed, that the steady state of this equation is the Gibbs state ωβ = e−βH/Tr[e−βH].

Note that Equation (2.17) is explicitly in the GKSL form (2.9).

Fermionic MME. If in the previous model we assume the system S to be in contact with a fermionicfield at thermal equilibrium instead of a bosonic one, the fermionic commutation rela-tions {b, b†_{} = 1}_{make the structure of Eq. (}_2.15_{) change in (e.g. on a qubit system)}

˙ ρ = Γ(1 − N ) σ−ρσ+− 1 2{σ+σ−, ρ} + ΓN σ+ρσ−− 1 2{σ−σ+, ρ} (2.18) with N being the mean value of hb†_(E)b(E)i_{on the fermionic bath equilibrium state; in case it is}

thermal,

N = 1

eβE_{+ 1}. (2.19)

This master equation can be used to derscibe thermalising effects involving fermionic systems like quantum dots [31].

In what follows we largely use a simplified version of Eq. (2.18), i.e. ˙

ρ = Γ(η − ρ) , (2.20)

which allows for a direct integration ρ(t) = ρ(0)e−Γt_{+ η(1 − e}−Γt_{). In case η = ω}

β the equation

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25 Chapter 2. Markovianity vs non-Markovianity (2.20) is not explicitly in the GKSL form (2.9), which however can be recovered considering the diagonalisation of η η =X i λi|iihi| X i λi= 1 λi≥ 0. (2.21)

and the GKSL operators2

A_(i,j)=√Γpλi|iihj| . (2.24)

An heuristic derivation of (2.20) can be obtained in a collisional model: suppose the system S is affected by some Poissonian noise, whose effect is to put into contact S with a large environment E. When in contact, S relaxes to a steady state η (dictated by E ) with a time scale so small to be instantaneous for our purposes. This can happen, for example, if E is a big gathering of copies of η, i.e. ρe = η⊗N (N 1), and when in contact with S , a swap can occur with given probability

p. If the Poissonian noise is such that ν contact events happen per unit time, the state of ρ after an infinitesimal time dt is

ρ(t +dt) = (1 − νp dt)ρ(t) + νp dt η. (2.25) Taking now the time derivative, we find, identifying Γ = νp, the MME (2.20).

For a comparison between Eq.s (2.18) and (2.20) we observe that the former can be translated for the Bloch vector3_~a_as ˙a1 = − Γ 2a1, (2.26a) ˙a2 = − Γ 2a2, (2.26b) ˙a3 = −Γa3− (2N − 1)Γ . (2.26c)

Instead (2.20) with η = ωβ yields

˙a1 = −Γa1 , (2.27a)

˙a2 = −Γa2 , (2.27b)

˙a3 = −Γa3− (2N − 1)Γ . (2.27c)

It is easy to see that both maps make ρ thermalise, the only difference is in the quicker dephasing of the latter, which could be achieved by an additional noisy channel ∼ ˙ρ = (Γ/4)(σz_ρσz_{− ρ)}[46].

As a final comment, we mention that a similar MME is used in [20,21] and derived in [31].

2 X (i,j) A(i,j)ρA † (i,j)− 1 2{A † (i,j)A(i,j), ρ} = Γ X i,j λi|iihj| ρ |jihi| − 1 2{λi|jihi| |iihj| , ρ} , (2.22)

and performing the sum first on j for the first addend, while on i for the second,

= Γ X i λi|iiTr[ρ] hi| − 1 2 X j {|jihj| , ρ} = Γ ηTr[ρ] − 1 2{1, ρ} (2.23)

which is exactly Eq. (2.20)

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Chapter 2. Markovianity vs non-Markovianity 26

2.2 non-Markovian measures

Once formally defined the set of Markovian dynamical maps, it can be important, from a theoretical and practical point of view, to quantify how non-Markovian is a map which does not satisfy the CP-divisibility property. This is the role of non-Markovian measures.

N_RHP. Given t > t0, the definition we have adopted (cf. Section 2.1and Ref.[53]) to identify a Markovian regime requires us to determine whether or not the connecting map Λt,t0 = Λ_t,0◦ Λ−1

t0_,0

is CP (Λ−1

t0_,0being the mathematical inverse of Λt0_,0). From Section1.3.1we know this is equivalent

to see whether or not the associated Choi-Jamiolkowski state is positive. Being Λ ⊗ 1 also trace-preserving, implies that

ρ_C−J ≥ 0 ⇒k ρ_C−J k₁=Tr[|ρ_C−J|] = 1 . (2.28) It is easy to convince oneself instead that in case ρC−J is not positive k ρC−J k1is greater than 1.

This property is useful to quantify the non-Markovianity of the evolution. That is, introducing [53,54]

g(t) := lim

ε→0+

k [Λ_t+ε,t_{⊗ 1](|Ψi hΨ|) k}₁−1

ε , (2.29)

gis null on the time-domain where Λ is divisible, hence Markovian. It is then possible to introduce the non-Markovianity measure

N_RHP = Z

dt g(t) ≥ 0 , (2.30) which is zero only for Markovian evolutions.

Other definitions and measures. For the sake of completeness, we want to remark that different definitions are present in the literature. For example, historically the absence of memory effects was commonly associated with the formulation of differential dynamical equations for ρ(t) with time-independent coefficients, as can be derived e.g. from collisional models. In this mind setting the Markovian evolutions are the ones given by quantum dynamical semigroups given in Eq. (2.11), that is, complete divisibility equipped with time-translation invariance.

Breuer, Laine and Pillo instead proposed a less-strict definition, by just asking for P-divisibility of the dynamics [7]. Thanks to the Lemma of Eq. (1.51), a simple quantifier of non-Markovianity in the BLP setup is, inspired by the Helstrom Theorem, given by

NBLP = max ρ(1)_,ρ(2) Z σ≥0 dt σ(ρ(1)_{, ρ}(2)_{, t) ≥ 0 ,} _(2.31) where σ(ρ(1), ρ(2), t) := d dt k ρ(1)(t) − ρ(2)(t) k1 . (2.32)

Between the RHP and the BLP definition it is possible to introduce an intermediate hierarchy [15] based on the k-divisibility property. Basically, a family of dynamical maps, {Λt2,t1, t2 ≥ t1≥ t0}is

k-divisible if Λt2,t1 ⊗ 1kis a positive map (here 1kdenotes the identity map acting on the space of

k×k matrices). Therefore, if the dimension of the quantum system is d, a k-divisible process with k ≥ dis what in this work has been called a divisible or Markovian process (we stand by the RHP

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27 Chapter 2. Markovianity vs non-Markovianity definition). The 1-divisible processes are the P-divisible processes, and the 0-divisible processes are processes where Λt2,t1 is not a positive operator for some choice of t1 ≤ t2. We can sum up the

general structure as follows (in red our choice)

{Semigroup def.}⊂{d-divisibility}⊂ . . . ⊂{k-divisibility} ⊂ . . . ⊂{1-divisibility}

≡RHP 1 < k < d ≡BLP

(2.33) Of course, when looking at non-Markovianity, the inclusion relations on the complementary sets are inverted, hence in general a non-zero non-Markovianity measure from one of the choices above, implies non-zero measures for all of the others on its left. For example, if NBLP > 0, this implies

the dynamics is non-Markovian also in the RHP sense, while the viceversa is not true in general. In this case we say NBLP is a witness of non-Markovianity (in the RHP sense), so we can use it to

detect the violation of the CP-divisibility.

A witness of non-Markovianity is, in general, a quantity Ω such that

Λis a Markovian evolution ⇒ Ω = 0 . (2.34) That is, if Ω 6= 0, non-Markovianity must be present - however Ω = 0 does not imply the evolution is Markovian. The monotonicity properties of the relative entropy, trace norm, mutual information etc. (cf. Eq.s (2.13,2.14)) can be used as witnesses.

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Chapter 3

Quantum Thermodynamics

Originally, Thermodynamics was born as a purely phenomenological science. Pioneers in the field tried to give definitions for quantities which were intuitively obvious to the observer, like pressure or temperature, and studied their interconnections. In spite of the fact that all physical systems are finally constructed out of basic subunits, the time evolution of which follows complicated cou-pled microscopical equations, the macrostate of the whole system is typically defined only by very few macroscopic observables. From a phenomenological point of view one finds simple relations between these macroscopic observables, essentially condensed into the fundamental laws of Ther-modynamics, other than various astonishingly simple bonds among macro variables of different physical systems at thermal equilibrium.

The idea that these phenomena might be linked, or even deduced from other fields of physics, like classical mechanics, began to rise at the end of the 19th century, when the atomic theory became popular. This gave rise to modern Thermodynamics, founded on statistical mechanics, a theory the validity of which is beyond any doubt today. Meanwhile, quantum theory, also initially triggered by the atomic hypothesis, made huge progress during the last century and is today acknowledged to be more fundamental than classical mechanics. Furthermore, thermodynamic principles seem to be applicable to systems that cannot even be described in classical phase space. These developments make it necessary to rethink the work done so far, and the fact that a basically classical approach apparently did so well may even be considered rather surprising.

Quantum Thermodynamics is the theory whose goal is to study these issues, the foundations of Thermodynamics and its applications for the manipulation and exploitation of quantum objects in-teracting with thermal sources.

On a very general point of view, we can identify 3 main aims of Quantum Thermodynamics: • the study of the emergence of Thermodynamics from the microscopic dynamics.

• the characterisation of thermodynamic quantities (like heat and work) in terms of quantum observables which have a clear operational meaning.

• the study of the performances of quantum thermal machines, i.e. of "working fluids" whose dynamics can involve coherences, discrete energy spectra, or other inherently quantum prop-erties.

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Chapter 3. Quantum Thermodynamics 30 There is an abundant literature focused on the first issue, especially the problem of thermalisation and equilibration (see for example the review [27]). We will take thermalisation as a given, and study instead the remaining, with a particular interest on optimal control, finite time effects, and memory effects (i.e. non-Markovian dynamics). Our main postulate will then be the existence of given systems - the thermal reservoirs Ei- with an infinitely large number of degrees of freedom at

equilibrium, and an associated temperature Ti. The contact of one of these reservoirs with a system

S, described by a density matrix and a local Hamiltonian (ρs, Hs), makes the system relax over a

certain characteristic time τRto the canonical Gibbs state (1.9)

ρs

Ei

−−−−−−−−→

thermal contact ωβi =

e−βiHs

Tr[e−βiHs_] βi=

1 Ti

. (3.1)

3.1 Dynamical configuration space

Figure 3.1: A pictorial representation of the configuration space. A zero-measure subset of the space is given by the thermal states. The point OOE is out of equilibrium. Examples of possible operations: the dashed arrow represents a thermalisation due to the contact with a bath at temperature T = 1/β, while the continuous arrow is the result of a quench, that is an instantaneous change of the Hamiltonian.

There are no isolated macrosystems: system and environment constitute the most fundamental partition of the physical world underlying any physical description, and this becomes paradigmatic in Thermodynamics. This basic realizations makes Open Quantum Systems Dynamics the natu-ral platform for the study of Quantum Thermodynamics. In particular has been introduced [3] a

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31 Chapter 3. Quantum Thermodynamics configuration space, that is the set of all pairs

C = (ρ, H) {ρ ∈ S(H_S) , H ∈ L(HS)} (3.2)

of density matrices and Hamiltonians on a given Hilbert space HS - S(HS) ⊂ L(HS)being

re-spectively the set of unitary-trace positive hermitian operators and hermitian operators. This is the space of evolution of a thermodynamic system, which allows a pictorial representation (Figure3.1). The typical setup will then be given by:

• A system S (the "engine") with an Hamiltonian HS(~λ(t))depending on a set of parameters

externally controlled.

• One (or more) thermal baths at a temperature T = 1/β, which can be put in contact with S and make the system thermalise (ρ, H) → (ωβ, H)(here ωβ = e−βH/Z(β, H)is the Gibbs

state).

The physical mapping of the above scheme is by the general situation in which an external "hu-man" control is able to manipulate a quantum systems (typically with a finite number of degrees of freedom), e.g. by changing its energy levels, and putting it in contact with thermal reservoirs, thus making possible the realization of thermodynamic cycles (cf. Section3.4) to exploit the thermal sources, for example for the extraction of work from them.

This framework allows us to define different kinds of transformation, i.e. we will refer to

• Isotherms, that is externally driven shifts in which the system is kept in contact with a bath and follows (or approximately follows) the Gibbs state curve of Figure3.1.

• Adiabats, as in classical Thermodynamics, transformations in which the system’s entropy is unaltered. With a little lexicon abuse, we will often refer to them as a synonym of quenches, a (proper) subset of adiabats formed by (instantaneous) transformations (ρ, H) 7→ (ρ, H0₎_in

which the system’s state is left unchanged.

• Thermalisations, i.e. the result of free thermal contact, with the Hamiltonian left unchanged, (ρ, H) → (ωβ(H), H).

3.2 Work and heat

The concept of internal energy U of a system has its natural extension to the quantum regime, that is, for a given state ρ with Hamiltonian H, we can identify U with its mean value,

U =Tr[ρH] . (3.3)

The microscopic definition of work and heat is more troubled, and still debated today [29]. In gen-eral, the principal ideas, borrowed from classical Thermodynamics, to split the variation of energy δTr[ρH] in two contributions are:

• the identification of a "disordered" contribution that has also the effect of changing the system entropy; one would like to label this share as "heat".

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Chapter 3. Quantum Thermodynamics 32 • the explicit consideration of a work storage system (e.g. often modelled as "unbounded" har-monic oscillator with very small gap, i.e. having a continuous, flat energy density spectrum and translation invariance with respect to interaction operations with the system).

• the identification of variation of energy due to external control with "work" performed by the experimenter.

In the context of open quantum systems dynamics, the more common definition [1,36,67], based on the configuration space framework, considers the variation of mean energy

dU =Tr[dρH] + Tr[ρdH] , (3.4) and identifies the first contribution with heat and the latter with work. This definition can be justi-fied in the configuration space framework introduced before, noting that:

the heat Tr[dρH] is given by the internal energy change due to reconfiguration of the system, which is also responsible for the change in entropy dS = Tr[−dρ ln ρ]. Moreover, in the absence of a contact with a thermal reservoir, the evolution is given by ˙ρ = −i[H, ρ] and Q nullifies.

the work Tr[dρH] is due to change in energy given by external modification of Hamiltonian con-trols, thus assimilable to work "pumped" from the experimental devices. The work supply is left implicit.

Note that with this definition work and heat exchange correspond (respectively) to horizontal and vertical displacementsin the configuration space (Figure3.1). We will stand from now on by this definition, which has no ambiguities as we saw in cases where thermal contact and work operations can be performed separately, extending it to the continuous scenario case, that is

Q = Z

dtTr[ ˙ρH] W = Z

dtTr[ρ ˙H] . (3.5) However, we mention here how other definitions are present in the literature, for example consid-ering explicitly the different contacts with the thermal reservoirs and with a work storage system [64,65], or by splitting the energy variation using the concept of "ergotropy" [45]. Other stud-ies worth to be mentioned analyse the energy exchanges in a bipartite quantum system in detail [26,34,68]. An abstract treatise of work from the point of view of resource theories is present in [17,24,25]. The problem of an abstract definition of work and heat is particularly delicate, and we shall remark here how any of the definitions for the first law should in any case be faced with the actual system analysed and the experimental setup.

3.3 Free energy, the 2nd law and Markovianity

Thermodynamic quantities first appeared in equilibrium, macroscopic scenarios. Some of them, e.g. the mean energy U, have a direct meaning and generalization to non-equilibrium situations (cf. Eq. (3.3)), while for others it could be less immediate. For example it would be very convenient to upgrade to the most general case the definitions of the so called thermodynamic potentials, defined

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33 Chapter 3. Quantum Thermodynamics from U by means of Legendre transform in classical Thermodynamics. Thermodynamic potentials are useful theoretical tools and have an important meaning from the resource point of view. For example, the free energy F = U − T Sthermo (Sthermo being the thermodynamic entropy of the

system) characterises, in classical equilibrium Thermodynamics, the amount of work it is possible to extract from the system with a given bath at temperature T . In general however irreversibility occurs and the bound cannot be filled, leading to an inequality

∆F ≤ ∆W . (3.6)

This is one of the possible statements of the 2nd law of Thermodynamics.

Exploiting the formal connection kB1SShannon = Sthermobetween the informational entropy and

the thermodynamic entropy [48], a generalisation of F into the QT setting for non-equilibrium scenarios can be obtained [18], so that the interpretation of F stays the same (i.e. (3.6) still holds and can be saturated). That is, by using the quantum-upgraded von Neumann entropy (cf. Section

1.2),

F (ρ, H) = U − T S =Tr[ρH] + T Tr[ρ ln ρ] . (3.7) In the context of OQS, these findings rely on the assumption of the Markovian character of the dynamics: consider the following equality2_{, where we write for simplicity ω}

β instead of ωβ(H),

T S(ρ k ωβ) = T Tr[ρ ln ρ − ρ ln(e−βH/Z)] =

=Tr[ρH] − T S(ρ) + T ln Z = F (ρ, H) − F (ωβ, H) . (3.10)

That is, the relative entropy S(ρ k ωβ) quantifies the excess (S(. . . k . . .) ≥ 0) of free energy

from the thermal equilibrium state. Now consider a Markovian generator Ltwhich has the thermal

equilibrium as a unique null eigenvector

Lt[ωβ(t)] = 0 (3.11)

(the time dependence of ωβ is due to H(t)). The inequality (2.14) implies that under a Markovian

evolution the relative entropy is continuously decreasing, thus taking the time derivative limit we can write

TrLt[ρ] ln ρ − Lt[ρ] ln ωβ =Tr[ ˙ρ ln ρ + β ˙ρH] ≤ 0 , (3.12)

where we used that ωβis steady, the contribution Tr[ρ ˙(ln ρ)]is null (cf. AppendixC), and Tr[ ˙ρ1] = 0.

We can read Eq. (3.12) as

1

TQ − ˙˙ S ≤ 0 , (3.13)

1_{Our convention for this manuscript is to impose k} B= 1 2_{We remember here that for an equilibrium thermal state ω}

β= e−βH/Zthe logarithm is simply

ln ωβ = −βH − 1 ln Z (3.8)

and the equilibrium free energy coincides with the logarithm of the partition function

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Chapter 3. Quantum Thermodynamics 34

Figure 3.2: Schematic representation of Quantum Carnot cycle in the configuration space. which is exactly the 2nd law of Thermodynamics. Adding the work exchange on both sides and multiplying by T we recover

˙

F ≤ ˙W . (3.14)

In case H is constant, no work is performed, the thermal state is fixed and the free energy decreases under the thermalisation d

dtS(ρ k ωβ) ≤ 0.

3.4 Quantum thermodynamic cycles

Given the general setup and definitions pointed out in Sections3.1and3.2, we can review here some standard thermodynamic cycles revisited in their quantum version, mainly on a qubit working fluid, albeit they can be easily generalised: as an example, the portrayed Carnot and Otto cycles performed on a two-level system can be immediately generalised to any system with an external control that can modulate the entire spectre by the same factor (H(t) = g(t)H0).

3.4.1 Quantum Carnot cycle

In the Quantum Thermodynamic framework we have introduced, a Quantum Carnot Cycle is iden-tified with a 4 steps process inspired directly from its classical counterpart, that is two isothermal

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35 Chapter 3. Quantum Thermodynamics strokes alternated with two isoentropic (adiabatic) strokes.

Consider for simplicity a two-level system with Hamiltonian H = Egap

2 σ

z_, _(3.15)

which can be coupled independently to two reservoirs with temperature TH > TC (here σz =is

the third Pauli matrix). In the ideal quasistatic limit the operations are performed slowly enough to allow the system to be in thermal equilibrium at every instant. The 4 steps are as in Figure3.2:

1) while being coupled to the cold reservoir, the energy gap is modified continuously from Egap = ∆to ∆∗ (if we want the heat to be released to the cold source we need ∆∗ > ∆

in order to make Tr[ ˙ρH] negative).

2) with the system isolated from the reservoirs, a quench is performed taking ∆∗ _{→ ∆}∗ TH TC.

3) while being coupled to the hot reservoir, the energy gap is modified continuously from ∆∗ TH TC

to ∆TH TC.

4) again isolating the system a quench is performed to restore ∆TH TC → ∆

TH TC

TC TH = ∆.

Note the factors TH TC and

TC

TH are chosen in order for ρ to be continuous during the quenches. In fact

in the quasistatic limit ρ is a function of βEgap(the canonical state ωβ(H)); thus for example during

the quench 2) one has 1 TC∆

∗ ₌ 1 TH∆

∗ TH TC.

In a Carnot cycle heat is exchanged only during the 1), 3) steps (absorbed from the hot source, released to the cold one), so we can compute the efficiency using the observation that over a cycle ∆Q + ∆W = 0 ηC = −∆W Qabs = QH + QC QH = 1 + QC QH = 1 − TC TH , (3.16)

where we have used in the last step that QC = Z βEgap=βC∆∗ βEgap=βC∆ Tr[dρH] = βC Z ∆∗ ∆ dE Tr_d(βE)dρ H , (3.17) while QH = βH Z ∆ ∆∗ dE Tr_d(βE)dρ H = −βH βC QC . (3.18) 3.4.2 Otto cycle

Again taking inspiration by the classical version translated in our setup, the Otto cycle is composed by two isoentropic (adiabatic) strokes alternated with two thermalisations (classically isochores). Considering the same qubit engine used for the description of the Carnot Cycle, the 4 steps can be summarized as in Figure3.3:

1) starting with a state ρh keeping fixed the gap Egap = ∆c the system is let thermalise in

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Chapter 3. Quantum Thermodynamics 36

Figure 3.3: Schematic representation of Quantum Otto cycle in the configuration space. 2) after isolating the system from the bath, a quench is performed taking ∆C → ∆H.

3) while the gap is fixed, the system is let thermalise in contact with the hot reservoir, the state of the system going back to ρc→ ρs.

4) a final quench restores ∆H → ∆C.

Note the states ρcand ρhin the quasistatic limit (infinite time for the thermalisation) are exactly the

the thermal states ωβc, ωβh, but in general they don’t need to.

The efficiency of an Otto engine depends only on the two gaps chosen, in fact being the Hamiltonian constant during the isochore steps, the heat can be simply written

QH =Tr[(ρh− ρc)∆H

σz

2 ] , QC =Tr[(ρc− ρh)∆C σz

2 ] . (3.19) Hence the efficiency can be computed

ηO= QH + QC QH = 1 + QC QH = 1 − ∆C ∆H . (3.20)

On a two-level system is always possible to identify an "instantaneous temperature" T0 _{= 1/β}0_by

parametrizing the energy level populations as follows p0 = [ρ]00=

eβ0Egap/2 eβ0_Egap/2