Non-Markovianity through entropy-based quantum thermodynamics J. M. Z. Choquehuanca1F. M. de Paula2and M. S. Sarandy1 1Instituto de F ısica Universidade Federal Fluminense

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Non-Markovianity through entropy-based quantum thermodynamics

J. M. Z. Choquehuanca,1F. M. de Paula,2and M. S. Sarandy1

1Instituto de F´ısica, Universidade Federal Fluminense,

Av. Gal. Milton Tavares de Souza s/n, Gragoat´a, 24210-346, Niter´oi, RJ, Brazil

2Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC,

Avenida dos Estados 5001, 09210-580, Santo Andr´e, SP, Brazil

(Dated: March 28, 2023)

We introduce a generalized approach to characterize the non-Markovianity of quantum dynamical maps via

breakdown of monotonicity of thermodynamic functions. By adopting an entropy-based formulation of quantum

thermodynamics, we use the relationship between heat and entropy to propose a measure of non-Markovianity

based on the heat ﬂow for single-qubit quantum evolutions. This measure can be applied for unital dynamical

maps that do not invert the sign of the internal energy. Under certain conditions, it can also be extended for other

thermodynamic functions, such as internal energy and work ﬂows. In this context, a natural connection between

heat and quantum coherence can be identiﬁed for dynamical maps that are both unital and incoherent. As appli-

cations, we explore dissipative and non-dissipative quantum dynamical processes, illustrating the compatibility

between our thermodynamic quantiﬁers and the well-establish measure deﬁned via quantum coherence.

I. INTRODUCTION

Quantum thermodynamics [1, 2] is a research ﬁeld in the

making, with fruitful conceptual implications and a potential

range of applications, mainly in the ﬁeld of quantum tech-

nologies (see, e.g., Ref. [3]). It aims at generalizing the laws

of thermodynamics to the quantum domain, rewriting thermo-

dynamic state functions and processes from the bottom up so

that the laws of thermodynamics can be consistently applied

to small systems and scaled up to the classical world [4–9].

Remarkably, the deﬁnition of thermodynamic variables, such

as work and heat, has been challenging, since they cannot be

taken as observables described by Hermitian operators [10].

In the standard framework of quantum thermodynamics [11],

the internal energy change due to state population rearrange-

ment induced by the external environment has been deﬁned

as heat. On the other hand, the internal energy change due

to energy gap variations induced by the coherent dynamics

has been deﬁned as work. This framework has recently been

revisited through an entropy-based formulation for heat and

work [12–14], with heat identiﬁed as the entropy-related con-

tribution to the internal energy change and work taken as the

part unrelated to the entropy change. In particular, this has

been applied to the study of the thermodynamics of a single

qubit far from equilibrium [15], where a notion of temperature

compatible with the classical limit is introduced. The alter-

native entropy-based view turns out to motivate an interplay

with quantum information, with the interpretation of thermo-

dynamic variables in terms of resources and correlations in a

quantum system.

Indeed, quantum thermodynamics is intrinsically connected

with quantum information theory, with quantum resources

strongly aﬀecting thermodynamic variables and processes.

This is manifested in a number of situations, such as mea-

surements and their inﬂuence on the work performed on/by

a quantum system [10, 16–19], eﬃciency of thermodynamic

tasks in quantum correlated devices [6, 20–22], coherence ef-

fects on work and heat [23–30], among others. By looking

at information-based quantiﬁers, it is known that quantum co-

herence plays a relevant role in the characterization of system-

bath memory eﬀects. In particular, quantum coherence can be

associated with a measure of non-Markovianity for incoherent

dynamical maps [31–34]. Notice that a memoriless (Marko-

vian) behavior is always an idealization, with non-Markovian

quantum dynamics being non-negligible in a number of diﬀer-

ent scenarios (see, e.g., Refs. [35–37]). From an applied point

of view, non-Markovian dynamics may be also a resource for

certain quantum tasks [38, 39].

In this work, we adopt the entropy-based formulation of

quantum thermodynamics to introduce a measure of non-

Markovianity based on the heat ﬂow for single-qubit quan-

tum evolutions. This measure can be applied for unital dy-

namical maps that do not invert the sign of the internal en-

ergy (energy sign-preserving maps), which is rooted in the

idea that von Neumann entropy itself can be used to build a

non-Markovianity measure for unital maps [40]. Under cer-

tain conditions, we can also extend this approach to other

thermodynamic functions, such as internal energy and work

ﬂows. Then, a natural connection between heat and quan-

tum coherence can be observed for dynamical maps that are

both unital and incoherent. In particular, we show that heat

and work are monotonically related to quantum coherence for

non-dissipative processes. Consequently, both heat and work

can witness non-Markovianity for non-dissipative unital maps

as well as non-dissipative incoherent maps. Moreover, we es-

tablish and illustrate the conditions over the system purity and

the Hamiltonian such that the thermodynamic variables can

be used to provide non-Markovianity measures throughout the

evolution in a decohering environment.

The paper is organized as follows. In Section II, we brieﬂy

introduce the entropy-based formulation of quantum thermo-

dynamics, expressing the ﬁrst law of thermodynamics and its

corresponding thermodynamic variables for a single qubit sys-

tem. In section III, we discuss the non-Markovian dynamics,

where we propose measures of non-Markovianity via general-

ized sign-preserving functions. In the section IV, we consider

applications in dissipative and non-dissipative evolutions, il-

lustrating the use of thermodynamic variables as measures of

non-Markovianity. In section V, we present our conclusions.

arXiv:2210.03767v2 [quant-ph] 27 Mar 2023

II. QUANTUM THERMODYNAMICS

According to the standard framework for quantum thermo-

dynamics [11], the internal energy Uof a system described by

a density operator ρis provided by the expected value of its

Hamiltonian H, i.e., U=tr ρH. In this formalism, the ﬁrst

law of thermodynamics emerges from an inﬁnitesimal change

in the internal energy dU =δQ+δW, with δQ=tr dρH

and δW=tr ρdHdeﬁning the heat absorbed by system and

the work performed on system, respectively. Considering the

density operator as expressed in its spectral decomposition,

we have ρ=Pkrk|rki hrk|, where |rkidenotes an eigenvector

of ρand rkthe corresponding eigenvalue. Then, the thermo-

dynamic quantities U,δQ, and δWcan be rewritten as

U=X

rkhrk|H|rki,(1)

δQ=X

drkhrk|H|rki+X

rk(hrk|H d |rki+h.c.),(2)

δW=X

rkhrk|dH |rki.(3)

Note that the ﬁrst term on the right-hand-side of Eq. (2) is

the responsible for changes in the von Neumann entropy, S=

−kBtr ρlnρ, since dS =−kBPkdrklnrk. Therefore, in order

to connect the heat ﬂow with the entropy change as in classical

thermodynamics, an entropy-based formulation of quantum

thermodynamics has recently been introduced in Refs. [12–

14]. In this framework, heat and work are redeﬁned through

δQ=δQ−δW∗(4)

and

δW=δW+δW∗,(5)

where δW∗is an additional work contribution given by

δW∗=X

rk(hrk|H d |rki+h.c.).(6)

The work δW∗is related to the variation d|rkiof the den-

sity operator eigenvectors. Notice that the entropy-based for-

malism satisﬁes the ﬁrst law of thermodynamics, i.e., dU =

δQ+δW, being then equivalent to the standard framework for

δW∗=0. Remarkably, it can be shown that the existence of

quantum coherence in ρin the energy eigenbasis {|hki}is a

necessary ingredient for a non-zero work δW∗if the energy

eigenvectors are ﬁxed (i.e., for d|hki=0, ∀k). Here, we will

deﬁne coherence through the l1-norm [31] of ρin the energy

eigenbasis, reading

C(ρ)=X

k,l

|hhk|ρ|hli| .(7)

Indeed, observe that, if His constant and ρand Hhave a com-

mon basis of eigenvectors, then d|hki=d|rki=0. This leads

to δW∗=0. Moreover, since a common basis for Hand ρ

implies that ρis diagonal in the energy eigenbasis {|hki}, we

will have C(ρ)=0.

For an arbitrary single qubit system, the density operator

can be written in the Pauli basis {I,~

σ}as

ρ=1

2I+~

r·~

σ,(8)

where ~

r=(x,y,z) is the Bloch vector. For the Hamiltonian,

we have H=−~

h·~

σ. We observe that ~

r=(x,y,z) can be in-

terpreted as a classical magnetic dipole moment immersed in

an external magnetic ﬁeld ~

h. Indeed, the quantities in Eqs. (1)

- (7) reduce to [15]:

U=−~

h·~

r,(9)

δQ=−~

h·d~

r,(10)

δW=−~

r·d~

h,(11)

δQ=Urdr,(12)

δW=rdUr,(13)

δW∗=−rh ˆ

h·dˆr,(14)

and

C=rr1−U2

h2,(15)

where r≡ |~

r|is the purity, Ur≡U/ris the internal energy per

unit of purity, h≡ |~

h|is the positive energy eigenvalue, ˆr≡~

r/r

is the dipole direction, and ˆ

h≡~

h/his the external ﬁeld direc-

tion. Note that heat and work in the entropy-based qubit for-

malism, namely, δQand δW, are necessarily associated with

changes in r(and consequently in S) and Ur, respectively. On

the other hand, the standard contributions for these thermo-

dynamic quantities, δQand δW, are required to be associated

with variations in ~

rand ~

h, respectively.

An interpretation for δW∗in terms of the behavior of ~

rcan

be obtained through the relation between δW∗and C(ρ). First,

a change in the eigenvectors of ρis required for non-zero δW∗.

By imposing dˆ

h=0 (i.e., a ﬁxed energy eigenbasis), we can

express δW∗in Eq. (14) as δW∗=hCdθ, where hC =|~

r×~

denotes the absolute value of the torque on ~

rinduced by ~

hand

θ=arcos(ˆ

h·ˆr) is the angle between ~

rand ~

h[15]. Therefore,

δW∗is equivalent to the energy cost required to rotate a mag-

netic dipole moment immersed in an external magnetic ﬁeld,

being proportional to coherence. More generally, δW∗repre-

sents the departure from the quasistatic dynamics [12, 13].

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Non-Markovianitythroughentropy-basedquantumthermodynamicsJ.M.Z.Choquehuanca,1F.M.dePaula,2andM.S.Sarandy11InstitutodeF´sica,UniversidadeFederalFluminense,Av.Gal.MiltonTavaresdeSouzas/n,Gragoat´a,24210-346,Niter´oi,RJ,Brazil2CentrodeCienciasNaturaiseHumanas,UniversidadeFederaldoABC,AvenidadosEstado...

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Non-Markovianity through entropy-based quantum thermodynamics J. M. Z. Choquehuanca1F. M. de Paula2and M. S. Sarandy1 1Instituto de F ısica Universidade Federal Fluminense.pdf

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Non-Markovianity through entropy-based quantum thermodynamics J. M. Z. Choquehuanca1F. M. de Paula2and M. S. Sarandy1 1Instituto de F ısica Universidade Federal Fluminense

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