Most robust and fragile two-qubit entangled states under depolarizing channels

0.19 MB·English

by  Chao-Qian Pang

Most robust and fragile two-qubit entangled states under depolarizing channels
Chao-Qian Pang,1 Fu-Lin Zhang,1,∗ Yue Jiang,1 Mai-Lin Liang,1 and Jing-Ling Chen2,3
1Physics Department, School of Science, Tianjin University, Tianjin 300072, China
2Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin, 300071, China
3Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
(Dated: March 5, 2012)
In the two-qubit system under the local depolarizing channels, the most robust and the most
fragile states for agiven concurrence ornegativity are derived. Fortheone-sided channel,with the
aidoftheevolutionequationforentanglementgivenbyKonradetal. [Nat. Phys. 4,99(2008)],the
purestates are proved to be the most robust. Based on a generalization of the evolution equation,
we classify the ansatz states in our investigation by the amount of robustness, and consequently
2 derivethemost fragile states. Forthetwo-sided channel,thepurestatesare provedtobethemost
1
robust for a fixed concurrence, but is the most fragile with a given negativity when the channel is
0
uniform. Under the uniform channel, for a given negativity, the most robust states are the ones
2
withthemaximalconcurrence,whicharealso themost fragile stateswhentheconcurrenceisgiven

r intheregionof[1/2,1]. Whentheentanglementapproacheszero,themostfragilestatesforagiven
a
negativitybecomethepurestates,buttheonesforconcurrenceturnintoamixtureoftwoseparable
M
purestates.

2 PACSnumbers: 03.67.Mn,03.65.Ud,03.65.Yz

]
h
I. INTRODUCTION tion with a noise environment. Vidal and Tarrach [12]
p
introduced the robustness of entanglement as a measure
-
t of entanglement, corresponding to the minimal amount
n Quantum coherent superposition makes quantum in-
a formation conceptually more powerful than classical in- ofmixing with separablestates whichwashedout allen-
u tanglement. Subsequently, Simon and Kempe [13] con-
formation, which is the essential distinction between a
q sidered the critical amount of depolarization, where the
quantum system and a classical one. Entanglement is
[
entanglementvanishes,asaquantitativesignatureofthe
a manifestation of the distinction in composite systems
2 [1] and is one of the key resources in the field of quan- robustness of the entanglement, when they studied the
v robustnessofmulti-partyentanglementunderlocaldeco-
tum information science [2]. However, the unavoidable
8 herence, modeled by partially depolarizing channels act-
couplings between a real quantum system and its envi-
9 ing independently on each subsystem. This definition
7 ronment can lead to the decoherence phenomenon. Si-
was adopted in the recent work [14], in which an inter-
2 multaneously, the entanglement between the subsystems
esting residual effect was pointed out, on the robustness
. is usually destroyed.
2 of a three-qubit system in an arbitrary superposition of
0 Because of the importance both in the fundamental
Greenberger-Horne-Zeilingerstate and W state.
2 theory and the application in quantum information, the
1 dynamics of entanglement in a quantum system under Wenoticethat,eveninthetwo-qubitsystem,thereare
:
v decoherencehasattractedwideattentioninrecentyears. stillsomequestionsaboutthe robustnesstoexplore. For
i Many significant and interesting results have been re- instance,theresultsof[14]showtherobustnessincreases
X
ported. For instance, the concept of sudden death of en- synchronouslywiththedegreeofentanglementforatwo-
r tanglement (ESD) has been presented by Yu and Eberly qubit pure state. Obviously, it should be very hard to
a
[3, 4], which means that the entanglement can decay to extend this conclusion to the mixed states, because the
zeroabruptlyinafinitetimewhilecompletedecoherence measurements of entanglement for the mixed states are
requiresaninfiniteamountoftime. Thisinterestingphe- not unique [15–17], which would influence the robust-
nomenonhas been recently observedin two sophistically ness jointly . Then, a question arises: For a given value
designedexperimentswithphotonicqubits[5]andatomic of entanglement under a measurement, which states are
band [6]. In [7], by utilizing the Jamiol kowski isomor- the most robust and which are the most fragile? Since,
phism, Konrad et al. presented a factorization law for a generally speaking, the value of entanglement in a two-
two qubit system, which described the evolution of en- qubit system correlated positively with the resource of
tanglement in a simple and general way. This result has the preparation [15, 16] and its capacity in quantum in-
been extended in many directions [8–11]. formation [18, 19], the entanglement is expected to be
This paper is concernedabout the topic of the robust- stable under the same conditions. Many similar inves-
ness of entanglement, when a quantum system interac- tigations have been reported for about ten years. For
example, the famous maximally entangled mixed states
[20, 21] canbe consideredto be the most residualentan-
glement under a global noise channel. In [22], Yu and
∗Email: fl[e-mail] Eberly studied the robustness and fragility of entangle-
2
ment in some exactly solvable dephasing models. In the Bloch sphere representation for the qubit density opera-
recentworkofNovotny`et al. [23],theypresentageneral tor, the transformation in Eqs. (1) and (2) is given by
andanalyticallysolvabledecoherencemodelwithoutany ρ (t)= 1(I +s~r ~σ )with~r being thethe initialBloch
i 2 i i i· i i
weak-couplingorMarkovianassumption,anddistinguish vector.
the robust and fragile states in the model. Forsimplicitybutwithoutlossofgenerality,wechoose
In the present work, we investigate the two-qubit sys- the decay constants κ = 1+∆ and κ = 1 ∆, with
1 2
−
tem under local depolarizing channels which is invariant the uniform parameters ∆ [0,1]. Then, the noise pa-
under local unitary operations, and focus on the most rameters can be written as∈s = s1+∆ and s = s1−∆,
1 2
robustentangledstates(MRES)andthemostfragileen- where s = e−t. Here, we introduce the measurement of
tangled states (MFES) for a given amount of initial en- the robustness for a state ρ under a nonuniform channel
tanglement. The model in [13] and [14] is generalized to as
the nonuniform case in our study. Namely, the strength
of the coupling of the two qubits with its environment (ρ)=1 scrit(ρ), (3)
R −
are different. This generalization is more close to some
practical cases , in which the entangled two qubits un- where scrit(ρ) denotes the critical value of the noise pa-
dergo different environment. Two measurements of the rameter s, at which occurs the ESD of the two-qubit
degree of entanglement,the concurrence[15, 16] and the system with the initial state ρ. When ∆ = 0, it returns
negativity [17], are adopted for comparison. In Sec. II, the definition in [13] and [14].
we review the definitions and introduce some notations
inthis paper. Consideringits relationwiththe evolution
equation for quantum entanglement in [7], we selectively B. Measurements of entanglement
givetheresultoftheone-sidedchannelinSec. III,which
is an extreme case of the nonuniform channels. In Sec. The concurrence of a pure two-qubit state ψ =
| i
IV,the uniformandnonuniformchannelsarestudiedto- c 00 + c 01 + c 10 + c 11 is given by (ψ ) =
1 2 3 4
| i | i | i | i C | i
gether. Conclusion and discussion are made in the last 2c c c c . For a mixed state, the concurrence is de-
1 4 2 3
| − |
section. fined as the average concurrence of the pure states of
the decomposition,minimized overall decompositionsof
ρ= p ψ ψ ,
II. NOTATIONS AND DEFINITIONS j j| jih j|
P
(ρ)=min p (ψ ). (4)
j j
A. Channels and robustness C C | i
j
X
Underlocaldecoherencechannels,withouttheinterac- It can be expressed explicitly as [15, 16] (ρ) =
C
tion between the two qubits, the dynamics of each qubit max 0,√λ1 √λ2 √λ3 √λ4 , inwhichλ1,...,λ4 are
is governed by a master equation which depends on its thee{igenvalu−esofth−eoper−atorR}=ρ(σy σy)ρ∗(σy σy)
⊗ ⊗
own environment [24]. The evolution of each reduced in decreasing order and σy is the second Pauli matrix.
density matrices is described by a completely positive Forabipartitesystemdescribedbythedensitymatrix
trace-preserving map: ρ (t) = ε (t)ρ (0), i = 1,2, and ρ, the negativity is defined as [17, 26]
i i i
for the whole state ρ(t)=ε (t) ε (t)ρ(0). In the Born-
1 2
⊗
Markov approximation these maps (or channels) can be (ρ)=2 λ , (5)
j
N | |
described using its Kraus representation j
X
M where λ are the negative eigenvalues of ρT2 and T de-
εi(t)ρi(0)= Ejiρi(0)Ej†i (1) notes thje partial transpose with respect to the se2cond
j=0 subsystem. For the two-qubit system, the partial trans-
X
pose density matrix ρT2 has at most one negative eigen-
where Eji satisfying Mj=−01EjiEj†i = I, are the Kraus value [21].
operators, M being the number of operators needed to
P
completelycharacterizethechannel. Forthedepolarizing
channels, the Kraus operators are C. Ansatz states
1 1
E0i = √3si+1Ii, Eji = √1 siσji, (2) To derive the MRES and MFES for a given value
2 2 −
of concurrence or negativity, we adopt the approach in
where σ , (j = 1,2,3) is the jth Pauli matrices for [20, 21]. Under a given value of the nonuniform parame-
ji
the ith qubit. We consider the noise parameter s = ter∆,werandomlygeneratetwo-qubitstatesandobtain
i
exp( κ t),whereκ isthe decayconstantdeterminedby their robustness and degree of entanglement by numer-
i i
−
the strength of the interaction of the ith qubit with its ical computation. Plotting them in the corresponding
environment and t is the interaction time [25]. In the robustness-entanglementplanes (suchasthe onesinFig.
3
1), we fortunately find, for any value of ∆, their regions
0.4
are always the same as the states H1L H2L H3L
ρansatz(r,θ)=r|ψ(θ)ihψ(θ)|+(1−r)|01ih01|, (6) Robustness 00..23
where r [0,1] and ψ(θ) = cosθ 00 +sinθ 11 with 0.1
∈ | i | i | i
θ [0,π/2]. For each value of ∆ 0,0.1,0.2,...1 , and H4L HaL
∈ ∈ { } 0.0
forboththeentanglementmeasures,thisissupportedby 0.0 0.2 0.4 0.6 0.8 1.0
300000 random states. Therefore, the MRES or MFES Concurrence
can be represented as the state (6) with a constraint, 0.4
H1L H2L H3L
which we will derive in the following sections. We call
cthaesOesnsteaotftehthi(n6eg)sttthoaetbeasenamsraeetnzctosirtoarneteesdpioninstdthihnisagtp,taoTp,hetreh.etwpoureexstrtaemtees Robustness 00..23 H4L
0.1
withthesamevaluesofnegativityandconcurrencewhen
HbL
r = 1, and the ones with the minimal negativity for a 0.0
0.0 0.2 0.4 0.6 0.8 1.0
given concurrence when θ =π/4, respectively [27].
Negativity
FIG. 1: (color online) Plots of the states ρ(c,p) in the (a)
III. ONE-SIDED NOISY CHANNEL robustness-concurrence and (b) robustness-negativity planes
under one-sided noise channel. The dotted lines correspond
to the parameter c = 1,0.8,0.6,0.4,0.2 from top to bottom,
A. Robustness vs concurrence
in both (a) and (b). The dashed lines refer to theparameter
(1) p = tan(0.1 × π/2), (2) p = tan(0.2 × π/2), (3) p =
When ∆=1, s1 =s2 and s2 =1, the channel reduces tan(0.3×π/2) and (4) p = tan(0.4×π/2). The solid line is
the one-sided depolarizing channel , in which only the theMFES in thetwo cases.
first qubit influenced by the environment. The central
result of [7] is the factorizationlaw for an arbitrarypure
state ψ under a one-sided noisy channel
| i
[($ I)ψ ψ ]= [($ I)ψ+ ψ+ ] (ψ ), (7) with1/γ =Tr[(I M)ρ (I M†)]andM =I+~a ~σ, (a
C ⊗ | ih | C ⊗ | ih |C | i ⊗ 0 ⊗ · ∈
[0,1]), the evolutions of their concurrence are related as
where ψ+ = ψ(π/4) and $ denotes an arbitrary
| i | i
channel not restricted to a completely positive trace- [($ I)ρ ] (ρ )= [($ I)ρ ] (ρ ). (11)
1 0 0 1
preservingmap. Anevidentcorollaryisthat,theESDof C ⊗ C C ⊗ C
the systems initial from any entangled pure states occur We don’t give the proof of the above relation, which is
at the same time, and consequently their robustness are exactlythesameastheprocessfor(7)in[7]. Asthecase
the same. of the pure states, we can conclude that the robustness
An application of (7) in [7] is the inequality for mixed of ρ0 and ρ1 in (10) have the same value.
states ρ ,
0
Then, we find the ansatz states (6) can be classi-
[($ I)ρ0] [($ I)ψ+ ψ+ ] (ρ0). (8) fied by the amount of robustness, with the aid of the
C ⊗ ≤C ⊗ | ih |C
generalized factorization law (11). Namely, choosing
From (7) and (8), one can find the fact that, under the
ρ = ρ (c,π/4) and ~a = (0,0,(1 p)/(1+p)), the
0 ansatz
same one-sided channel, with the same initial concur- −
ansatz states can be written as
rence, the ESD of a mixed state comes not later than
the one of a pure state. Hence, the MRES with a fixed ρ (r,θ)=ρ(c,p)=γ(I M)ρ (I M†), (12)
ansatz 0
concurrence is the pure sates. Under the depolarizing ⊗ ⊗
channel considered in the present work, we can derive wheretherelationsbetweenthetwopairsparametersare
the value of robustness
r rcos2θ
1 c= − , p=tanθ. (13)
= (ψ )=1 , (9) 1 rcos2θ
RMRES R | i − √3 −
The robustness of the ansatz state is only dependent of
which is a constant, independent of the concurrence. the parameter c in ρ , which is given by
0
To obtain the MFES, we generalize the factorization
law(7)inthe arbitrarytwo-qubitstates case. Namely,if 2 c
[ρ(c,p)]=1 − , (14)
two states satisfy R − 2+c
r
ρ =γ(I M)ρ (I M†) (10) corresponding the result for the pure states in (9) when
1 0
⊗ ⊗
4
c=1. The concurrence can be obtained as 0.5
2cp 0.4
[ρ(c,p)]= =rsin2θ. (15)
In Fig. 1C(a), we ploct+th(e2−ancs)apt2z states with some dis- Robustness 00..23
cretevaluesofcorpintherobustness-concurrenceplane. 0.1
OnecanfindtheMFES,withtheminimalrobustnessfor
0.0
0.0 0.2 0.4 0.6 0.8 1.0
a given concurrence, achieves the maximal concurrence
Concurrence
when the value of robustness is fixed. It is easy to de-
termine the maximalvalue of concurrence in (15) with a
FIG. 2: (color online) The relation between the robust-
fixed c. Then the constraint on the ansatz sates for the ness and the concurrence of a two-qubit system in a pure
MFES can be obtained as entangled state. From top to bottom the curves refer to
∆=1,0.75,0.5,0.25,0.
c(1+p2) 2p2 =0, (16)
−
with 0 p 1, or equivalently rencegivenin(16)and(17),whenθ 0,the proportion
≤ ≤ →
r drops to 1/2, which has the maximal entropy in the
2rcos2θ =1, (17)
family of the ansatz states.
with 0 θ π/4 and 1/2 r 1. It is interesting that
≤ ≤ ≤ ≤
the condition leads to the lengthes of the Bloch vectors
IV. TWO-SIDED NOISY CHANNEL
for the two qubit are r = 2(1 r) and r = 0, which
1 2
−
reach the maximal difference with a fixed r.
Wefailingeneralizingtherelationsin(10)and(11)to
the two-sided channel case. Therefore, we calculate the
B. Robustness vs negativity ESD critical condition of the ansatz states directly
Itiseasilytoprovethe thefactthat,the MRESunder P =4r2sin22θs21s22−[1+(1−2r)s1s2]2
an arbitrary one-sided noise channel for a givennegativ- +[rcos2θ(s s )+(1 r)(s +s )]2 =0,(20)
1 2 1 2
− −
ity are also the pure states, with the aid of the results
for the concurrence case. Suppose ρ denotes an arbi- which is one of the main basis to determine the MRES
trary entangled two-qubit state, ψ and ψ are two and MFES in this section. When r = 1, one can obtain
1 2
| i | i
pure states, and (ρ) = (ψ ) and (ρ) = (ψ ). the relation of the entanglement of pure states and their
1 2
N N | i C C | i
Tworelationscanbeobtainedintheabovesubsectionas critical noise parameters
(ρ) (ψ ) and (ψ ) = (ψ ), from which we
2 1 2
dReduc≤e tRhe|coniclusionR. | i R | i 2(ψ )= 2(ψ )= (1−s21)(1−s22) . (21)
In Fig. 1 (b), one can notice the region of the ansatz C | i N | i 4s21s22−(s1−s2)2
states, of course the one of the arbitrary states, in the
robustness-negativity plane is similar to the one in Fig. For the Bell state ψ+ , the above equation gives the
| i
1 (a) where the entanglement measured by concurrence. robustness (ψ+ )=1 1/√3, whichis independentof
R | i −
Thus,todeterminethe MFESistofindthe statesρ(c,p) thenonuniformparameter∆. Inthissense,ourdefinition
withthemaximalnegativityforafixedc. Thenegativity of robustness is reasonable. When ∆ = 0, (ψ ) =
R | i
of the ansatz states is 1 1/ 2 (ψ )+1, which is the result of the uniform
− C | i
channels given in [14]. In Fig. 2, we plot the relation of
p
[ρ (r,θ)]= r2sin22θ+(1 r)2 (1 r).(18) the robustness and the entanglement of the pure state.
ansatz
N − − −
Underafixed∆,therobustnessisamonotoneincreasing
q
Inserting the relations (13), one can find the maximal function of the entanglement. A nonmaximal entangled
negativity occurs when pure state becomes more fragile as ∆ decreases.
2√2 c( 1+c)c3/2+2c2 c3
p= − − − . (19) A. Robustness vs concurrence
s 8+24c 20c2+5c3
− −
The corresponding relation of the parameters r and θ is Because of the invariance of the depolarizing channels
verycloseto the numericalresultsfor∆=0.99inFig. 5 underlocalunitary(LU) transformations,the MRESfor
(a) in the following section. concurrence can be proved to be the pure states under
In the MFES, when the parameter θ 0, the en- thetwo-sidedchannels. Accordingtotheproceduregiven
→
tanglement of the entangled composition decreases, its byWootters[16],onecanalwaysobtainadecomposition
proportion r 1. However, in the MFES for concur- φ minimizing the average concurrence in Eq. (4),
i
→ {| i}
5
0.4 0.4 family of the ansatz states. In the other, when 1/2,
Robustness 000...123 D=0.75 Robustness 000...123 D=0.5 tahfiexMedFcEoSncaurerrtehnecest[a2t7e]s. with the minimal negatCiv≥ityfor
0.0 0.0 When 0 < ∆ < 1, the MFES for the arbitrary depo-
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
larizing channels is given by the simultaneous equations
Concurrence Concurrence
of (20) and (24), which is shown in Fig. 3 for some val-
0.4 0.4
Robustness 000...123 D=0.25 Robustness 000...123 D=0 uthees orofb∆u.stIntescsanofbMe nRoEtiSceadndthMatF, EthSeddeiffcreeraesnecsewbietthwetehne
decrease in the nonuniform parameter ∆. When β 0,
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 the concurrence 0, the proportion of the entan→gled
Concurrence Concurrence C →
state in MFES approaches 1/2.
Considering the absence of a brief expression of the
FIG. 3: (color online) The relation between the robustness
MFES, we present a family of quasi-MFES. Namely, we
and the concurrence of the MRES and MFES under dif-
replace the critical depolarizing parameter s2 [1/3,1]
ferent nonuniform parameters, ∆ = 0.75,0.5,0.25,0. The
∈
dots on the lines denote the quasi-MFES, whose parameters in (24) by its mean value 2/3, and obtain Ω Ω =
β =0.1,0.2,0.3,0.4,withthevaluesofconcurrenceincreasing (2/3)∆−1−1 . It is obviously, Ω = 1 and Ω→ =
. (2/3)−∆−1−1 |∆=0 |∆=e1
0.The positions of the quasi-MFES in the robustness-
concurrence plane are shown ineFig. (3), whicheare very
ρ = t φ φ , in which t = 1 and all the ele- close to the MFES. Our numerical result shows, for a
m0ents haivei|thiiehsia|me value of coinicurrence as the mixed given β, the fidelity of the quasi-MFES and the MFES
state ρP0. The elements are eqPuivalent under LU trans- F(ρquasi−MFES,ρMFES) ≥ 1 −10−4 with F(ρ1,ρ2) =
2
formation to the same state |ψi Tr √ρ1ρ2√ρ1 , which is supported by one million
pairs of randomly generated quasi-MFES and MFES.
φ =UA UB ψ , (22) (cid:2) (cid:0)p (cid:1)(cid:3)
| ii i ⊗ i | i
with the concurrence (ψ ) = (ρ ). After passage
0 B. Robustness vs negativity
C | i C
through a basis independent local channel $ $ , the
1 2
⊗
concurrence [($ $ )φ φ ] = [($ $ )ψ ψ ]. It
C 1⊗ 2 | iih i| C 1⊗ 2 | ih | To determine the MRES andMFES for the negativity
then immediately follows, by convexity, that
under the two-sideddepolarizing channels,we derive the
condition,
[($ $ )ρ ] [($ $ )ψ ψ ], (23)
1 2 0 1 2
C ⊗ ≤C ⊗ | ih |
∂ ∂ ∂ ∂
which leaves us with the claim at the beginning of this N P N P =0, (26)
∂r ∂θ − ∂θ ∂r
paragraph.
Similar with the case of the one-sided depolarizing where the negativity of an ansatz state reaches an ex-
channels,theMFESinthepresentcasearealsothestates tremum for a fixed robustness. The form of the the
achieve the maximal concurrence for a given value of ro- MRES and MFES are given by the physically accepted
bustness, which canbe noticed in Fig. 3. From (15) and solutionsto the simultaneousequationsof (20)and(26).
(20), we find, for a given pair of the critical noise para-
When ∆=0, one cancheck the two solutions θ =π/4
ments s1 and s2, or equivalently the robustness and and r = 1, corresponding to the MRES and MFES
R
thenonuniformparameter∆,theMFEScanbeobtained
respectively. In other words, under the uniform chan-
applying the restraint nel, the MRES for the entanglement measurement neg-
ativity are ρ = ρ (r,π/4), and the MFES
1 MRES ansatz
α= 1+ 8Ω(2β 1)β+1 (24) ρMFES =ρansatz(1,θ) are the pure states. The result is
4(cid:20) − (cid:21) contrary to the concurrence case, where the pure states
p
are the most robust, and ρ (r,π/4) are the most
on the ansatz states (6), where α = rcos2θ [0,1/2], ansatz
β = rsin2θ ∈ [0,1/2] and Ω = ss2122((11−−ss2221)). Whe∈n ∆ = 1, bfruasgtinlees(swihseninflCu≥enc1e/d2)b.yTbhoethdiffoefrethnecetwinodimcaetaesuthreesroo-f
Ω=0,therestraint,α=1/2,returnstotheresultofthe entanglement. The pure states, which are the MRES
one-sided channels in (16) and (17). for concurrence, have the maximal negativity when the
For the uniform noise channels, where ∆=0 and Ω= concurrenceisfixed. Theyarealsothestateswithamin-
1, the MFES can be determined as imalconcurrence for a givennegativity, andthus are the
MFES for negativity. The states ρ (r,π/4) are the
ansatz
1[ψ(θ) ψ(θ) + 01 01], <1/2, states having the minimal negativity for a fixed concur-
ρMFES = r2ψ|+ ψih+ +(|1 |r)i0h1 |01, C 1/2,(25) rence, and also the states with maximal concurrence for
(cid:26) | ih | − | ih | C ≥
a given negativity. Therefore, they appear to be more
by using the relation in (24). In the first region where fragile than the other states with the same concurrence,
< 1/2, the MFES reach the maximal entropy in the butmorerobustthantheotherstateswiththesameneg-
C
6
0.4 r=0.2 r=0.4 1.0 æææææææææææææææ æ æ æ æ æ ìææòà
bustness 00..23 r=0.6 r=0.8 r 00..89 àìòôàìàòôìàòàìôàòìôàìòàôìòàôìòàìôòàìôòàìôòàìôòàìôòàìôòàôìòàôìòôàòìôòæìàôòæìôàòDô=ì0.ò0ô0
o
R 0.7 à à D=0.25
0.1 ì ì D=0.50
HaL 0.6 ò ò D=0.75
HaL ô ô D=0.99
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.5 0 Π Π 3Π Π
Negativity 16 8 16 4
Θ
Π
0.4 Π 2 ô ô ô ô ô ô ô ô ô ô ôôôôôô ô ô ô ô
0.3 Θ=0.3‰Π Θ=0.4‰2 38Π ìò ìò ìò ìòìòìòìòìòìòìòìòìòìòìòìòìòìòìòìòìò
Robustness 0.2 Θ=0.2‰Π2 2 Θ Π4 æà æà æà æà æà æà æà æàæàæàæàæàæàæàæààæàæààæDà=æ0.à25æàìæòôà
Π Π ì ì D=0.50
0.1
Θ=0.1‰ 8 ò ò D=0.75
2 HbL HbL æ æ D=0.00 ô ô D=0.99
0.0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Negativity r
FIG.5: (coloronline)Relationsoftheparametersr andθfor
FIG. 4: (color online) Plots of the states ρ(r,θ) in the
the MFES (a) and MRES (b) with different inhomogeneous
robustness-negativityplaneswiththeMRESandMFESwhen
parameters, ∆=0.99,0.75,0.5,0.25,0. The robustness takes
the nonuniform parameters ∆ = 0.5. The parameters r/θ
a set of regularly spaced values.
takes a set of discrete values in (a)/(b), which are labeled
nearby thetangent pointswith the MRES/MFES.
rived for a given amount of entanglement, measured by
ativity. concurrence and negativity. Withe a numerical simula-
tion, we find a family of ansatz states (6), which contain
It is impossible to derive an explicit expression of
the MRES and MFES. The ansatz state is a mixture of
the solution to (20) and (26) for the arbitrary ∆
∈ two mutually orthogonalpure states, one of which is en-
(0,1). Thus we plot the ansatz states in the robustness-
tangled and the other separable.
negativityplaneforvariousvaluesofthe nonuniformpa-
rameter ∆, one of which is shown in Fig. 4. One can Under the one-sided channels, the pure states are
notice that, the MRES can be expressed as the ansatz proved to be the MRES, for both the two entanglement
states with a constrain as θ = θ(r), with r varying from measurements, by utilizing the factorization law for the
0 to 1. And the constrain for MRES is r =r(θ) , where evolution of concurrence given by [7]. A generalized fac-
thedomainisθ [0,π/4]. Thenwesearchtheconstrains torization law is presented in (10) and (11). Based on
by numericalcom∈putation, andplotthem inFig. 5. Ob- this result, we classify the ansatz states by the values of
viously,the proportionofthe entangledcompositionr in robustnessandderivetheMRESforthetwomeasuresof
the MFES is always more than 3/4. The proportion ap- entanglement. In the MRES for concurrence, the length
proaches 1, when the entanglement 0. It increases of the Bloch vector for the first qubit under the noise
with decrease in the nonuniform parNam→eter ∆. Under a reached its maximum, and the one for the free qubit is
fixed∆,theparameterθalterslittlewhentherobustness zero.
varies in a vary large region. It decreases to π/4 quickly, Underthetwo-sidedchannels,foragivenconcurrence,
when the r approaches1, andthe robustness approaches the pure states are prove to be the most robust. For the
themaximum1 1/√3. When∆ 1,thedomainofthe uniformchannel,theMFESarethestateswiththemini-
robustness beco−mes a point = 1→ 1/√3, where r = 1 mal negativity when 1/2,but the ansatz states with
and θ varies from π/2 to π/4R. − the maximal entropyCw≥hen < 1/2. The proportion of
C
theentangledpurestatetheMFESapproaches1/2,when
the concurrence 0. In contrast, the MFES for the
C →
negativity under the same channels become a separable
V. CONCLUSION AND DISCUSSION
pure state when the entanglement approaches zero. In
the uniform channel, the pure states are the most frag-
In conclusion, we investigate the robustness of the en- ile for a fixed negativity. And the ansatz states with
tangled states in the two-qubit system, under the local θ = π/4 are more robust than the other states with the
depolarizing channels. The MRES and MFES are de- samenegativity. Thereasontothedifferencebetweenthe
7
resultsforconcurrenceandnegativityisthefactthat,the evolution equations. On the other hand, the capacity of
robustness is affected by both of the two entanglement an entangled qubit pair in quantum information [18, 19]
measurements. In addition to the concurrence and the doesnotonlydependonthe amountofentanglement. It
negativity, the degree of mixture is a possible factor af- isnecessarytoinvestigatetherobustnessofthecapacities
fecting the robustness, as shown by the MFES in the in a specific quantum information process. Our scenario
uniform channel a given concurrence in [0,1/2]. The lo- to derive the MRES or MFES when an analytic proof is
cal properties, such as the Bloch vectors of each qubit, absence can be applied directly to these topics.
presentasignificantaffectionontherobustnesswhenthe
channel becomes nonuniform.
Basedtheclassificationoftheansatzstatesunderone-
Acknowledgments
sided channels, it is easy to study the relation of the
robustness and entanglement. Then, can the arbitrary
two-qubit states be classified by robustness? The solu- F.L.Z. is supported by NSF of China (Grant No.
tiontothisquestionwouldleaveusanexplicitexpression 11105097). J.L.C. is supported by National Basic Re-
of robustness under one-sided channel. The evolution search Program (973 Program) of China under Grant
equations of entanglement given in [7] have been gen- No. 2012CB921900,NSFofChina(GrantNos. 10975075
eralized in many directions [8–11]. It is interesting to and11175089)andalsopartlysupportedbyNationalRe-
study robustness of entanglement in for multi-qubit or searchFoundationandMinistryofEducation,Singapore
multi-qudit systems, as an applicationof the generalized (Grant No. WBS: R-710-000-008-271).
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