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Statistical portrait of the entanglement decay of twoqubit memories
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Physical Review A · October 2012
DOI: 10.1103/PhysRevA.86.042325 · Source: arXiv
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a r X i v : 1 2 1 0 . 5 6 4 5 v 1 [ q u a n t  p h ] 2 0 O c t 2 0 1 2
A statistical portrait of the entanglement decay of twoqubit memories
Karen M. FonsecaRomero,
1,
∗
Juli´an Mart´ınezRinc´on,
1,2
and Carlos Viviescas
1
1
Universidad Nacional de Colombia  Bogot´ a, Facultad de Ciencias,Departamento de F´ısica, Carrera 30 Calle 4503, C.P. 111321, Bogot´ a, Colombia
2
Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
(Dated: December 15, 2013)We present a novel approach to the study of entanglement decay, which focuses on collectiveproperties. As an example, we investigate the entanglement decay of a twoqubit system, producedby local identical reservoirs acting on the qubits, for three experimentally and theoretically relevantcases. We study the probability distributions of disentanglement times, a quantity independentof the measure used to quantify entanglement, and the timedependent probability distribution of concurrence. Analytical results are obtained for initially uniformly distributed pure states. Thecalculation of these probability distributions gives a complete insight on how diﬀerent decoherencechannels aﬀect the entanglement initially contained in the set of twoqubit pure states. Numericalresults are reported for randomly distributed initial mixed states. Although the paper focusesin Markovian noisy channels, we show that our results also describe nonautonomous and nonMarkovian channels.
PACS numbers: 03.65.Ud,03.65.Yz,03.67.Mn
I. INTRODUCTION.
Entanglement is a key resource for quantum information processing and communication, which can be manipulated, broadcasted, controlled and distributed [1].A better understanding of the eﬀects of decoherence onentanglement could accelerate the development of technological applications of quantum information protocols.Its study under realistic, dissipative conditions, is of fundamental importance to assess the resilience of quantuminformation processing. The dynamics of quantum coherence and entanglement can be strikingly diﬀerent. Under the inﬂuence of a noisy environment, coherences usually decay asymptotically in time, while entanglementcan do so in a ﬁnite time [2, 3]. Extensive work [4–11],
including several experimental demonstrations [12–14],
has been devoted to this remarkable phenomenon, sometimes called entanglement sudden death (ESD). A complex, yet partial picture of the entanglement dynamics,has emerged. Recently, a complete characterization of the dynamics of a twoqubit system in which only oneof the qubits is coupled to a noisy channel [15] has beenpresented and general approaches, based on geometricalarguments [16] and quantum trajectories [17], have been
proposed.In this work, adopting a broader perspective, we rendera comprehensive description of the entanglement dynamics in twoqubit systems when corrupted by environmentinduced decoherence. Our approach relies on a rathernovel tool in the area: the statistical distributions of concurrence and of the ESD times. Obtaining analytic expressions for them is far from being a symple task. Infact, previous analytic work was restricted to averages
∗
kmfonsecar@unal.edu.co
[18] and often relied on approximations [19]. Other re
ported explorations of entanglement evolution in bipartite [2] and multipartite systems [20], similar in spirit
to our approach, are both numerical. Our analytic formulation, employed before in a kinematical context [21],allows us to depict a highly detailed image of the entanglement time evolution.This paper is organized as follows. The dynamical evolution of a twoqubit system coupled to local channelsis introduced in section II, where the probability densities of the disentanglement times and of concurrenceare deﬁned. We consider three models of decoherencefor a single qubit and assume identical environments foreach qubit. In the next four sections the results for theprobability densities of the disentanglement times and of concurrence are evaluated for the chosen environments,assuming initial pure states uniformly distributed. Theresults for mixed states, a single noisy channel, and nonautonomous systems are brieﬂy presented in sections VII,VIII and IX, respectively. Section X is devoted to the
demonstration that, although we concentrate on Markovian processes with constant decay rates, our results alsoapply to nonMarkovian processes. Some conclusions aredrawn in the last section of this paper.
II. DISENTANGLEMENT PROBABILITYDISTRIBUTIONS
We investigate the
global
entanglement dynamics of twoqubit systems coupled to local reservoirs (channels), a situation likely to hold for spatially separatedqubits. The qubits are assumed to be quantum memories; they do not follow independent unitary dynamics nor do they interact with each other. Whenever the initial systemenvironment state is separablewith vanishing quantum discord [22, 23], the dynam
2ical evolution of the state of the twoqubit system isgiven by a completely positive map [24, 25]. Then,
ρ
(
t
), the system state at the physical time
t
, is givenby
ρ
(
t
) =
ij
E
(1)
i
⊗
E
(2)
j
ρ
(0)
E
(1)
†
i
⊗
E
(2)
†
j
=Λ
t
(
ρ
(0))
,
where the timedependent Kraus operators
E
(
a
)
i
=
E
(
a
)
i
(
t
), acting on the
a
th qubit, satisfy thetracepreserving condition
i
E
(
a
)
†
i
E
(
a
)
i
=
I
(
a
)
,
a
= 1
,
2.These operators can be experimentally determined by aquantum process tomography [26].We chose pairs of identical environments to act on thequbits, and regard the experimentally relevant cases of depolarizing (D), amplitudedamping (AD), and phasedamping (PD) channels. These generally nonMarkovianchannels are valid for ﬁnite or inﬁnite environments andweak or strong coupling [27]. The Kraus operators forthese noisy environments, in terms of the (reparameterized) time
q
, are (0
≤
q
≤
1):
E
0
(
q
) =
1
−
3
q/
4
I
and
E
i
(
q
) =
q/
4
σ
i
,
i
= 1
,
2
,
3
,
for case D;
E
0
=

0
0

+
√
1
−
q

1
1

and
E
1
=
√
q

0
1

for case AD;and
E
0
=
√
1
−
q
I
,
E
1
=
√
q

0
0

, and
E
2
=
√
q

1
1

for case PD. Here
σ
1
,σ
2
, and
σ
3
denote, as usual, thethree Pauli matrices. Unless otherwise stated, we focuson the Markovian scenario, with constant decay rates,when
q
(
t
) = 1
−
exp(
−
γt
),
γ
being a positive constant.The motivation behind our selection of channels lies intheir qualitative diﬀerent longtime behavior,
q
→
1, determining the dynamical evolution of entanglement on agross level [16]. All states will suﬀer ESD if the asymptotic state belongs to the interior of the set of separablestates
S
(case D). Some states will separate asymptotically if the asymptotic state belongs to
∂
S
, the borderbetween
S
and
E
, the set of entangled states (case AD).The case PD displays a set of stationary separable states,diagonal in the standard basis, which includes 4 purestates in
∂
S
.Rather than compute the concurrence (or any otherentanglement measure) for speciﬁc initial conditions, wepursue a statistical approach to entanglement dynamicsin open systems. Although randomly chosen normalizedpure states

ψ
=
ψ
00

00
+
ψ
01

01
+
ψ
10

10
+
ψ
11

11
are used for our calculations, other choices might besensible. We choose pure states because they generally oﬀer higher quantum correlations than mixed states,a key point for quantum information protocols. Moreover, they are a convenient idealization of the almostpure states that can be produced under experimentalconditions. If minimal prior knowledge about the stateof the system is assumed [28], or invariance under unitary transformations is required, it becomes natural touse the uniform (Haar) measure for all possible purestates of the system. Hence, the probability to ﬁnd astate in a small volume
d
2
ψ
=
ij
dψ
ij
dψ
∗
ij
around
ψ
is
p
(
ψ
)
d
2
ψ
= Γ(4)
δ
(1
−
ψ

ψ
)
/π
4
d
2
ψ
, where Γ(
x
) denotes,as usual, the Gamma function.We characterize the disentanglement process throughthe reparameterized separation time probability density,
p
(
q
S
) =
d
2
ψδ
(
q
S
−
q
S
(
ψ
))
p
(
ψ
)
,
(1)where
q
S
(
ψ
) is the ESD time, i.e., the time at which
ρ
(
q
) = Λ
q
(

ψ
ψ

), becomes separable. This distribution is independent of the measure used to quantify theentanglement of the system. However,
p
(
q
S
) may overestimate entanglement if many states sustain small valuesof entanglement for long times, before undergoing ESD.Therefore, a complete characterization of entanglementdecay also requires its quantiﬁcation.The global evolution of entanglement is characterizedby using the concurrence probability density,
p
(
C
;
q
) =
d
2
ψδ
(
C
−
C
(
q
;
ψ
))
p
(
ψ
)
,
(2)where
C
(
q
;
ψ
) is the Wootter’s concurrence [29]of the state
ρ
(
q
) = Λ
q
(

ψ
ψ

). Concurrence isgiven by
C
(
q
;
ψ
)
≡
max
{
0
,
√
λ
1
−
4
i
=2
√
λ
i
}
,where
λ
i
are the eigenvalues of the matrix
ρ
(
q
)
σ
(1)2
⊗
σ
(2)2
ρ
(
q
)
∗
σ
(1)2
⊗
σ
(2)2
, being
λ
1
thelargest one. Here
ρ
∗
is the complex conjugate of
ρ
in thestandard basis, and
σ
2
is the second of Pauli matrices.Then, for our chosen ensemble of initial states,
p
(
C
;
q
)
dC
is the probability to ﬁnd a state with concurrence valuein an interval
dC
around
C
at time
q
. In particular, atthe initial time
q
= 0, eq. (2) provides the concurrencedistribution over an ensemble of uniformly distributedpure states,
p
(
C
;0) = 3
C
√
1
−
C
2
[21, 30]. The con
tribution of the states with vanishing concurrence tothe distribution
p
(
C
;
q
) can be singled out by writing
p
(
C
;
q
) = ˜
p
(
C
;
q
) +
S
(
q
)
δ
(
C
), where ˜
p
(
C
;
q
) is theprobability density for strictly positive concurrence and
S
(
q
) = 1
−
C
M
0
dC
˜
p
(
C
;
q
) is the probability to ﬁnd aseparable state at a time
q
, where
C
M
=
C
M
(
q
) is themaximum concurrence at that time.We will show that
p
(
C
;
q
) and
p
(
q
S
) provide a complete portrait of the entanglement behavior for all times,enabling us to monitor entanglement degradation, concentration, and even loss. Despite the dissimilar dynamics of the D, AD, and PD channels, in all cases we ﬁndanalytical expressions for the disentanglement time andthe concurrence, for initial pure states, and use them toevaluate the probability distributions of separation times
p
(
q
s
) and of concurrence
p
(
C
;
q
) as a function of the reparameterized time. Our results were checked using purelynumerical methods.
3
III. PROBABILITY DISTRIBUTIONS OF THEDISENTANGLEMENT TIMES.
The disentanglement times,
q
(D)
S
= 1
−
1
/
1 + 2
C
0
,
(3)
q
(AD)
S
=
C
0
/
2

ψ
11

2
,
(4)
q
(PD)
S
= 1
−
C
20
+
d
2
−
C
0
/d
(5)for identical depolarizing, amplitudedamping and phasedamping channels, respectively, are found from the condition of separability,
C
(
q
S
,ψ
) = 0. Here, we haveset
d
= 4

ψ
00
ψ
01
ψ
10
ψ
11

1
/
2
, and
C
0
= 2

ψ
00
ψ
11
−
ψ
01
ψ
10

=
C
(0
,ψ
), and have used superscripts indicating the case/channel we are considering. We use Eq. (1)and these expressions for the separation time to obtainthe following probability distributions of disentanglementtime,
p
(D)
(
q
S
) = 3
q
S
(2
−
q
S
)
(
q
2
S
−
2
q
S
+ 2)(3
q
2
S
−
6
q
S
+ 2)4(1
−
q
S
)
7
, p
(AD)
(
q
S
) =
q
2
S
−
12
q
S
(1 +
q
2
S
)
2
+arctan(
q
S
)2
q
2
S
+2+
π
8
δ
(
q
S
−
1)
, p
(PD)
(
q
S
) =
dθdsdrP
J
(
s,r
)
δ
(
q
S
−
q
(PD)
S
)
/
2
π.
In the latter case the dependency on the angle
θ
comes from the initial concurrence
C
0
=
√
2
s
2
+
r
2
+ (
s
2
−
r
2
)cos(
θ
). Here,
P
J
(
s,r
) =
24(
s
2
−
r
2
)
√
1
−
4
r
2
K
1
−
4
s
2
1
−
4
r
2
stands for the joint probability distribution of
r
and
s
, with
s
+
r
= 2

ψ
00
ψ
11

and
s
−
r
=2

ψ
01
ψ
10

. The complete elliptic integral of the ﬁrst kindand elliptic modulus
k
is denoted by
K
(
k
2
).The probability distributions of the separation times(Fig. 1) display completely diﬀerent behaviors. Allpresent a rather conspicuous peak at around
q
S
= 0
.
38,
q
S
= 1, and
q
S
= 0
.
59 for cases D, AD, and PD, respectively. However, the ﬁrst is the widest, and the secondis the more narrow and is actually a delta distribution.While in the ﬁrst case all states become separable before
q
S
= 1
−
1
/
√
3
≈
0
.
42, only
6
−
π
8
≈
35
.
73% of the statesexperience ESD in the second case, and a zeromeasureset of states separate asymptotically in the third case.The three cases we have considered can be tell apart bytheir disentanglement time probability distributions, andthe fraction of entangled states at a given time can becalculated. However, it is not clear how much entanglement remains at a given time. This is especially important in the AD case, because the entanglement of moststates (64
.
27%) decay asymptotically. In the next threesections, we consider the dynamics of the concurrenceprobability distribution for each case.
0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1Dimensionless Separation Time q
S
P r o b a b i l i t y d i s t r i b u t i o n p ( q
S
)
FIG. 1. (Color online) Disentanglementtime probability distributions: solid line (black) for local depolarizing channels(D); dashed line (blue), including a delta function contribution, at
q
s
= 1, for local amplitude damping channels (AD);and dotted line (red) for local phase damping channels (PD).
IV. DEPOLARIZING CHANNELS.
We start by investigating the case in which each qubitis coupled to a depolarizing channel. As time goes by,the depolarizing channels induce decoherence on the twoqubits and their initial entanglement is lost. The concurrence at time
q
,
C
(D)
(
q
;
C
0
) = max
0
,C
0
(1
−
q
)
2
−
12
q
(2
−
q
)
,
(6)depends only on (time and) the concurrence of the initial state
C
0
. The loss of entanglement in the system isuniform, and initial maximally entangled states remainas the states with the highest concurrence for all times,
C
M
(
q
) = 1
−
32
q
(2
−
q
). For an ensemble of uniformly distributed initial pure states, the concurrence probabilitydensity at a time
q
˜
p
(D)
(
C
;
q
) = 3
C
+
12
q
(2
−
q
)(1
−
q
)
4
1
−
C
+
1
2
q
(2
−
q
)(1
−
q
)
2
2
,
follows from Eq. (6) and
p
(
C
;0) = 3
C
√
1
−
C
2
. In theinset of Fig. 2 we draw ˜
p
D
(
C
;
q
) for diﬀerent values of time,
q
.In Fig. 2, entanglement degradation is apparent. Indeed, due to the loss of entanglement, the whole distribution moves towards zero and the average concurrence
C
q
=
C
M
0
dC C
˜
p
(D)
(
C
;
q
) decreases. At the sametime entanglement concentrates around its mean valuebecause the standard deviation (
C
2
q
−
C
q
2
)
1
/
2
falls as
q
increases. Finally, entanglement is lost: the probabilityto obtain an entangled state
C
M
0
dC
˜
p
(D)
(
C
;
q
) declineswith time.Entanglement degradation, concentration and loss aredisplayed in all the examples considered in this work,
4
0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5
C o n c u r r e n c e
Dimensionless Reparameterized Time q 0 1 2 3 0 0.2 0.4 0.6 0.8 1
Concurrence C
C o n c u r r e n c e d i s t r i b u t i o n
FIG. 2. (Color online) Identical but independent depolarizingchannels. The solid blue lines represent the average concurrence plus a standard deviation (darkest line), the averageconcurrence, and the average concurrence minus a standarddeviation (lightest line); maximum concurrence is plotted inred (dashed line), and the value of concurrence at which thedistribution peaks is plotted in green (dotted line). Inset:From right to left, concurrence distributions for
q
= 0
,
0
.
2and 0
.
4.
and seem typical for open markovian systems, whoseasymptotic states belong to
S
and
∂
S
. This asymptotic behavior srcinates also entanglement concentrationdue to the shrinking average distance between the statesthat remain entangled. Entanglement is lost when statescross the border between entangled and separable states.Thus, the concurrence probability distribution providesa complete picture of the process of entanglement decay. However, the most salient features of this processare contained in the evolution of the concurrence maximum, its average, its deviation and the separationtimeprobability distribution.
V. AMPLITUDEDAMPING CHANNELS.
In our second example each qubit experiences dissipation under the action of an amplitudedamping channel. For an initial twoqubit pure state, the concurrenceevolves in time as [17]
C
(AD)
(
q
;
C
0
,ψ
11
) = max
0
,
(1
−
q
)
C
0
−
2

ψ
11

2
q
.
(7)We notice that the set of initially entangled states separates into two classes: those states for which
C
0
<
2

ψ
11

2
, becoming separable at a ﬁnite time; while allthe others states, for which
C
0
>
2

ψ
11

2
, reaching separability only asymptotically. Hence, contrary to the previous case, the loss of concurrence is not uniform: whencomparing states with the same initial concurrence, thosewith smaller

ψ
11

are more robust. At a given time
q
, themaximum concurrence
C
M
= 1
−
q
corresponds to initialstates of the form (

01
+ exp(
iφ
)

10
)
/
√
2, with arbitrary phase
φ
. Since
C
(AD)
(
q
) depends on both the ini
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
C o n c u r r e n c e
Dimensionless Reparameterized Time q 0 2 4 6 0 0.2 0.4 0.6 0.8 1
C o n c u r r e n c e d i s t r i b u t i o n
Concurrence C
FIG. 3. (Color online) Identical but independent amplitudedamping channels. The solid blue lines represent the average concurrence plus a standard deviation (darkest line), theaverage concurrence, and the average concurrence minus astandard deviation (lightest line); maximum concurrence isplotted in red (dashed line), and the value of concurrenceat which the distribution peaks is plotted in green (dottedline). Inset: From right to left, concurrence distributions for
q
= 0
,
0
.
4
,
0
.
8 and 0
.
9.
tial concurrence and the state component in the

11
direction, the joint probability distribution
P
J
(

ψ
11

2
,C
0
)over all twoqubit pure states is calculated ﬁrst in orderto evaluate the concurrence distribution ˜
p
(AD)
(
C
). This joint probability either vanish or is equal to 3
C
0
log((1+
1
−
C
20
)
/z
), where
z
= max(1
−
1
−
C
20
,
2

ψ
11

2
) for2

ψ
11

2
≤
1+
1
−
C
20
. The concurrence probability distribution ˜
p
(AD)
(
C
;
q
) =
d

ψ
11

2
dC
0
δ
(
f
)
P
J
(

ψ
11

2
,C
0
)(where
f
=
C
−
C
(AD)
(
q
;
C
0
,ψ
11
)) can be calculated analytically, but the expression is cumbersome and not illuminating.The concurrence probability distribution is depicted inthe inset of Figure 3, for some values of time
q
. As in theprevious case the degradation of concurrence (the disentanglement) is more important for smaller (larger) times.The separation between these processes is more conspicuous here because most states disentangle asymptotically.The asymptotic massive disentanglement is not evidentin the graphic of average concurrence.
VI. PHASEDAMPING CHANNELS.
In our last example each qubit evolves under the actionof a dephasing channel, which washes away the coherences in the
σ
3
representation. The concurrence at time
q
of an initial pure twoqubit state under the inﬂuence of