PHYSICAL REVIEW A 88, 012105 (2013)

Probing the degree of non-Markovianity for independent and common environments

Felipe F. Fanchini,1 Goktug Karpat,2 Leonardo K. Castelano,3 and Daniel Z. Rossatto3

1Faculdade de Ciencias, UNESP–Universidade Estadual Paulista, Bauru, Sao Paulo, 17033-360, Brazil2Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul, 34956, Turkey

3Departamento de Fısica, Universidade Federal de Sao Carlos, Sao Carlos, Sao Paulo, 13565-905, Brazil(Received 22 January 2013; published 8 July 2013)

We study the non-Markovianity of the dynamics of open quantum systems, focusing on the cases of independentand common environmental interactions. We investigate the degree of non-Markovianity quantified by twodistinct measures proposed by Luo, Fu, and Song and Breuer, Laine, and Pillo. We show that the amount ofnon-Markovianity, for a single qubit and a pair of qubits, depends on the quantum process, the proposed measure,and whether the environmental interaction is collective or independent. In particular, we demonstrate that whilethe degree of non-Markovianity generally increases with the number of qubits in the system for independentenvironments, the same behavior is not always observed for common environments. In the latter case, ouranalysis suggests that the amount of non-Markovianity could increase or decrease depending on the propertiesof the considered quantum process.

DOI: 10.1103/PhysRevA.88.012105 PACS number(s): 03.65.Yz, 42.50.Lc

I. INTRODUCTION

The concept of non-Markovianity is a prominent aspectof the dynamics of open quantum systems and has beenattracting both theoretical and experimental attention in thelast few years [1,2]. Moreover, it has been shown that non-Markovianity can be used as a tool in quantum protocols [3],can be employed to advantage in quantum metrology [4],and can be exploited in quantum key distribution [5]. Furtherconcepts behind the non-Markovian dynamics have also beeninvestigated, for example, the influence of environment size[6] and the possibility to pursue new quantum technologiesby using non-Markovian effects [3]. Although all thoseefforts to understand the connection between non-Markoviandynamics and quantum information theory have been made,measuring non-Markovianity is complicated and generallyonly small systems have been considered [7]. However, the realusefulness of a quantum system for computation or simulationis appreciable only in the limit of large-scale informationprocessing. Therefore, it is fundamental to understand theproperties of non-Markovianity for multipartite systems.

Recently, various measures for quantifying the degreeof non-Markovianity of the dynamics of an open quantumsystem [8–11] have been introduced in the literature; however,there is no consensus on what precisely determines thenon-Markovianity of a dynamical quantum process. It hasbeen demonstrated that the conclusions drawn from differentmeasures might not agree, depending on the consideredphysical model. The most widely used measure of non-Markovianity was introduced by Breuer, Laine, and Piilo(BLP) [8]. In their seminal paper, they claimed that informationflows only from the system into the environment for aMarkovian process and the information flow can be measuredby the trace distance of two arbitrary quantum states, whichprobes the distinguishability between them. To implement thismeasure, one needs to perform an optimization by checkingthe dynamics of the trace distance for a huge number ofinitial sates. Thus, this procedure is very demanding andalmost impracticable when dealing with multipartite systems.Moreover, Rivas, Huelga, and Plenio (RHP) have constructed

a measure of non-Markovianity that quantifies the deviationfrom divisibility for a dynamical map [9], which is alsodifficult to implement in general. In order to overcome thesedifficulties, we use an efficient method to evaluate a measureof non-Markovianity, recently proposed by Luo, Fu, and Song(LFS), based on the nonmonotonic behavior of the quantummutual information for non-Markovian processes [10]. Thismeasure coincides with other important ones such as the BLPmeasure for quite general cases and can be straightforwardlyextended for studying multipartite systems.

When dealing with the interaction between a quantumsystem and its environment, there are two important physicalprocesses that must be considered: relaxation and decoherence(here called dephasing). While relaxation is associated witha process involving loss of energy, dephasing is associatedwith the loss of purity without any exchange of energybetween the system and its surroundings. In this work, weexplore both processes, considering the amplitude dampingchannel to describe dissipative processes, and taking intoaccount two different kinds of interaction to describe phasedamping processes, namely, a super-Ohmic dephasing channeland the phase damping case employed to describe impurityatoms coupled to a Bose-Einstein condensate. For all thedifferent scenarios, we analyze the effect of both independentand common environmental interactions on the behavior ofnon-Markovianity by investigating two distinct quantifiers ofthe degree of non-Markovianity given by the LFS and the BLPmeasures. For zero-temperature environments, we link the LFSmeasure to the rate of change of the system entropy S(ρs(t))and the environment entropy S(ρe(t)). We show that, for theLFS measure, a quantum process is non-Markovian if the timederivative of S(ρs(t)) is greater than the time derivative ofS(ρe(t)). We present a detailed analysis of the evaluation ofthe LFS measure for a single qubit, and discuss the behaviorof the optimal initial states as a function of the parameters ofthe environments. Moreover, we demonstrate that the degreeof non-Markovianity, for both proposed measures, increasesin general as a function of the number of qubits in thesystem for the case of independent environments. On the otherhand, for global environments, we show that the amount of

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FANCHINI, KARPAT, CASTELANO, AND ROSSATTO PHYSICAL REVIEW A 88, 012105 (2013)

non-Markovianity depends on the quantum process and theproposed measure.

The paper is organized as follows. In Sec. II, we introducethe measures of non-Markovianity that will be used in ourinvestigation. Section III outlines several system-environmentmodels describing the dynamics of open quantum systems.Sections IV and V include our findings related to the behaviorof the LFS and BLP measures under the considered models,respectively. Section IV covers the summary of the resultsobtained in this work.

II. MEASURING NON-MARKOVIANITY

A. LFS measure

The definition of the LFS measure of non-Markovianity[10] is based on the following: Suppose that we have a quantumsystem in a Hilbert space H , and a quantum process �(t)governing the dynamical evolution of the considered system. Ifan arbitrary ancilla system in a Hilbert space Ha is introduced,the composite state of the main system and ancilla ρsa pertainsto the Hilbert space H ⊗ Ha . In this case, assuming a trivialdynamics on the ancillary, the time evolution of the total systemis given by ρsa(t) = [�(t) ⊗ I ]ρsa(0), where I is the identityoperator acting on the state space of the ancillary. The amountof total correlations in a bipartite state ρsa can be quantifiedthrough the quantum mutual information

I (ρsa) = S(ρs) + S(ρa) − S(ρsa), (1)

where ρa = trsρsa and ρs = traρsa represent the reduceddensity operators of the system and the ancilla, respectively.S(ρ) = −trρ log2 ρ is the von Neumann entropy. Exploitingthe fact that quantum mutual information decreases monoton-ically as a function of time for a Markovian process, LFS haveproposed a new quantity for measuring the non-Markovianityof the dynamical process �(t) from an informational perspec-tive:

N (�) = supρsa(0)

∫(d/dt)I (ρsa(t))>0

d

dtI (ρsa(t))dt, (2)

where the supremum is taken over all possible initial statesρsa(0). Even though this measure has an interesting meaningfor the quantification of non-Markovianity, its evaluation ishard to perform due to the potentially complex optimizationproblem.

As described in Eq. (2), a dynamical quantum processis said to be non-Markovian if d

dtI (ρsa(t)) > 0. Note that

the ancilla, unlike the system, does not interact with theenvironment. In other words, the state of the ancilla is timeindependent. Therefore, the time derivative of the quantummutual information can be written as

d

dtI (ρsa(t)) = d

dtS(ρs(t)) − d

dtS(ρsa(t)). (3)

For a zero-temperature environment, an interesting result canbe obtained from this equation. Since we take ρsa(0) as a purestate, and the environment starts in the state ρe(0) = |0〉〈0| forthe zero-temperature case, the total quantum state composedof the system, the ancilla, and the environment is a pure stateat any time, leading to S(ρsa(t)) = S(ρe(t)). Consequently,following the LFS measure, we obtain a non-Markovianity

criterion without the need of an ancilla; therefore, a quantumprocess is non-Markovian if and only if

d

dtS(ρs(t)) >

d

dtS(ρe(t)). (4)

This condition links the non-Markovianity measure to therate of change of the system and the environment entropies.Nonetheless, some peculiar aspects should be noted: theenvironment is initially in a pure state by assumption, butno restrictions were imposed on the system. If the systemis initially in a pure state as well, we have ρse pure andtherefore S(ρs(t)) = S(ρe(t)). Such a result does not meanthat the process is actually Markovian because we need tomaximize over all possible initial conditions of the systemto be able to determine the degree of non-Markovianity. Anequivalent equation for Eq. (2) can be deduced without thenecessity of an ancilla:

NT =0 K(�) = supρs (0)

∫(d/dt)�Sse(t)>0

d

dt�Sse(t)dt, (5)

with �Sse(t) = S(ρs(t)) − S(ρe(t)). The advantage of theabove equation over Eq. (2) is the fact that the maximizationis just over the initial conditions of the system instead of theinitial conditions of the composite state of the system and theancilla, which is required to calculate N (�). It is importantto emphasize that the time derivative of the entropy is theimportant quantity for a non-Markovian process. Furthermore,to calculate S(ρe(t)), we do not need to worry about the stateof the environment. The idea here is to maximize NT =0 K overall possible initial system states and, for each choice, purifyit including an extra subsystem. Because the environment isset in a pure state at t = 0, the entropy of the system plus thepurifier subsystem is equal to the entropy of the environmentat any time. An important observation that deserves to bementioned is that, to evaluate Eqs. (2) and (5), it is possible tosuppress the calculation of the integrals and time derivativesof the integrands. In fact, it is straightforward to note that wecan rewrite the LFS measure as

N (�) = supρsa(0)

∑i

[I (ρsa(bi)) − I (ρsa(ai))]. (6)

To compute this quantity, we first determine the time intervals(ai,bi) in which the mutual information increases; then wesum up the contribution of each interval to obtain N (�).

On the other hand, LFS presented a significant simplifi-cation for Eq. (2) in Ref. [10]. Assuming that Ha = H andρsa(0) = |�〉〈�| where |�〉 is any maximally entangled purestate of the system and the ancilla, they obtain an easilycomputable measure of non-Markovianity:

N0(�) =∫

(d/dt)I (ρsa(t))>0

d

dtI (ρsa(t))dt, (7)

with ρsa(t) = [�(t) ⊗ I ]|�〉〈�|. In the Appendix, we ex-plicitly show that N0(�), despite its utility as a witness fornon-Markovianity, may be misleading and give an inaccurateconclusion about the degree of non-Markovianity of a quantumprocess. Furthermore, we demonstrate that N (�) does notdepend on the amount of entanglement shared between thesystem and ancilla because two distinct initial states withthe same degree of entanglement can give different results.

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PROBING THE DEGREE OF NON-MARKOVIANITY FOR . . . PHYSICAL REVIEW A 88, 012105 (2013)

Actually, the optimal state is not maximally entangled ingeneral.

B. BLP measure

The BLP measure of non-Markovianity [8] employs thetrace distance D12(t) = 1/2tr|ρ1(t) − ρ2(t)| between two arbi-trary reduced density matrices ρ1(t) and ρ2(t) in order to checkthe distinguishability between them. Such reduced densitymatrices result from the calculation of the partial trace withrespect to the environment part of the total density matrixthat describes a system coupled to an environment. WhendD12(t)/dt < 0, the distinguishability between the reduceddensity matrices decreases and there is a flow of informationfrom the system to the environment. On the other hand,information flow from the environment back to the system canhappen if dD12(t)/dt > 0. In such situations, the processesare said to be non-Markovian. Furthermore, the BLP measureof non-Markovianity [8] is mathematically defined as follows:

NBLP(�) = maxρ1(0),ρ2(0)

∫[dD12(t)/dt]>0

dD12(t)

dtdt, (8)

where the maximum value is taken over all pairs of initialreduced states ρ1(0) and ρ2(0). It is important to emphasizethat to numerically implement this measure, it is necessaryto evaluate the dynamics of the trace distance D12(t) for ahuge number of initial states, which makes this procedurevery demanding when dealing with multipartite systems. Wenote that the above equation can also be rewritten as

NBLP(�) = maxρ1(0),ρ2(0)

∑i

[D12(bi) − D12(ai)], (9)

where the time intervals (ai,bi) correspond to the regions inwhich dD12(t)/dt > 0 and the maximum value is taken overall pairs of initial reduced states ρ1(0) and ρ2(0).

III. OPEN-SYSTEM DYNAMICS

A. Phase-damping channel

We consider a spin-boson-type Hamiltonian HIPD thatdescribes a pure dephasing type of interaction between a qubitand a bosonic environment:

HIPD = ω0

2σz +

∑k

ωka†kak +

∑k

σz(gka†k + g∗

k ak), (10)

where the first and the second terms of Eq. (10) are responsiblefor the free evolution of the qubit and the environment,respectively. The third term of Eq. (10) accounts for theinteraction between the qubit and its environment. We firstnote that [H,σz] = 0, which immediately implies the absenceof transitions between different energy levels. Thus, the popu-lation terms in the density matrix of the system are conservedquantities. Here, ω0 is the transition frequency of the qubitand ωk is the field frequency of the kth environmental fieldmode. The constant gk controls the strength of the couplingbetween the qubit and each field mode of the environment.While the qubit operator is given by the usual Pauli σz matrix,the creation operator ak and the annihilation operator a

†k ,

satisfying the bosonic commutation relations [ak,a†k′] = δk,k′ ,

represent the environment. It is worth stressing that this

qubit-plus-environment model admits an exact solution [12].We assume that the composite state of the qubit and theenvironment is initially factorized, that is, there exist nocorrelations between the system and the environment at t = 0;furthermore, the environment is initially in its vacuum stateρe(0) = |0〉〈0| at zero temperature. We consider a sufficientlylarge environment; therefore, we can replace the sum over thediscrete coupling constants by an integral over a continuousdistribution of frequencies of the environmental modes, i.e.,∑

k |gk|2 → ∫ ∞0 dωJ (ω). In addition, we suppose that the

spectral density of the environmental modes is Ohmic-like:

J (ω) = ηωs

ωs−1c

e−ω/ωc , (11)

with ωc being the cutoff frequency and η a dimensionlesscoupling constant. Depending on the parameter s, the spectraldensity is called sub-Ohmic (s < 1), Ohmic (s = 1), or super-Ohmic (s > 1). Under these conditions, the dynamics of asingle qubit can be obtained in the operator-sum representationas

ρ(t) =2∑

i=1

Ki(t)ρ(0)K†i (t), (12)

where the Kraus operators Ki(t) are given by

K1(t) =(

1 0

0 r(t)

), K2(t) =

(0 0

0√

1 − r2(t)

), (13)

with∑2

i=1 K†i (t)Ki(t) = I for all values of t , where I denotes

the 2 × 2 identity matrix. Here, the dephasing parameter r(t)is

r(t) = exp

[−

∫ t

0γ (t ′)dt ′

], (14)

where the dephasing rate γ (t) takes the form

γ (t) = ηωc[1 + (ωct)2]−s/2�(s) sin[s arctan(ωct)], (15)

with �(s) being the Euler Gamma function.

B. Amplitude-damping channel

In order to discuss the relaxation process, we consider thefollowing model Hamiltonian:

HIAD = ω0σ+σ− +∑

k

ωka†kak + (σ+B + σ−B†), (16)

where B = ∑k gkak with gk being the coupling constant. The

first two terms of Eq. (16) describe the free evolution ofthe qubit and the environment, respectively, while the thirdterm accounts for the interaction between the qubit and theenvironment. The transition frequency of the qubit is ω0, andσ± denote the raising and lowering operators related to thequbit. The index k is used to label the different environmentalfield modes with frequencies ωk , which are mathematicallydescribed by the annihilation and creation operators givenby ak and a

†k , respectively. Restricting ourselves to the case

of a single excitation, the modes of the environment can be

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FANCHINI, KARPAT, CASTELANO, AND ROSSATTO PHYSICAL REVIEW A 88, 012105 (2013)

described by an effective spectral density of the form

J (ω) = 1

2π

γ0λ2

(ω0 − ω)2 + λ2, (17)

where λ defines the spectral width of the coupling and it isalso connected to the correlation time of the environment τB

by the relation τB ≈ 1/λ. γ0 is the time scale τR over whichthe state of the system changes by τR ≈ 1/γ0. For this form ofa spectral density, it is not hard to distinguish the weak- andthe strong-coupling regimes. The case τR > 2τB correspondsto the weak-coupling regime where the decoherence processis Markovian because the relaxation time is greater than thecorrelation time of the environment. On the other hand, the caseτR < 2τB corresponds to the strong-coupling regime where thenon-Markovian nature of the environment becomes evident.We note that at zero temperature this Hamiltonian with theconsidered spectral density (known as the damped Jaynes-Cummings model in the literature) represents one of the fewexactly solvable models for open quantum systems. In thestrong-coupling regime, the time evolution of a single qubitcan be expressed in the operator-sum representation as

ρ(t) =2∑

i=1

Mi(t)ρ(0)M†i (t), (18)

where the corresponding Kraus operators Mi(t) are given by

M1(t) =(

1 0

0√

p(t)

), M2(t) =

(0

√1 − p(t)

0 0

), (19)

satisfying the condition∑2

i=1 M†i (t)Mi(t) = I for all values

of t . The damping parameter p(t) reads

p(t) = e−λt

[cos

(dt

2

)+ λ

dsin

(dt

2

)]2

, (20)

with d =√

2γ0λ − λ2.

C. Impurity atoms coupled to a Bose-Einstein condensate

The third model considered in this work deals with twoatoms interacting with an ultracold bosonic rubidium gas in aBose-Einstein condensate (BEC) state [13,14]. Here, the qubitis represented by an impurity atom in a double-well potentialof an optical superlattice of wavelength λ, where the size of thequbit is the distance between the lattice sites, L = λ/4. Thesuperlattice is immersed in the BEC environment, where therubidium gas is assumed to be in the weak-coupling regime,justifying the validity of the Bogoliubov approach. For moredetails on the model, see Refs. [13,14].

In Ref. [13], the authors studied the non-Markovianity ofthis model from the points of view of the BLP and RHPmeasures, and they showed that it is possible to tune from acommon environment to an independent one by adjusting thespatial separation of the qubits. Moreover, the authors observedthat whereas the BLP measure is superadditive when the qubitsare very close to each other (common environment regime), itis subadditive when the qubits are sufficiently far enough fromeach other (independent environment regime).

The dynamics of the model is given by a Lindblad-typemaster equation with time-dependent decay rates [13,14]

dρ

dt= γ1(t) − γ2(t)

2

[(σ (1)

z − σ (2)z

)ρ(σ (1)

z − σ (2)z

)− 1

2

{(σ (1)

z − σ (2)z

)(σ (1)

z − σ (2)z

),ρ

}]+ γ1(t) + γ2(t)

2

[(σ (1)

z + σ (2)z

)ρ(σ (1)

z + σ (2)z

)− 1

2

{(σ (1)

z + σ (2)z

)(σ (1)

z + σ (2)z

),ρ

}], (21)

where σ (n)z is the usual Pauli matrix for the nth atom (n = 1,2),

and

γ1(t) = g2SEn0

hπ2

∫ ∞

0dkk2e−k2σ 2/2 sin

(Ek

2h t)

cos(

Ek

2h t)

(εk + 2gEn0)

×(

1 − sin (2kL)

2kL

)(22)

γ2(t) = g2SEn0

2hπ2

∫ ∞

0dkk2e−k2σ 2/2 sin

(Ek

2h t)

cos(

Ek

2h t)

(εk + 2gEn0)

×(

sin[2k(D + L)]

2k(D + L)+ sin[2k(D − L)]

2k(D − L)

− 2sin(2kD)

2kD

), (23)

where gE = 4πh2aE/mE is the boson-boson coupling fora BEC environment with scattering length aE and atomicmass mE , and gSE = 2πh2aSE/mSE is the coupling be-tween the system and the environment with scattering lengthaSE and reduced mass mSE = mSmE/(mS + mE). Ek =√

2εkn0gE + ε2k is the energy of the kth Bogoliubov mode,

n0 is the condensate density, εk = h2k2/2mE , and σ is thevariance parameter of the lattice site. Finally, 2D � 8L isthe distance between the atoms. As in Ref. [13], we consider23Na impurity atoms immersed in a 87Rb condensate withλ = 600 nm and n0 = 1020 m−3. The scattering length of theatoms is aRb = 99a0 where a0 is the Bohr radius, and weassume aSE = 55a0. Finally, we choose the scattering lengthof the BEC environment as aE = 0.5aRb.

IV. DEGREE OF NON-MARKOVIANITY: LFS MEASURE

A. Single qubit

Before starting to elucidate the properties of the LFSmeasure for multipartite systems at zero temperature, weconsider the case of a single-qubit system. In this case, themaximization in Eq. (5) can be numerically evaluated becausethe general form of the density matrix ρs(0) depends just onthree real variables. Explicitly,

ρs(0)

=(ρ11(0) Re[ρ12(0)] + iIm[ρ12(0)]

Re[ρ12(0)] − iIm[ρ12(0)] 1 − ρ11(0)

).

For both dephasing processes, our numerical analysis showsthat the maximum in Eq. (2) is reached for a maximallymixed initial state. In other words, the optimal state of the

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PROBING THE DEGREE OF NON-MARKOVIANITY FOR . . . PHYSICAL REVIEW A 88, 012105 (2013)

0

0.002

0.004

0.006

0 0.2 0.4 0.6 0.8 1non-

Mar

kovi

anity

ρ11

(c)

0 0.2 0.4 0.6 0.8

1

0 0.2 0.4 0.6 0.8 1non-

Mar

kovi

anity

ρ11

(b)

0

0.002

0.004

0.006

0 0.2 0.4 0.6 0.8 1non-

Mar

kovi

anity

ρ11

(a)

FIG. 1. (Color online) Non-Markovianity for one qubit as afunction of the density matrix element ρ11(0) for (a) the super-Ohmicdephasing process with s = 3, wc = 1 and η = 2, (b) the relaxationprocess with γ0 = 1 and λ = 0.1, and (c) an impurity atom coupledto a BEC environment with σ = 45 nm.

composite system ρsa(0) is maximally entangled, justifyingthe simplification proposed by LFS, as described in Eq. (7).However, contrary to what one might expect, the maximumvalue in Eq. (2) for the relaxation process is obtained for adiagonal initial state whose system-plus-ancilla density matrixis not maximally entangled.

As suggested by our numerical investigation, we first setto zero the off-diagonal elements of the density matrix, i.e.,ρ12(0) = ρ21(0) = 0. In Figs. 1(a) and 1(c) we plot the possiblevalues of the degrees of non-Markovianity quantified by theLFS measure N (�) for the super-Ohmic phase-damping (PD)channel with s = 3, wc = 1, and η = 2, and for the BECenvironment with σ = 45 nm, respectively, as functions ofthe density matrix population ρ11(0). As can be seen fromFigs. 1(a) and 1(c), the LFS measure corresponds to havingρ11(0) = 0.5, implying that the optimal initial composite stateis maximally entangled. On the other hand, Fig. 1(b) presentsthe results of the same analysis performed for the amplitude-damping (AD) channel with γ0 = 1 and λ = 0.1. It is shownthat the LFS measure is obtained for ρ11(0) ≈ 0.4, meaningthe optimal composite state is not maximally entangled in thiscase. In particular, this result points out that the LFS measuredoes not generally depend on the initial entanglement betweenthe system and the ancilla.

Another interesting point is related to the dependence ofthe optimal initial state of the system on the parameters ofthe considered environmental model. In Figs. 2(a), 2(b), and2(c) we plot the density matrix element ρ11(0) of the optimalinitial state for the super-Ohmic PD channel, the AD channel,and the BEC environment, respectively, as a function of the

0.4 0.45 0.5

0.55 0.6

10 20 30 40 50 60 70ρ 1

1σ (nm)

(c)

0.1 0.2 0.3 0.4 0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ρ 11

λ(γ0)

(b)

0.4 0.45 0.5

0.55 0.6

2 4 6 8 10 12 14

ρ 11

ωc

(a)

FIG. 2. (Color online) The density matrix element ρ11(0) of theoptimal initial state of a single-qubit system as a function of the bathparameters for (a) the super-Ohmic dephasing process with s = 3 andη = 2, (b) the relaxation process, and (c) an impurity atom coupledto a BEC environment.

bath parameters ωc, λ, and σ . The results of this analysisdemonstrate an important point, that is, whereas the optimalinitial states for the super-Ohmic PD channel and the BECenvironment do not depend on the bath parameters ωc and σ ,the AD channel is highly sensitive to the bath parameter λ. Infact, we have found that as the parameter λ gets smaller, theoptimal state tends to be a maximally mixed one.

B. Multiple qubits: Independent environments

In this section, we apply the LFS measure to the multipartitecase, taking into account only independent environments. Firstof all, we consider a quantum state which is composed of asystem of n qubits and an ancilla that purifies the system state.Assuming that only n qubits are subjected to the environmentalnoise and the ancillary system evolves freely, the dynamics ofthe composite system ρsa(t) can be obtained, for the AD andsuper-Ohmic PD channels, as

ρsa(t) =∑

i={1,2}(E⊗n

i ⊗ I )ρsa(0)(E⊗ni ⊗ I )†, (24)

where Ei are the Kraus operators describing the AD orsuper-Ohmic PD channels for a single qubit, n is the numberof qubits, I denotes the identity matrix with dimensions ofthe ancillary system, and the sum over the index i = {1,2}runs over all possible permutations of the Kraus operatorsE⊗n

i . For the case of n atoms independently coupled to a BECenvironment, on the other hand, we evaluate the dynamicsby numerically extending Eq. (21) under the assumption thatD/L → ∞ in Eq. (23). In this situation, since the dynamics of

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FANCHINI, KARPAT, CASTELANO, AND ROSSATTO PHYSICAL REVIEW A 88, 012105 (2013)

0.005 0.01

0.015 0.02

0.025 0.03

1 2 3 4 5

N

number of qubits

(c)

1.0

2.0

3.0

4.0

1 2 3 4 5

N

number of qubits

(b)

0.005 0.01

0.015 0.02

0.025

1 2 3 4 5

N

number of qubits

(a)

FIG. 3. (Color online) Non-Markovianity N (�) for (a) the super-Ohmic dephasing process with s = 3, wc = 1, and η = 2, (b) therelaxation process with γ0 = 1 and λ = 0.1, and (c) impurity atomscoupled to a BEC environment with σ = 45 nm, as a function of thenumber of qubits.

each atom are independent of each other, it is straightforwardto consider more than two atoms.

Performing the optimization required for the calculationof the LFS measure N (�) becomes a very difficult taskfor a system of two or more qubits due to the significantlyincreasing number of variables involved in such cases. Toovercome this difficulty, we limit our study to diagonal productinitial states of the form ρs(0)⊗n for multipartite systems.Indeed, this is a reasonable choice given the results acquiredfor the case of a single qubit. Moreover, by using suchdiagonal states we are able to obtain a lower bound for thedegree of non-Markovianity of a quantum process. Whilewe plot the LFS measure as a function of the number ofqubits for super-Ohmic dephasing environments with s = 3,wc = 1, and η = 2 in Fig. 3(a), similar results are displayedfor the relaxation process with γ0 = 1 and λ = 0.1 in Fig.3(b), and for the BEC environment with σ = 45 nm in Fig.3(c). These figures indicate a linear increase in the degreeof non-Markovianity for the considered initial states, whichproves that, for independent environments, the LFS measureis at least additive, i.e.,

N (�⊗n) � n[N (�)]. (25)

This peculiar behavior of the LFS measure, despite beingnonintuitive, is simple to understand in this context becauseboth the system and the environment entropies are additivequantities when independent environments are considered.

C. Two qubits: Common environment

In this section, we turn our attention to the behavior ofnon-Markovianity when the system is globally interacting witha common environment. Due to the difficulty of calculating thedynamics for many qubits when considering their interactionwith a common environment, we restrict our analysis to onlytwo qubits. However, the analysis of the two-qubit case is stillinteresting to infer the main characteristics of the degree ofnon-Markovianity as a function of the system scale. We firstconsider the case where both qubits are coupled to a commondephasing bath described by the following Hamiltonian:

HCPD = 1

2

∑n

ω(n)0 σ (n)

z +∑

k

ωka†kak

+∑n,k

σ (n)z (gka

†k + g∗

k ak), (26)

where the index n denotes the terms related to the first(n = 1) and second (n = 2) qubits. We also consider onlydiagonal two-qubit initial states to evaluate the degree of non-Markovianity measured by the LFS measure. The results ofour analysis suggest that the LFS measure N (�) suffers a verysignificant decay as compared to the single-qubit case, havinga value of the order of 10−7. Note that such a finding is clearlydifferent from what is observed for independent environments,where the degree of non-Markovianity increases at leastlinearly.

Next, we focus on the scenario where a system of two qubitsglobally interacts with a common relaxation environment:

HCAD =∑

n

ω(n)0 σ

(n)+ σ

(n)− +

∑k

ωka†kak

+∑

n

(σ (n)+ B + σ

(n)− B†). (27)

In this case our investigation reveals that, by assumingdiagonal initial states, the degree of non-Markovianity issignificantly amplified when compared to the single-qubitcase, approximately turning out to be N (�) ≈ 6.21. Inter-estingly, among the two-qubit diagonal states that we haveconsidered, the optimal one is always the maximally mixedstate, independently of the bath parameters. Such a finding israther surprising because the optimal initial state is stronglydependent on the parameters of the environment for thesingle-qubit case.

Comparing the results obtained for the super-Ohmic de-phasing process with those obtained for the relaxation process,one could believe that while the degree of non-Markovianityincreases for relaxation processes, it decreases for dephasingprocesses, as compared to the single-qubit case. However, asshown below, this is not generally true. Although the situationwhere two impurity atoms are coupled to a BEC environmentdoes not involve any energy exchange between the systemand its surroundings, the degree of non-Markovianity stillincreases when compared to the case of a single qubit. In orderto demonstrate this behavior, we first note thatN (�) ≈ 0.0055for a single qubit. Next, we calculate the degree of non-Markovianity for two qubits by taking the distance betweenthe pair of impurity atoms in Eq. (23) as D = 600 nm. Inthis case, BEC environment interacts collectively with the

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TABLE I. Degree of non-Markovianity of a single qubit fordifferent kinds of quantum processes considering the BLP measure.

Quantum process NBLP

Dephasing 0.0432Amplitude damping 0.5041BEC environment 0.0019

pair of impurity atoms and we find N (�) ≈ 0.0260. Thisresult corroborates those obtained in Ref. [13], where thedegree of non-Markovianity is studied by considering the BLPmeasure. In fact, such an outcome suggests that the amountof non-Markovianity is not only connected to the exchangeof energy between the system and environment but alsointimately related to the spectral density of the reservoir modesand the dynamics of the coherence terms.

V. DEGREE OF NON-MARKOVIANITY: BLP MEASURE

In this section, we aim to study the degree of non-Markovianity quantified by the BLP measure. Since thismeasure has been broadly studied in the literature, we repeatsome of the earlier results [2,8,13] here for the purpose ofcompleteness of our work. Given the difficulty of calculatingthe BLP measure even numerically, we avoid the analysis ofmultiqubit systems and focus on the cases of having one- andtwo-qubit systems separately.

A. Single qubit

We begin our investigation by examining a single-qubitsystem interacting with a reservoir. This problem was firstaddressed in Ref. [15] for the dephasing process, in Ref. [8]for the relaxation process, and in Refs. [2,13] for the case ofimpurity atoms coupled to a BEC environment. We maintainthe same parameters that we have used in the previous sections,i.e., s = 3, ωc = 1, and η = 2 for the super-Ohmic dephasingchannel, γ0 = 1 and λ = 0.1 for the amplitude-dampingchannel, and σ = 45 nm for impurity atoms coupled to a BECenvironment. The results for the BLP measure for a singlequbit considering different types of quantum processes aredisplayed in Table I. These results will be useful in order tounderstand the BLP measure for a pair of qubits, as discussedbelow.

B. Two qubits: Independent environments

In this section, we employ the BLP measure to study thedegree of non-Markovianity for two qubits that are indepen-dently interacting with uncorrelated environments. As alreadydiscussed in Sec. IV B the amount of non-Markovianityquantified by the LFS measure increases at least linearly dueto the additivity of the von Neumann entropy. Following thesame reasoning, our goal is to comprehend the degree ofnon-Markovianity quantified by the BLP measure for a systemof two qubits. In Table II, we present the results of our analysisfor the super-Ohmic dephasing process, the relaxation process,and for impurity atoms coupled to a BEC environment.

By comparing Tables I and II, we observe that thebehavior of the degree of non-Markovianity, for independent

TABLE II. Degree of non-Markovianity of two qubits subjected toindependent environments and different kinds of quantum processes,considering the BLP measure.

Quantum process NBLP

Dephasing 0.0432Amplitude damping 1.0476BEC environment 0.0038

environments, depends on the quantum process. For the caseof BEC environments, the BLP measure is approximately0.0019 for one qubit, and 0.0038 for two qubits subjectedto independent environments. As already shown in Ref. [13],the degree of non-Markovianity for this process is subadditive.On the other hand, for independent relaxation environments,the BLP measure turns out to be superadditive, which isconceptually different from the results obtained for two qubitssubjected to independent BEC environments. In particular,this result points out that the additivity properties of the BLPmeasure depend on the specific quantum process to whichthe qubits are subjected. Furthermore, we emphasize that theoutcomes of our analysis for the BLP measure do not generallyagree with the results obtained for the LFS measure concerningthe degree of non-Markovianity for a single qubit and a pair ofqubits. This feature is highlighted when we analyze the super-Ohmic dephasing process. If we compare the single-qubit caseto the two-qubit case subjected to independent super-Ohmicdephasing environments, our numerical analysis points outan interesting result: the BLP measure remains constant forboth cases. This result was verified by means of an exhaustivenumerical analysis where we considered 106 pairs of initialconditions encompassing both pure and mixed states.

To understand the difference between the two processes,we need to make a detailed analysis of the dynamics for oneand two qubits subjected to independent environments. Forone qubit, the pair of states that maximize Eq. (8) is givenby ρ1(0) = |+〉 and ρ2(0) = |−〉 [2], where |+〉 and |−〉 arethe eigenvectors of σx Pauli matrix. In Fig. 4 we plot thedensity matrix coherence as a function of time, when the initialstate is given by ρ1(0) and the quantum state is subjected to asuper-Ohmic dephasing process (a) and to a BEC environment(b). For both cases, we see that NBLP is exactly given by 2�,where � is the amount of recoherence.

For two qubits subjected to super-Ohmic independentenvironments, our numerical analysis shows that the pair ofstates that maximize Eq. (8) is given by ρ1(0) = |↓+〉 andρ2(0) = |↓−〉, where |↓〉 is one of the eigenvectors of theσz Pauli matrix. In such a case, the only nonzero coherenceelements of the density matrix are ρ34 and ρ43 of the reduceddensity matrix,

ρ =

⎛⎜⎜⎜⎝

ρ11 ρ12 ρ13 ρ14

ρ21 ρ22 ρ23 ρ24

ρ31 ρ32 ρ33 ρ34

ρ41 ρ42 ρ43 ρ44

⎞⎟⎟⎟⎠ . (28)

Furthermore, the dynamics of the element ρ34 is the sameas that imposed on the coherence of one qubit. This meansthat the recoherence, and consequently the trace distance,

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0.485

0.486

0.487

0.488

0.2 0.6 1 1.4

ρ 12(

t)

time (ps)

(b)

Δ

0.16

0.17

0.18

0.19

0.2

0.21

0.22

5 10 15 20 25

ρ 12(

t)

time

(a)

Δ

FIG. 4. (Color online) Dynamics of the coherence term for (a) onequbit as a function of time for the super-Ohmic dephasing processwith s = 3, wc = 1, and η = 2, and (b) the impurity atoms coupledto a BEC environment with σ = 45 nm.

are equivalent to those in the one-qubit case, when the pairof states used to calculate the BLP measure are given byρ1(0) = |↓+〉 and ρ2(0) = |↓−〉. Although the explanationfor the super-Ohmic dephasing process is clear, why doesthe degree of non-Markovianity for the BEC environmentincrease? The fact that explains such behavior is relatedto a different pair of initial conditions whose amount ofrecoherence is greater, ρ1(0) = 1√

2(|↓↑〉 − |↑↓〉) and ρ2(0) =

1√2(|↓↑〉 + |↑↓〉). Thus, the amount of recoherence given by

the element ρ23 is greater than the amount of recoherencegiven by the element ρ34 and consequently the BLP measurefor the BEC model is bigger than the BLP measure for thesuper-Ohmic dephasing model.

C. Two qubits: Common environment

We explore the situation where two qubits are collectivelyinteracting with a common environment. We once again con-sider the super-Ohmic dephasing and the relaxation processessupposing a totally correlated environment, and the case of twoqubits coupled to a BEC assuming that the distance betweenthe pair of impurity atoms is equal to D = 600 nm. In sucha configuration, the BEC interacts with the two qubits with ahigher degree of correlation and thus we simulate a commonenvironment in practice. In Table III, we display the resultsobtained for all three quantum processes.

Taking into account the values displayed in Tables I andIII, we see that non-Markovianity quantified by the BLP

TABLE III. Degree of non-Markovianity for two qubits subjectedto common environments and different kinds of quantum processes,considering the BLP measure.

Quantum process NBLP

Dephasing 0.0002Amplitude damping 7.8320BEC environment 0.0091

measure decreases for the super-Ohmic dephasing processwhen compared to the single-qubit case. On the other hand,the BLP measure increases for both the relaxation process andthe case of impurity atoms coupled to a BEC environment. Inparticular, the BLP measure is superadditive for these twoprocesses. In fact, these findings confirm that the amountof non-Markovianity is not only related to the exchangeof energy between the system and the environment butalso fundamentally connected to the spectral density of thereservoir modes and the dynamics of the coherence terms.

VI. SUMMARY

We study the degree of non-Markovianity of independentand common dephasing and relaxation processes for a singlequbit and a pair of qubits. Whereas we utilize the ampli-tude damping channel to represent dissipative processes, weconsider the super-Ohmic dephasing channel and the caseof impurity atoms interacting with a BEC environment todescribe phase-damping processes. We develop our investi-gation by analyzing two conceptually different measures ofnon-Markovianity, a recently introduced quantity called theLFS measure, and the well-known BLP measure.

Considering zero-temperature environments, we show thatno ancillary system is required to evaluate the degree ofnon-Markovianity quantified by the LFS measure since thequantity N (�) can be directly calculated by the differenceof the time derivatives of the system and the environmententropies. This simplification provides an efficient method forcalculating the degree of non-Markovianity due to the factthat the Hilbert space, where the maximization is evaluated,does not include an additional ancillary system. We provide anextensive analysis of the LFS measure for a single qubit anddetermine the optimal initial states of the system required forthe evaluation of this particular measure, as a function of theparameters of the environment.

When it comes to the degree of non-Markovianity forindependent-environmental interactions, we demonstrate thatthe LFS measure might indeed increase with the number ofqubits in the system for all considered quantum processes.In particular, we obtain a lower bound to the LFS measurefor multipartite systems, namely, N (�⊗n) � n[N (�)], whichimplies that the LFS measure is at least additive. On the otherhand, while the BLP measure is found to be superadditivefor the relaxation process, it turns out to be subadditive forthe case of impurity atoms coupled to a BEC environment.More interestingly, for the super-Ohmic dephasing process,our numerical analysis suggests that the BLP measure remainsinvariant, independent of whether we consider a systemconsisting of one or two qubits.

Furthermore, we examine the behavior of non-Markovianity for a system of two qubits interacting witha common reservoir. In this scenario, the LFS and BLPmeasures agree on the general behavior of non-Markovianity.In particular, the degree of non-Markovianity for the relaxationprocess is found to be superadditive for both of the measures.However, for a common-environmental interaction, dependingon the considered quantum process, the amount of non-Markovianity can be amplified or diminished as comparedto the case of a single qubit. In fact, although the super-Ohmic

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dephasing process and the case of impurity atoms coupled toa BEC environment do not involve any exchange of energybetween the system and its surroundings, both LFS andBLP measures indicate that the degree of non-Markovianityis fundamentally different for these two processes. That is,while the degree of non-Markovianity for two impurity atomscoupled to a BEC environment is superadditive, the amountof non-Markovianity for the super-Ohmic process decreasesvery significantly when compared to that for a single qubit.Indeed, such an outcome points to the fact that the degree ofnon-Markovianity is significantly dependent on the spectraldensity of the reservoir modes and dynamics of the coherenceterms.

ACKNOWLEDGMENTS

We thank Mauro Paternostro, Bogna Bylicka, L. G. E.Arruda, and Sabrina Maniscalco for fruitful discussions. Thiswork is supported by FAPESP and CNPq through the NationalInstitute for Science and Technology of Quantum Information(INCT-IQ) and by the Scientific and Technological ResearchCouncil of Turkey (TUBITAK) under Grant No. 111T232.

APPENDIX: COMPARISON OF N (�) AND N0(�)

We compare the results obtained for the LFS measureN (�)and its simplified version N0(�) for multipartite systems,considering independent environments. While the formerinvolves a difficult maximization over all possible initial states,the latter can be directly calculated by choosing a specificinitial state, which we take as a Greenberger-Horne-Zeilingertype of state. Due to this restriction, it is clear that N0(�)underestimates the degree of non-Markovianity, and conse-quently N0(�) � N (�). In Fig. 5(a) we plot the logarithmof the simplified LFS measure, ln[N0(�)], as a function of

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

1 2 3 4 5

ln(N

0)

number of qubits

(a)

0.5

0.6

0.7

0.8

1 2 3 4 5

N 0

number of qubits

(b)

FIG. 5. (Color online) Logarithm of non-Markovianity ln[N0(�)]for (a) the dephasing process with s = 3, wc = 1, and η = 2, andnon-Markovianity N0(�) for (b) the relaxation process with γ0 = 1and λ = 0.1, as functions of the number of qubits.

the number of qubits for the super-Ohmic dephasing processwith s = 3, wc = 1, and η = 2. It can be observed that thedegree of non-Markovianity measured by N0(�) decays expo-nentially as a function of the number of qubits. Furthermore,we see that even for very small systems (two qubits), thedegree of non-Markovianity diminishes very significantly.In Fig. 5(b) we make the same analysis for the relaxationprocess considering the parameters γ0 = 1 and λ = 0.1. Ourfindings demonstrate that, unlike in the case of super-Ohmicdephasing, the simplified LFS measure N0(�) might increasefor the relaxation process. As a result, comparing Figs. 5(a)and 5(b) to Figs. 3(a) and 3(b), we conclude that N0(�)might be a misleading quantity for determining the degreeof non-Markovianity, despite the fact that it is an easilycomputable witness of non-Markovianity.

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