PHYSICAL REVIEW A 86, 052310 (2012)

Shielding quantum discord through continuous dynamical decoupling

Felipe F. Fanchini,1 Emanuel F. de Lima,2 and Leonardo K. Castelano3

1Faculdade de Ciências, UNESP–Universidade Estadual Paulista, Bauru, São Paulo, 17033-360, Brazil
2Instituto de Geociências e Ciências Exatas, UNESP–Universidade Estadual Paulista, Rio Claro, São Paulo, 13506-900, Brazil

3Departamento de Fı́sica, Universidade Federal de São Carlos, São Carlos, São Paulo, 13565-905, Brazil
(Received 12 July 2012; published 6 November 2012)

This work investigates the use of dynamical decoupling to shield quantum discord from errors introduced by
the environment. Specifically, a two-qubit system interacting with independent baths of bosons is considered.
The initial conditions of the system were chosen as pure and mixed states, while the dynamical decouplingwas
achieved by means of continuous fields. The effect of the temperature on the shielding of quantum discord is also
studied. It is shown that although the quantum discord for particular initial states may be perfectly preserved over
some finite time window in the absence of any protective field, the effectiveness of the dynamical decoupling with
continuous fields depends essentially on the time scale required to preserve quantum discord. It is also shown
that for these particular initial states the time for which the shielding of the quantum discord becomes effective
decreases as the temperature increases.

DOI: 10.1103/PhysRevA.86.052310 PACS number(s): 03.67.Pp, 03.65.Ud, 03.65.Yz

I. INTRODUCTION

One of the major obstacles in building an efficient
quantum computer is the presence of errors introduced by
the environment. For over 20 years, different methods have
been developed aiming to protect the quantum information,
such as quantum error-correction codes [1], decoherence-
free subspaces or subsystems [2], and dynamical decoupling
[3–6]. Among these strategies, dynamical decoupling has the
advantage of not requiring the use of extra qubits to encode
the logical qubits, actively protecting quantum information
by means of a sequence of ultrafast pulses or high-frequency
fields.

Quantum entanglement has been considered one of the
main concepts concerning the measurement of quantum cor-
relations. However, several alternative measures of quantum
correlations have drawn considerable attention in the last few
years. Among these plethora of new correlation measurements
[8], quantum discord [9,10] has been one of the most
employed. Quantum discord (QD) is intimately connected
to the entanglement of formation [11–13], the conditional
entropy [13] and the mutual information. Moreover, quantum
discord is related to many important protocols such as the
distribution of entanglement [14], quantum locking [15],
entanglement irreversibility [16], and many others [17].

For some specific initial conditions [18–20], QD has the
property of not being affected by the environment during an
initial finite time. Such a finite time is known as the transition
time because it delimits a sudden transition from the classical
to the quantum decoherence regime. Recently, Luo et al.
[21] investigated the possibility of extending the transition
time by means of a dynamical decoupling (DD) technique
that uses bang-bang pulses to control the decoherence of
two qubits coupled to independent spin baths. Their results
demonstrated that the transition time can indeed be extended
by this kind of protection. In the present work, we theoretically
investigate the protection of the QD through the application of
a different type of DD technique that uses continuous fields,
which is a more realistic scheme than using instantaneous
pulses [22]. The interest of the scientific communityin this

kind of protection has been increasing, and very recently it
has been experimentally implemented to protect the quantum
information in nitrogen-vacancy systems [23,24]. It has been
shown that using this kind of protection, the coherence time
can be prolonged about 20 times [24]. Furthermore, it can be
applied to a wide variety of physical systems including trapped
atoms and ions, quantum dots, and nitrogen-vacancy centers
in diamond.

In order to probe the protection method, we solve the
Redfield master equation, taking into account the interaction
of each qubit with its own boson bath. We study the effects
caused by temperature and two kinds of initial conditions: pure
and mixed states. We show that protecting QD by means of the
DD technique can be tricky depending on the initial condition
of the system and on the time scale required to preserve
QD. In contrast to entanglement and fidelity, the protection
of quantum discord by means of DD is not efficient for some
initial states with special symmetries before the transition time.
However, if one is interested in the long-term behavior of QD,
then the DD protective scheme should be considered and the
control field should be turned on well before the transition
time. Such a result is very different from the one obtained
through the instantaneous pulse approach [21]. This difference
occurs because instantaneous pulses applied to the system do
not break the symmetry of the reduced density matrix, which is
an essential ingredient to keep QD intact before the transition
time [18–20]. On the other hand, a continuous field applied
to the system induces a dynamics that breaks the symmetry
of the reduced density matrix, thereby inhibiting the sudden
transition.

II. DYNAMICAL DECOUPLING

The Hamiltonian for a two-qubit system and the environ-
ment can be written as

H = H0 + HE + Hint, (1)

where H0 is the environment-free Hamiltonian, which includes
the qubits’ Hamiltonian and the time-dependent fields applied
to protect the quantum discord. HE is the Hamiltonian

052310-11050-2947/2012/86(5)/052310(7) ©2012 American Physical Society

http://dx.doi.org/10.1103/PhysRevA.86.052310


FANCHINI, DE LIMA, AND CASTELANO PHYSICAL REVIEW A 86, 052310 (2012)

describing the environment and the Hamiltonian Hint takes
into account the interaction between the qubits and the
environment. Here,

HE =
2∑

i=1

∑
k

ωi
ka

i†
k ai

k (2)

represents two independent baths of harmonic oscillators (one
for each qubit) with ωi

k given by the frequency of the kth normal
mode of the thermal bath and ai

k (ai†
k ) the usual annihilation

(creation) operator for the ith qubit, i.e., ai
k (ai†

k ) is the operator
that annihilates (creates) a bath quantum in the kth mode of the
ith qubit. We used units in which h̄ = 1 in Eq. (2). Henceforth,
we adopt this convention.

The interaction Hamiltonian Hint describing two qubits
coupled to their baths is written as

Hint = L1σ
(1)
z + L2σ

(2)
z , (3)

where Li , for i = 1,2, are operators that act on the environ-
mental Hilbert space. σ (i)

z is the Pauli z matrix acting on the
ith qubit. This model is called a dephasing noise and it is a
typical class of errors in quantum dots [25,26] and Josephson
junction [26]. Explicitly,Li = Bi + B

†
i , Bi = ∑

k gi
ka

i
k , where

gi
k is a complex coupling constant for the kth normal mode with

frequency dimension.
For the sake of simplicity, we assume that there exists a

global, static control field that exactly cancels the intrinsic
system Hamiltonian, i.e., H0 represents only the active
protection and it is given by a continuous applied field defined
as

H0 = nxω
(
σ (1)

x + σ (2)
x

)
, (4)

where ω = 2π/tc, tc is the period of U0(t) = exp(−iH0t),
and nx is an integer number. The general prescription for the
dynamical decoupling [3,7] is based on finding a controlled
unitary operation such that∫ tc

0
U

†
0 (t)HintU0(t)dt = 0. (5)

This is a sufficient condition to shield the effect of the
environment over the system and, as exposed in Ref. [27],
perfect shielding can be reached when nx tends to infinity.

III. MASTER EQUATION

In this work, we assume that the interaction between the
qubit and its environment is sufficiently weak that linear-
response theory holds. In the interaction picture, the master
equation is written as

dρI (t)

dt
= −

∫ t

0
dt ′TrE{[HI (t),[HI (t ′),ρE(0)ρI (t)]]}, (6)

while HI (t) reads

HI (t) = U
†
0 (t)U †

E(t)HintUE(t)U0(t), (7)

where

UE(t) = exp

(
−i

HEt

h̄

)
(8)

and

U0(t) = exp

(
−i

H0t

h̄

)
. (9)

In Eq. (6), ρE(0) is the initial density matrix of the thermal
bath, which is chosen as the Feynman-Vernon state,

ρE(0) = 1

Z
exp(−βHE), (10)

where Z is the partition function given by

Z = TrE[exp(−βHE)], (11)

and β = 1/(kBT ), where kB is the Boltzmann constant and T is
the temperature of the environment. Defining U

†
E(t)LiUE(t) ≡

L̃i(t) and �i(t) ≡ U
†
0 (t)σ (i)

z U0(t), we can write the HI Hamil-
tonian in the interaction picture as follows:

HI (t) = �1(t)L̃1(t) + �2(t)L̃2(t). (12)

Thus, substituting Eqs. (10) and (12) into the master equa-
tion (6), we obtain

dρI (t)

dt
=

2∑
i=1

∫ t

0
dt ′Di(t,t

′)[�i(t),ρI (t)�i(t
′)]

+
∫ t

0
dt ′D∗

i (t,t ′)[�i(t
′)ρI (t),�i(t)], (13)

where Di(t,t ′) = Ii(t − t ′) + Ti(t − t ′) with

Ii(t − t ′) = TrE{B̃i(t)ρ̃E(t)B̃†
i (t ′)}

=
∑

k

∣∣gi
k

∣∣2
ni

k exp
[−iωi

k(t − t ′)
]

(14)

and

Ti(t − t ′) = TrE{B̃†
i (t)ρ̃E(t)B̃i(t

′)}
=

∑
k

∣∣gi
k

∣∣2(
ni

k + 1
)

exp
[−iωi

k(t − t ′)
]
, (15)

with B̃i(t) = U
†
E(t)BiUE(t), ρ̃E(t) = U

†
E(t)ρE(0)UE(t), and

ni
k the average occupation number of the kth mode of the ith

qubit, ni
k = 1/[exp(βωi

k − 1)]. Since each qubit is subjected
to identical environments, the i index of gi

k , ni
k , and ωi

k can
be omitted. Thus, defining the spectral function as J (ω) ≡∑

k |gk|2δ(ω − ωk), we can replace the above summations by
the following integrals:

I(t) =
∫ ∞

0
dωJ (ω)n(ω) exp(−iωt) (16)

and

T (t) =
∫ ∞

0
dωJ (ω) exp(iωt)[n(ω) + 1]. (17)

We assume that the environment has an Ohmic spectral density
J (ω) = ηω exp(−ω/ωc), where ωc is a cutoff frequency and
η is a damping constant. Therefore, we can explicitly evaluate
the above integrals, which yield

D(t,t ′) = ηω2
c

[1 + iωc(t − t ′)]2

+ 2η

β2
Re{
(1st)(1 + 1/(βωc) − i(t − t ′)/β)}, (18)

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SHIELDING QUANTUM DISCORD THROUGH CONTINUOUS . . . PHYSICAL REVIEW A 86, 052310 (2012)

where 
(1st) is the first polygamma function. The first term
of the right-hand side of Eq. (18) represents the vacuum,
while the second term accounts for the effects of finite
temperature. In particular, for T = 0 we have D(t,t ′) =
ηω2

c/[1 + iωc(t − t ′)]2.

IV. FIDELITY, CONCURRENCE, AND
QUANTUM DISCORD

In this section, we present the definitions of three distinct
measures for quantum systems.

A. Superfidelity

To calculate the dissipative dynamics fidelity, we use an
alternative measure known in the literature as superfidelity
[28]. We calculate the superfidelity between two distinct
(possibly mixed) density matrix states ρ(t) and σ (t). Here,
ρ(t) represents the open quantum system dynamics, where
the two qubits interact with the environment, and σ (t) gives
the ideal dynamics, whose interaction with the environment is
disabled. The superfidelity is a function of linear entropy and
the Hilbert-Schmidt inner product between the given states.
Remarkably, the superfidelity is jointly concave and satisfies
all Jozsa’s axioms. Explicitly it is given by

F (ρ,σ ) = Tr(ρσ ) +
√

1 − Tr(ρ2)
√

1 − Tr(σ 2), (19)

and it is valid for pure or mixed states.

B. Concurrence

To measure the entanglement between the two qubits, we
use the well-known concurrence [29]. It is defined as the
maximum between zero and �(t),

�(t) = λ1 − λ2 − λ3 − λ4, (20)

where λ1 � λ2 � λ3 � λ4 are the square roots of the eigen-
values of the matrix ρ(t)σy ⊗ σyρ

∗(t)σy ⊗ σy , and ρ∗(t) is the
complex conjugation of ρ(t) and σy is the y Pauli matrix.

C. Quantum discord

The quantum discord was independently defined by
Handerson and Vedral [9] and Ollivier and Zurek [10] about
ten years ago. It was proposed as a measure of the quantum
correlations and, given two subsystems A and B, it is defined
as

δ←
AB = IAB − J←

AB (21)

where IAB = SA + SB − SAB is the mutual information, with
SA = −Tr(ρA log2 ρA) given by the von Neumann entropy of
the subsystem A, and similarly for B and AB. J←

AB is the
classical correlation, which can be explicitly written as

J←
AB = max

{�k}

[
SA −

∑
k

pkS(ρA|k)

]
, (22)

where ρA|k = TrB (�kρAB�k) /TrAB (�kρAB�k) is the re-
duced state of A after obtaining the outcome k in B. Here,
{�k} is a complete set of positive operator-valued measures
that results in the outcome k with probability pk . The quantum

discord measures the amount of the mutual information that is
not locally accessible [13,30] and generally is not symmetric,
i.e., δ←

AB �= δ←
BA.

V. DISSIPATIVE DYNAMICS

In order to illustrate the singular features involving the
protection of the quantum discord, we study two kinds of initial
conditions, pure and mixed states. Additionally, we consider
the effects of finite temperatures on the quantum discord. The
quantum discord dynamics is compared to the concurrence and
the fidelity of quantum systems. In the following calculations,
we have set the parameters of Eq. (18) to η = 1/16 and
ωc = 2π .

A. Pure-state initial condition

In order to analyze the dissipative dynamics for pure states,
we choose for the initial condition a maximally quantum
correlated state ρ0

AB(0) = 1√
2
(|00〉〈00| + |11〉〈11|). Figure 1

shows the concurrence (a) and the fidelity (b) dynamics when
the control fields are turned off (nx = 0) and at (nx = 2,3,4)
for a zero-temperature environment, T = 0 K. The solid line
corresponds to nx = 0 in Eq. (4) and the dotted, double-dot-
dashed, and dot-dashed lines give the concurrence and the
fidelity when the control fields are turned on at t = 0 with
nx = 2,3,4, respectively. When the protective field amplitude
increases both the fidelity and the concurrence increase [31].

In Fig. 2, the quantum discord is plotted considering the
same parameters used for the fidelity and concurrence. The
solid line corresponds to nx = 0 in Eq. (4) and the dashed,
dotted, and dot-dashed lines give the quantum discord when
the control fields are turned on at t = 0 with nx = 2,3,4,

 0.3

 0.4

 0.5

 0.6

 0.7

 0.8

 0.9

 1

 0  0.2  0.4  0.6  0.8  1

C
on

cu
rr

en
ce

t / τ

(a)

 0.65

 0.7

 0.75

 0.8

 0.85

 0.9

 0.95

 1

 0  0.2  0.4  0.6  0.8  1

F
id

el
ity

t / τ

(b)

FIG. 1. (Color online) Concurrence and fidelity as a function of
time considering a pure initial state ρ0

AB (0) and T = 0 K. (a) The solid
(black) line gives the concurrence when the control field is turned
off and the dotted (red), double-dot-dashed (blue), and dot-dashed
(green) lines give the concurrence when the control fields are turned
on at t = 0, using nx = 2,3,4, respectively. (b) The solid (black)
line gives the fidelity when the control field is turned off and the
dotted (red), double-dot-dashed (blue), and dot-dashed (green) lines
give the fidelity when the control fields are turned on at t = 0, using
nx = 2,3,4, respectively.

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FANCHINI, DE LIMA, AND CASTELANO PHYSICAL REVIEW A 86, 052310 (2012)

 0

 0.1

 0.2

 0.3

 0.4

 0.5

 0.6

 0.7

 0.8

 0.9

 1

 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1

Q
D

t / τ

FIG. 2. (Color online) Quantum discord as a function of time
considering a pure initial state ρ0

AB (0) and T = 0 K. The solid (black)
line gives the quantum discord when the control field is turned off and
the dotted (red), double-dot-dashed (blue), and dot-dashed (green)
lines give the quantum discord when the control fields are turned on
at t = 0, using nx = 2,3,4, respectively.

respectively. For the initial pure-state case, the concurrence,
fidelity, and quantum discord present a similar behavior, i.e., as
the amplitude of the protective field increases, the protection
of the quantum information is enhanced [31]. It is important
to emphasize that, despite the restrictive initial state, the
protection of fidelity, concurrence, and quantum discord is
reached for all pure initial states qualitatively in the same way,
that is, the higher the amplitude of the protective field, the
better the protection. As shown in the following, the quantum
discord does not necessarily behave in the same qualitative way
as the fidelity and concurrence with respect to the application
of the protective field for mixed states.

B. Mixed-state initial condition

Consider as initial condition a mixed state given by
ρ1

AB(0) = 0.5(|ψ1〉〈ψ1| + |ψ2〉〈ψ2|), where |ψ1〉 = 1√
4
(|00〉 +

|11〉 + |01〉 + |10〉) and |ψ2〉 = 1√
2
(|00〉 − |11〉). Figure 3

shows the dynamics of QD as function of time, considering
nx = 0,2,3,4. One can see that, as in the pure-initial-state
case, the control field is efficient in protecting QD. In brief,
the protection protocol has the same efficiency for the initial
state ρ1

AB(0) as the one found for pure initial states. This
characteristic is also observed for the great majority of initial
mixed states.

On the other hand, the scenario changes if the initial condi-
tion is given by some special states, for instance, ρ2

AB(0) =
0.8|�〉〈�| + 0.2|
〉〈
|, where |�〉 = 1√

2
(|00〉 + |11〉) and

|
〉 = 1√
2
(|01〉 + |10〉). In Fig. 4, we show the concurrence

and the fidelity dynamics for some frequencies of the protective
field. There is no qualitative difference involving the protection
of the concurrence and the fidelity: the greater the amplitude of
the protective field, the better the protection. Nevertheless, we
observe a different behavior when dealing with the quantum
discord.

 0

 0.05

 0.1

 0.15

 0.2

 0.25

 0.3

 0.35

 0.4

 0.45

 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1

Q
D

t / τ

FIG. 3. (Color online) Quantum discord as a function of time
considering a mixed initial state ρ1

AB (0) and T = 0 K. The solid
(black) line gives the quantum discord when the control field is turned
off and the dotted (red), double-dot-dashed (blue), and dot-dashed
(green) lines give the quantum discord when the control fields are
turned on at t = 0, using nx = 2,3,4, respectively.

In Fig. 5, an abrupt decrease can be seen, the sudden
transition, in the quantum discord around t = 0.4τ for the
nonprotected case. Up to this sudden transition, the results
obtained with the protection turned on at t = 0 are less
efficient in maintaining the quantum discord than with the
protection turned off. Actually, for t � 0.4τ , the protective
dynamics is equivalent to the nonprotected dynamics only in
the limit that nx goes to infinity. However, there is a finite
time when the protected quantum discord surpass that of the

 0.1

 0.2

 0.3

 0.4

 0.5

 0.6

 0  0.2  0.4  0.6  0.8  1

C
on

cu
rr

en
ce

t / τ

(a)

 0.91

 0.92

 0.93

 0.94

 0.95

 0.96

 0.97

 0.98

 0.99

 1

 0  0.2  0.4  0.6  0.8  1

F
id

el
ity

t / τ

(b)

FIG. 4. (Color online) Concurrence and fidelity as functions of
time considering a mixed initial state ρ2

AB (0) and T = 0 K. (a) The
solid (black) line gives the concurrence when the control field is turned
off and the dotted (red), double-dot-dashed (blue), and dot-dashed
(green) lines give the concurrence when the control fields are turned
on at t = 0, using nx = 2,3,4, respectively. (b) The solid (black)
line gives the fidelity when the control field is turned off and the
dotted (red), double-dot-dashed (blue), and dot-dashed (green) lines
give the fidelity when the control fields are turned on at t = 0, using
nx = 2,3,4, respectively.

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SHIELDING QUANTUM DISCORD THROUGH CONTINUOUS . . . PHYSICAL REVIEW A 86, 052310 (2012)

 0.1

 0.15

 0.2

 0.25

 0.3

 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1

Q
D

t / τ

 0.2

 0.4

 0.6

 0.8

 2  3  4  5  6  7  8
nx

t e
 / 

τ

FIG. 5. (Color online) Quantum discord as a function of time
considering a mixed initial state ρ2

AB (0) and T = 0 K. The solid
(black) line gives the quantum discord when the control field is
turned off and the closed points (magenta), dotted (red), double dot-
dashed (blue), and dot-dashed (green) lines give the quantum discord
when the control fields are turned on at t = 0, using nx = 1,2,3,4,
respectively. In the inset we plot the time for efficiency as a function
of nx .

nonprotected case, given by the crossing of the dotted lines
with the solid line in Fig. 5. The effectiveness of the protective
field occurs only above these times te(nx) which are greater
than the time of sudden transition of the quantum discord,
te(nx) � 0.4τ . Thus, for this particular initial condition, the
time scale needed to protect QD should be the determinant in
deciding for the use of the DD protective scheme. This peculiar
aspect involving the protection of the quantum discord occurs
because the plateau given in the free evolution exist for some
special symmetries of the initial state [18,19]. The dynamical
decoupling strategy is an active method of protection and
needs to continuously modify the density matrix during the
dynamics. On the other hand, the density matrix reaches its
original and special symmetry just at some periodic times.
These unavoidable density matrix modifications involving the
protection by means of DD decrease the quantum discord,
which reaches a lower value than the initial one. However,
when the quantum discord dynamics reaches the sudden
transition time, it starts to decrease, while the protective
scheme becomes efficient. It is important to emphasize the
difference between our results and the ones obtained by the
DD technique that employs instantaneous pulses [21]. If each
pulse is considered to be applied instantaneously, the changes
imposed on the system by this kind of pulse never alter the X
structure of the initial reduced density matrix. Therefore, the
sharp transition in the quantum discord is preserved, whereas
the continuous decoupling technique presented here leads to a
more gradual decay.

Based on the above results, one could be led to conclude
that if the initial state has a special symmetry such that the QD
presents a plateau for the free evolution of the system, then
the best strategy would be to turn on the protective field only
after the sudden transition time. However, as we show in the
following, this may not be the best strategy. Figure 6 presents
the evolution of the QD when the protective field is turned on at

Q
D

Q
D

t/τ t/τ 

FIG. 6. (Color online) Quantum discord as a function of time
considering a mixed initial state ρ2

AB (0) and T = 0 K. The solid
(black) lines give the quantum discord when the control field is
turned on at t = 0.4τ , while the dashed (red) curves correspond to
the QD when the control field is turned on at t = 0. We probe both
situations by using nx = 1 in (a), nx = 2 in (b), nx = 3 in (c), and nx =
4 in (d).

different moments. The dashed curves probe the situation with
the protective field turned on at t = 0, while the solid curves
probe the dynamics for a control field turned on at t = 0.4τ .
It is clearly seen that if one intends to enhance the protection
of QD in the long term, one should turn on the protective field
at t = 0 and not wait until the sudden transition.

It is interesting to analyze the case of protection for
completely unknown states, where an initial state is chosen
randomly. In this case, the density matrix will most likely never
reach the necessary structure to result in a frozen discord [20].
Thus, considering the long-term evolution, the best strategy is
to turn on the protection at t = 0. Nevertheless, it is important
to note that a state that presents a plateau in the QD is a
mixture of Bell states. These states are a very important class of
states and are used in many problems in quantum information
theory [32]. Furthermore, experimental realization of these
states has been observed in the context of quantum optics
experiments [33] and nuclear magnetic resonance [34] as well
as in solid state physics [35].

We now consider how a finite-temperature environment
influences the scenario of QD protection. The first relevant ob-
servation involving finite-temperature environments is that the
interaction with the environment becomes more deleterious.
In Fig. 7, we show the temperature effects on the protection
method. We take nx = 4 and present the time for effectiveness
of the protective field as a function of the temperature. As
one can observe, te(nx) decreases when the temperature is
increased. Therefore, for higher temperatures, the protective
scheme becomes efficient sooner. This characteristic occurs
because when the temperature increases, the density matrix
coherence decays faster and, consequently, the time of sudden
transition is reduced. Thus, the plateau of quantum discord
is smaller and the effectiveness of the protective scheme is
achieved sooner. We note that for T = 5 K the time for
efficiency approaches 0.05τ .

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FANCHINI, DE LIMA, AND CASTELANO PHYSICAL REVIEW A 86, 052310 (2012)

 0

 0.1

 0.2

 0.3

 0.4

 0  1  2  3  4  5

t e
 / 

τ

Temperature (K)

FIG. 7. (Color online) The time for efficiency as a function of the
temperature for nx = 4.

VI. SUMMARY

In summary, we have investigated the protection of QD
from errors introduced by the environment by means of

DD with continuous fields. We have compared two kinds of
initial condition, pure and mixed states, and also included
finite-temperature effects. It has been shown that, in contrast
with the fidelity and concurrence, the effectiveness of the
protective scheme depends on the initial state and on the
time scale required to protect QD. For initial states with a
particular symmetry such that the QD presents a plateau in
the free evolution of the system, one should not turn on
the protective field if the time scale of interest is below the
transition time. However, if one is interested in the long-term
evolution of QD, the protective field should be turned on well
before the transition time. Finally, we have verified that for
these particular initial states, the greater the temperature, the
faster is efficiency of the protective method achieved, which
evidences the relevance of the temperature in the protection
of QD.

ACKNOWLEDGMENTS

This work is supported by FAPESP and CNPq through the
National Institute for Science and Technology of Quantum
Information (INCT-IQ).

[1] P. W. Shor, Phys. Rev. A 52, R2493 (1995); A. M. Steane, Phys.
Rev. Lett. 77, 793 (1996); A. R. Calderbank and P. W. Shor, Phys.
Rev. A 54, 1098 (1996); A. M. Steane, Proc. R. Soc. London,
Ser. A 452, 2551 (1996); M. S. Byrd, L.-A. Wu, and D. A. Lidar,
J. Mod. Opt. 10, 2449 (2004), and references cited therein.

[2] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997);
D. A. Lidar, I. L. Chuang, and K. B. Whaley, ibid. 81, 2594
(1998); E. Knill, R. Laflamme, and L. Viola, ibid. 84, 2525
(2000); P. E. M. F. Mendonça, M. A. Marchiolli, and R. d. J.
Napolitano, J. Phys. A 38, L95 (2005), and references cited
therein.

[3] L. Viola and S. Lloyd, Phys. Rev. A 58, 2733 (1998).
[4] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999).
[5] L. Viola, S. Lloyd, and E. Knill, Phys. Rev. Lett. 83, 4888

(1999).
[6] L. Viola, J. Mod. Opt. 10, 2357 (2004), and references cited

therein.
[7] P. Facchi, S. Tasaki, S. Pascazio, H. Nakazato, A. Tokuse, and

D. A. Lidar, Phys. Rev. A 71, 022302 (2005).
[8] M. Horodecki, K. Horodecki, P. Horodecki, R. Horodecki,

J. Oppenheim, A. Sen De, and U. Sen, Phys. Rev. Lett. 90,
100402 (2003); M. Horodecki, P. Horodecki, R. Horodecki,
J. Oppenheim, A. Sen, U. Sen, and B. Synak-Radtke, Phys.
Rev. A 71, 062307 (2005); S. Luo, ibid. 77, 022301 (2008);
J. Maziero, L. C. Celeri, and R. M. Serra, arXiv:1004.2082;
K. Modi, T. Paterek, W. Son, V. Vedral, and M. Williamson,
Phys. Rev. Lett. 104, 080501 (2010).

[9] L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001).
[10] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001).
[11] M. Koashi and A. Winter, Phys. Rev. A 69, 022309 (2004).
[12] F. F. Fanchini, M. F. Cornelio, M. C. de Oliveira, and A. O.

Caldeira, Phys. Rev. A 84, 012313 (2011).
[13] F. F. Fanchini, L. K. Castelano, M. F. Cornelio, and M. C. de

Oliveira, New J. Phys. 14, 013027 (2012).

[14] V. Madhok and A. Datta, Phys. Rev. A 83, 032323 (2011);
D. Cavalcanti, L. Aolita, S. Boixo, K. Modi, M. Piani, and
A. Winter, ibid. 83, 032324 (2011); A. Streltsov, H. Kamper-
mann, and D. Bruss, Phys. Rev. Lett. 108, 250501 (2012); T. K.
Chuan, J. Maillard, K. Modi, T. Paterek, M. Paternostro, and
M. Piani, ibid. 109, 070501 (2012).

[15] S. Boixo, L. Aolita, D. Cavalcanti, K. Modi, and A. Winter, Int.
J. Quantum Inf. 9, 1643 (2011).

[16] M. F. Cornélio, M. C. de Oliveira, and F. F. Fanchini, Phys. Rev.
Lett. 107, 020502 (2011).

[17] A. Streltsov, H. Kampermann, and D. Bruss, Phys. Rev. Lett.
106, 160401 (2011); T. Werlang, C. Trippe, G. A. P. Ribeiro,
and G. Rigolin, ibid. 105, 095702 (2010).

[18] L. Mazzola, J. Piilo, and S. Maniscalco, Phys. Rev. Lett. 104,
200401 (2010).

[19] J. Maziero, L. C. Celeri, R. M. Serra, and V. Vedral, Phys. Rev.
A 80, 044102 (2009).

[20] B. You and L.-X. Cen, Phys. Rev. A 86, 012102 (2012).
[21] Da-Wei Luo, Hai-Qing Lin, Jing-Bo Xu, and Dao-Xin Yao, Phys.

Rev. A 84, 062112 (2011).
[22] F. F. Fanchini, J. E. M. Hornos, and R. d. J. Napolitano, Phys.

Rev. A 75, 022329 (2007); F. F. Fanchini and R. d. J. Napolitano,
ibid. 76, 062306 (2007).

[23] X. Xu et al., Phys. Rev. Lett. 109, 070502 (2012).
[24] J.-M. Cai et al., arXiv:1111.0930.
[25] F. F. Fanchini, L. K. Castelano, and A. O. Caldeira, New J. Phys.

12, 073009 (2010).
[26] M. Thorwart and P. Hanggi, Phys. Rev. A 65, 012309 (2001).
[27] F. F. Fanchini, J. E. M. Hornos, and R. d. J. Napolitano, Phys.

Rev. A 75, 022329 (2007).
[28] P. E. M. F. Mendonca, R. J. Napolitano, M. A. Marchiolli,

C. J. Foster, and Y. C. Liang, Phys. Rev A. 78, 052330 (2008);
J. A. Miszczak, Z. Puchala, P. Horodecki, A. Uhlmann, and
K. Zyczkowski, Quantum Inf. Comput. 9, 0103 (2009).

052310-6

http://dx.doi.org/10.1103/PhysRevA.52.R2493
http://dx.doi.org/10.1103/PhysRevLett.77.793
http://dx.doi.org/10.1103/PhysRevLett.77.793
http://dx.doi.org/10.1103/PhysRevA.54.1098
http://dx.doi.org/10.1103/PhysRevA.54.1098
http://dx.doi.org/10.1098/rspa.1996.0136
http://dx.doi.org/10.1098/rspa.1996.0136
http://dx.doi.org/10.1080/09500340408231803
http://dx.doi.org/10.1103/PhysRevLett.79.3306
http://dx.doi.org/10.1103/PhysRevLett.81.2594
http://dx.doi.org/10.1103/PhysRevLett.81.2594
http://dx.doi.org/10.1103/PhysRevLett.84.2525
http://dx.doi.org/10.1103/PhysRevLett.84.2525
http://dx.doi.org/10.1088/0305-4470/38/6/L04
http://dx.doi.org/10.1088/0305-4470/38/6/L04
http://dx.doi.org/10.1103/PhysRevA.58.2733
http://dx.doi.org/10.1103/PhysRevLett.82.2417
http://dx.doi.org/10.1103/PhysRevLett.83.4888
http://dx.doi.org/10.1103/PhysRevLett.83.4888
http://dx.doi.org/10.1080/09500340408231795
http://dx.doi.org/10.1080/09500340408231795
http://dx.doi.org/10.1103/PhysRevA.71.022302
http://dx.doi.org/10.1103/PhysRevLett.90.100402
http://dx.doi.org/10.1103/PhysRevLett.90.100402
http://dx.doi.org/10.1103/PhysRevA.71.062307
http://dx.doi.org/10.1103/PhysRevA.71.062307
http://dx.doi.org/10.1103/PhysRevA.77.022301
http://arXiv.org/abs/1004.2082
http://dx.doi.org/10.1103/PhysRevLett.104.080501
http://dx.doi.org/10.1088/0305-4470/34/35/315
http://dx.doi.org/10.1103/PhysRevLett.88.017901
http://dx.doi.org/10.1103/PhysRevA.69.022309
http://dx.doi.org/10.1103/PhysRevA.84.012313
http://dx.doi.org/10.1088/1367-2630/14/1/013027
http://dx.doi.org/10.1103/PhysRevA.83.032323
http://dx.doi.org/10.1103/PhysRevA.83.032324
http://dx.doi.org/10.1103/PhysRevLett.108.250501
http://dx.doi.org/10.1103/PhysRevLett.109.070501
http://dx.doi.org/10.1142/S0219749911008301
http://dx.doi.org/10.1142/S0219749911008301
http://dx.doi.org/10.1103/PhysRevLett.107.020502
http://dx.doi.org/10.1103/PhysRevLett.107.020502
http://dx.doi.org/10.1103/PhysRevLett.106.160401
http://dx.doi.org/10.1103/PhysRevLett.106.160401
http://dx.doi.org/10.1103/PhysRevLett.105.095702
http://dx.doi.org/10.1103/PhysRevLett.104.200401
http://dx.doi.org/10.1103/PhysRevLett.104.200401
http://dx.doi.org/10.1103/PhysRevA.80.044102
http://dx.doi.org/10.1103/PhysRevA.80.044102
http://dx.doi.org/10.1103/PhysRevA.86.012102
http://dx.doi.org/10.1103/PhysRevA.84.062112
http://dx.doi.org/10.1103/PhysRevA.84.062112
http://dx.doi.org/10.1103/PhysRevA.75.022329
http://dx.doi.org/10.1103/PhysRevA.75.022329
http://dx.doi.org/10.1103/PhysRevA.76.062306
http://dx.doi.org/10.1103/PhysRevLett.109.070502
http://arXiv.org/abs/1111.0930
http://dx.doi.org/10.1088/1367-2630/12/7/073009
http://dx.doi.org/10.1088/1367-2630/12/7/073009
http://dx.doi.org/10.1103/PhysRevA.65.012309
http://dx.doi.org/10.1103/PhysRevA.75.022329
http://dx.doi.org/10.1103/PhysRevA.75.022329
http://dx.doi.org/10.1103/PhysRevA.78.052330


SHIELDING QUANTUM DISCORD THROUGH CONTINUOUS . . . PHYSICAL REVIEW A 86, 052310 (2012)

[29] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[30] W. H. Zurek, Phys. Rev. A 67, 012320 (2003).
[31] We are not considering the anti-Zeno effect, where the

protective field could decrease both the fidelity and the
concurrence.

[32] M. Zwerger, W. Dur, and H. J. Briegel, Phys. Rev. A 85, 062326
(2012); A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and
V. Scarani, Phys. Rev. Lett. 98, 230501 (2007); C. H. Bennett,

G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters,
ibid. 70, 1895 (1993).

[33] J.-S. Xu et al., Nat. Commun. 1, 7 (2010); M. F. Cornelio et al.,
arXiv:1203.5068.

[34] R. Auccaise, L. C. Celeri, D. O. Soares-Pinto, E. R. de Azevedo,
J. Maziero, A. M. Souza, T. J. Bonagamba, R. S. Sarthour, I. S.
Oliveira, and R. M. Serra, Phys. Rev. Lett. 107, 140403 (2011).

[35] X. Rong et al., Phys. Rev. B 86, 104425 (2012).

052310-7

http://dx.doi.org/10.1103/PhysRevLett.80.2245
http://dx.doi.org/10.1103/PhysRevA.67.012320
http://dx.doi.org/10.1103/PhysRevA.85.062326
http://dx.doi.org/10.1103/PhysRevA.85.062326
http://dx.doi.org/10.1103/PhysRevLett.98.230501
http://dx.doi.org/10.1103/PhysRevLett.70.1895
http://arXiv.org/abs/1203.5068
http://dx.doi.org/10.1103/PhysRevLett.107.140403
http://dx.doi.org/10.1103/PhysRevB.86.104425