Markovian evolution of classical and quantum correlations in transverse-field XY model

0.42 MB·English

by  Amit Kumar Pal

Markovian evolution of classical and quantum correlations in transverse-field XY
model
Amit Kumar Pal∗ and Indrani Bose†
Department of Physics, Bose Institute, 93/1, Acharya Prafulla Chandra Road, Kolkata - 700009, India
Thetransverse-fieldXY modelinonedimensionisawell-knownspinmodelforwhichtheground
state properties and excitation spectrum are known exactly. The model has an interesting phase
diagramdescribingquantumphasetransitions(QPTs)belongingtotwodifferentuniversalityclasses.
These are the transverse-field Ising model and the XX model universality classes with both the
models being special cases of the transverse-field XY model. In recent years, quantities related
to quantum information theoretic measures like entanglement, quantum discord (QD) and fidelity
havebeenshowntoprovidesignaturesofQPTs. Anotherinterestingissueisthatofdecoherenceto
whichaquantumsystemissubjectedduetoitsinteraction,representedbyaquantumchannel,with
anenvironment. Inthispaper,wedeterminethedynamicsofdifferenttypesofcorrelations present
in a quantum system, namely, the mutual information I(ρAB), the classical correlations C(ρAB)
and the quantum correlations Q(ρAB), as measured by the quantum discord, in a two-qubit state.
2 Thedensity matrix of thisstateis given bythenearest-neighbour reduceddensitymatrix obtained
1
from the ground state of the transverse-field XY model in 1d. We assume Markovian dynamics
0
for the time-evolution due to system-environment interactions. The quantum channels considered
2
include the bit-flip, bit-phase-flip and phase-flip channels. Two different types of dynamics are

n identified for the channels in one of which the quantumcorrelations are greater in magnitude than
u theclassical correlations in afinitetimeinterval. Theorigins ofthedifferenttypesof dynamicsare
J further explained. For the different channels, appropriate quantities associated with the dynamics

3 of the correlations are identified which provide signatures of QPTs. We also report results for
1 further-neighbourtwo-qubit states and finite temperatures.

] PACSnumbers: 75.10.Pq,64.70.Tg,03.67.-a,03.65.Yz
h
p
- I. INTRODUCTION parameterforthetransitionishσxi,themagnetizationin
t
n the x direction, which has a non-zero expectation value
a Thefullyanisotropictransverse-fieldXY modelinone for λ > 1. The magnetization in the z direction, hσzi,
u
q dimension (1d) describes an interacting spin system for is non-zero for all values of λ with its first derivative ex-
[ which many exact results on ground and excited state hibiting a singularity at the critical point λc =1. In the
properties including spin correlations are known [1–3]. interval0<|γ|≤1,thecriticalpointtransitionatλc =1
2 The Hamiltonian describing the model is given by belongs to the Ising universality class. For λ ∈ [0,1],
v
there is another QPT, termed the anisotropy transition,
0
05 HXY =−λ2 L (1+γ)σixσix+1+(1−γ)σiyσiy+1 adtifftehreenctriutinciavlerpsoailnittyγcl=ass0.anTdhesetpraarnastiteisontwboelfoenrrgosmtoaga-
2 Xi=1(cid:8) (cid:9) netic phases with orderings in the x and y directions re-
. L spectively[1–5]. Figure1showsthephasediagramofthe
12 − σiz (1) transverse-fieldXY chain in the (λ,γ) parameter space.
1 Xi=1 The transverse-field XY model and its special case,
1 where L denotes the number of sites in the 1d lattice, the transverse-field Ising model (TIM), are well-known
v: σx,y,z are the Pauli spin operators defined at the lat- statistical mechanical models which illustrate QPTs. A
i
i tice site i, γ is the degree of anisotropy (−1 ≤ γ ≤ 1) QPT occurs at T = 0 and is brought about by tun-
X
and λ the inverse of the strength of the transverse mag- ing a non-thermal parameter, e.g., exchange interaction
r netic field in the z direction (λ > 0). The Hamiltonian strength, external magnetic field etc. In recent years,
a
is translation invariant and satisfies periodic boundary quantuminformation-relatedmeasureslikeentanglement
conditions. Two specific cases of the XY model are the [6–9], discord [10, 12] and fidelity [13] have been used as
transversefieldIsingmodelcorrespondingtoγ =±1and indicators of QPTs. A first-order QPT is characterized
the isotropic XX model (γ = 0) in the transverse mag- by a discontinuity in the first derivative of the ground
netic field. For all values of the anisotropy parameter, state energywith respectto the tuning parameter. Simi-
the Hamiltonian can be diagonalizedexactly in the ther- larlyinasecond-orderQPT,termedacriticalpointQPT,
modynamic limit L → ∞ [1–3] through the successive thesecondderivativeofthegroundstateenergybecomes
applicationsoftheJordan-WignerandBogoliubovtrans- discontinuousordivergentatthecriticalpoint. Entangle-
formations. For non-zero values of γ, the model exhibits ment and quantum discord (QD) provide different mea-
a second-order quantum phase transition (QPT) at the sures of quantum correlations in the mixed state of an
critical point λ = 1 separating a ferromagnetic ordered interacting many-body system. The QD has so far been
c
phase from a quantum paramagnetic phase. The order defined in the case of a bipartite quantum system. At
a first-order QPT point, appropriate entanglement mea-
sures and the QD are known to become discontinuous
[14, 15] whereas a critical-point QPT is signaled by the
∗ [email protected] discontinuity or divergence of the first derivative of the
† [email protected] quantum correlation measures with respect to the tun-
2
1 Ising (γ=1) time. The sudden changes in the decay rates of correla-
Critical Ising tions and their constancy in certain time intervals have
Y been demonstrated in actual experiments [26, 27].
X
al In this paper, we consider a two-qubit system each
c
Criti qubit of which interact with an independent reservoir.
The density matrix of the two-qubit system is described
Critical XX
γ 0 XX (γ=0) by the reduced density matrix derived from the ground
state of the transverse-field XY model in 1d. A simi-
Y lar study has earlier been carried out for the TIM in 1d
X
al [24]. WeinvestigatethedynamicsoftheQDaswellasthe
c
Criti classicalcorrelationsunderMarkoviantimeevolutionand
Critical Ising identify newfeaturesclosetothequantumcriticalpoints
-1 Ising (γ=-1) of the model. The study is further extended to the cases
0 1
λ offurther-neighbourtwo-qubitstatesandfinite tempera-
tures. In Sec. II, the calculationalscheme for computing
Figure1. (Coloronline)Criticalityofthetransverse-fieldXY C(ρAB)andQ(ρAB)isdescribed. Wefurtherdiscussthe
modelinthe(λ,γ)parameterspace. Thetransverse-fieldIsing methodology for studying the dynamics of the classical
model (γ = ±1) has a QCP at λc = 1. The line λ = 1 andquantumcorrelationsfordifferentquantumchannels
represents criticality of theXY model with universality class representing the system-environment interactions. Sec.
thesameasthatofthetransverse-fieldIsingmodel. Theγ =0 III presents the major results obtained and the discus-
line (XX model) represents theanisotropy transition line for
sionsthereof. Sec. IVcontainssomeconcludingremarks.
λ ∈[0,1] with the transition belonging to a new universality
class.
II. CLASSICAL AND QUANTUM
ing parameter [15]. The correlations between the differ- CORRELATIONS: DEFINITION AND
ent constituents of an interacting quantum system have DYNAMICS
two distinct components: classical and quantum. While
entanglement provides the most well-known example of
Inclassicalinformationtheory,thetotalcorrelationbe-
quantum correlations, the QD captures quantum corre-
tween two random variables A and B is quantified by
lations which are more general than those measured by
their mutual information I(A,B) = H(A) + H(B) −
entanglement [16, 17]. In fact, there are separable mixed
H(A,B). The random variables A and B take on the
states whichby definition areunentangledbut havenon-
values ‘a’ and ‘b’ respectively with probabilities given
zero QD.
by the sets {p } and {p }. H(A) = − p log p and
a b a a 2 a
Quantum systems, in general, are open systems be- H(B) = − bpblog2pb are the ShanPnon entropies for
causeoftheinevitableinteractionofasystemwithitsen- the variablesPAandB. H(A,B)=− a,bpa,blog2pa,b is
vironment. Thisresultsindecoherenceaccompaniedbya thejointShannonentropyforthevariPablesAandB with
decay of the quantum correlations in composite systems. p being the joint probability that the variables A and
a,b
A number of recent studies [18–24] explore the dynamics B have the respective values a and b. Generalization to
of entanglement and QD when the system-environment thequantumcaseisstraightforwardwiththedensityma-
interactionsaretakenintoaccount. Aninterestingresult trix ρ replacing the classicalprobability distribution and
yieldedbysuchstudiesisthattheQDismorerobustthan the von Neumann entropy S(ρ) = −Tr(ρlog ρ) replac-
2
entanglement in the case of Markovian(memoryless)dy- ingtheShannonentropy. Theexpressionforthequantum
namics. The QD decays in time vanishing only asymp- mutual information is given by
totically [20, 22–25] whereas entanglement undergoes a
‘sudden death’[18, 19], i.e., a complete disappearance at I(ρ )=S(ρ )+S(ρ )−S(ρ ) (2)
AB A B AB
afinitetime. Theclassicalcorrelations,C(ρ ),andthe
AB
QD, Q(ρ ), have mostly been computed only for two- An equivalent expression for the classical mutual infor-
AB
qubit systems described by the density matrix ρ with mation is J(A,B) = H(A) − H(A|B) where the con-
AB
A and B denoting the subsystems. Three general types ditional entropy H(A|B) is a measure of our ignorance
of Markoviantime evolutionhavebeen observed[21]: (i) aboutthestateofAwhenthatofB isknown. Theequiv-
C(ρ ) is constant as a function of time and Q(ρ ) alence with the earlier expression for the classical mu-
AB AB
decays monotonically, (ii) C(ρ ) decaysmonotonically tual information can be shown via the Bayes’ rule. The
AB
over time till a parametrized time p is reached and re- quantum versions of J(A,B) and I(A,B) are not, how-
sc
mainsconstantthereafter. Q(ρ )hasanabruptchange ever, equivalent because the magnitude of the quantum
AB
in the decay rate atp , andhas magnitude greater than conditional entropy is determined by the type of mea-
sc
thatofC(ρ )inaparametrizedtimeinterval,and(iii) surement performed on the subsystem B to gain knowl-
AB
bothC(ρ )andQ(ρ )decaymonotonically. Mazzola edge of its state. Different measurement choices yield
AB AB
et al. [22] have obtained a significant result that under different values for the conditional entropy. We consider
Markovian dynamics and for a class of initial states the von Neumann-type measurements on B represented by
QDremainsconstantinafinitetimeinterval0<t<t˜. In a complete set of orthogonal projectors, ΠB , corre-
k
thistimeinterval,theclassicalcorrelationsC(ρAB)decay sponding to the set of possible outcomes(cid:8)k. T(cid:9)he final
monotonically. Beyond t =t˜, C(ρ ) becomes constant state of the composite quantum system after measure-
AB
whereasthe QDdecreasesmonotonicallyasafunction of ment on B is given by ρ = I⊗ΠB ρ I ⊗ΠB /p
k k AB k k
(cid:0) (cid:1) (cid:0) (cid:1)
3
0.3 ing structure
elations 00 0..12.255 |c(p)|,|c(p)|12 000 000...012 ...0123555 0 psc 0.(2a) 0.4γp||=cc120 0((.pp.76)),|| λ= 00..87 1Correlations 00 00..01.. 12055 0 0(.2b)γ= 0-0.4.7QCp I((((Bρρρ 0AAAP.6BBBF)))), λ 0=.80.7 1 where A and B cρoArrBes=pondfa00to0z0bth00zbe ftd00wo individual qub(i7ts)
Corr γ=0.7 (BPF), -0.7 (BF) and z and f are real numbers. The eigenvalues of ρAB
0.1 λ=0.7 are [10]
0.05 QCI(((ρρρAAABBB))) λ0 = 14(cid:26)(1+c3)+q4c24+(c1−c2)2(cid:27)
0 0 psc 0.2 0.4 p 0.6 0.8 1 λ1 = 14(cid:26)(1+c3)−q4c24+(c1−c2)2(cid:27)
1
Figure 2. (Color online) Bit-phase-flip channel: Decay of λ2 = (1−c3+c1+c2)
mutual information I(ρAB) (solid line), classical correlations 4
Cas(ρaABfu)n(cdtiaosnheodflitnhee) apnadratmheetrQizDedQt(iρmAeB)p(d=ot-1d−ashee−dθtlinfoe)r λ3 = 14(1−c3−c1−c2) (8)
λ =0.7,γ =0.7. Also, psc =0.114. (inset) (i)The variations with
of |c1(p)| (solid line) and |c2(p)| (dashed line) as functions of
p. The crossing of |c1(p)| with |c2(p)| sets the parametrized c1 =2z+2f
timepointpsc. (ii)DecayofmutualinformationI(ρAB)(solid
line), classical correlations C(ρAB) (dashed line) and quan- c2 =2z−2f
tum discord Q(ρAB) (dot-dashed line) as a function of the c3 =a+d−2b
parametrized time p=1−e−θt for λ=0.7,γ =−0.7.
c4 =a−d (9)
The mutual information I(ρ ) can be written as [10]
AB
3
I(ρ )=S(ρ )+S(ρ )+ λ log λ (10)
AB A B α 2 α
with pk = Tr I⊗ΠBk ρAB I ⊗ΠBk . I denotes the αX=0
identityoperat(cid:0)o(cid:0)rforthe(cid:1)subsy(cid:0)stemAa(cid:1)n(cid:1)dpk istheprob- where
ability for obtaining the outcome k. The quantum con-
ditional entropy is given by 1+c4 1+c4
S(ρ )=S(ρ )=− log
A B 2
2 2
S ρ ΠB = p S(ρ ) (3) 1−c4 1−c4
AB k k k − log (11)
(cid:0) (cid:12)(cid:8) (cid:9)(cid:1) Xk 2 2 2
(cid:12)
With expressions for I(ρ ) and C(ρ ) given in Eqs.
leading to the following quantum extension of the classi- AB AB
(10), (11) and (5), the QD, Q(ρ ), (Eq. (6)) can in
cal mutual information AB
principle be computed. The difficulty lies incarryingout
J ρ , ΠB =S(ρ )−S ρ ΠB (4) the maximization procedure needed for the computation
AB k A AB k
(cid:0) (cid:8) (cid:9)(cid:1) (cid:0) (cid:12)(cid:8) (cid:9)(cid:1) of C(ρAB). It is possible to do so analytically when ρAB
The projective measurements on the sub(cid:12)system B re- is of the form given in (7) resulting in the following ex-
moves all the non-classical correlations between the sub- pressions for the QD [28]:
systems AandB. HendersonandVedral[17]havequan-
tifiedthetotalclassicalcorrelationsC(ρ )bymaximiz-
AB
ing J ρAB, ΠBk w.r.t. ΠBk , i.e., Q(ρAB)=min{Q1,Q2} (12)
(cid:0) (cid:8) (cid:9)(cid:1) (cid:8) (cid:9)
C(ρ )= max J ρ , ΠB (5) where
AB AB k
{ΠBk}(cid:0) (cid:0) (cid:8) (cid:9)(cid:1)(cid:1) a
Q1 =S(ρB)−S(ρAB)−alog2 a+b
ThedifferencebetweenthetotalcorrelationsI(ρ )(Eq.
AB b d b
(2)) and the classical correlations C(ρAB) (Eq. (5)) de- −blog2 −dlog2 −blog2 (13)
a+b d+b d+b
fines the QD, Q(ρ ),
AB
and
Q(ρ )=I(ρ )−C(ρ ) (6)
AB AB AB
Q2 =S(ρB)−S(ρAB)−∆+log2∆+
−∆ log ∆ (14)
− 2 −
Though the concept of the QD is well-established, its
computationisrestrictedtomostlytwo-qubitstates. For with ∆± = 21(1±Γ) and Γ2 =(a−d)2+4(|z|+|f|)2
the two-qubit X states, analytic expressions for the QD For spin Hamiltonians with certain symmetries, the
can be derived when the density matrix, defined in the two-spin reduced density matrix ρ has a form similar
ij
computationalbasis{|11i,|10i,|01i,|00i},hasthefollow- to that given in Eq. (7) (the qubits A and B now repre-
4
sentthespinsatsitesiandj respectively)wherethema-
trix elements of ρij can be written in terms of single-site (a) 1 ||cc12||γγ==00..77
magnetizationandtwo-sitespincorrelationfunctions. In 0.8 |c1|γ=0.8
0.8 |c2|γ=0.8
tt[5hr,iex6c,eal1es1em]oenfttsheofXthYemtwood-eslitienraedturcaendsvdeernses-itfiyelmd,atthriexmaare- c(0)|2 0.6 Rc 000...246 γγ==00..78
a= 1 + hσzi + hσizσjzi (0)|, |1 0.4 0.2 0.4 0.6 0.8 1
4 2 4 c λ
|
1 hσzi hσzσzi
i j
d= − + 0.2
4 2 4
1
b= 1−hσzσzi
4 i j 0
0.2 0.4 0.6 0.8 1
(cid:0) (cid:1)
1 λ
z = hσxσxi+hσyσyi
4 i j i j 0.5
1(cid:0) (cid:1) (b) 1
f = hσxσxi−hσyσyi (15) 0.9
4 i j i j
(cid:0) (cid:1) 0.4 Rc 0.8
The finite temperature single-site magnetization hσzi in 0.7
[t5h,e1c1a]se of the transverse-field XY model is given by c(0)|2 0.3 00..56 λλ==00..78
hσzi=−1 πdφtanh βωφ (1+λcosφ) (16) c(0)|, |1 0.2 -1 -0.5 γ0 0.5 1
|
π Z0 (cid:18) 2 (cid:19) ωφ
where 0.1 ||||cccc1212||||λλλλ====0000....8877
ω = (γλsinφ)2+(1+λcosφ)2 (17) 0
φ q -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
γ
describes the energy spectrum and β = 1 with k being
kT
the Boltzman constant and T the absolute temperature. Figure 3. (Color online) (a) Variations of |c1(0)| and |c2(0)|
Thespin-spincorrelationfunctionsareobtainedfromthe with λ for γ = 0.7 (|c1(0)| solid line and |c2(0)| dashed line)
determinants of Toeplitz matrices [2, 3, 11] as and γ = 0.8 (|c1(0)| dotted line and |c2(0)| dot-dashed line).
(inset) Variation of Rc (=|c2(0)|/|c1(0)|) with λ for γ =0.7
G−1 G−2 ··· G−r (solid line) and γ = 0.8 (dot-dashed line). (b)Variations
(cid:12) G0 G−1 ··· G−r+1 (cid:12) of |c1(0)| and |c2(0)| with γ for λ = 0.7 (|c1(0)| dotted
(cid:10)σixσix+r(cid:11)=(cid:12)(cid:12)(cid:12)(cid:12) ... ... ... ... (cid:12)(cid:12)(cid:12)(cid:12) lliinnee aanndd ||cc22((00))|| ddoats-hdeadshleinde)l.ine()inasnedt)λVa=ria0t.i8on(|co1f(0R)c| swoiltidh
(cid:12) Gr−2 Gr−3 ··· G−1 (cid:12) γ for λ = 0.7 (dot-dashed line) and λ = 0.8 (solid line).
(cid:12) (cid:12)
(cid:12)(cid:12) G1 G0 ··· G−r+2 (cid:12)(cid:12) Rc = |c2(0)|/|c1(0)| for γ > 0 (BPF channel) and Rc =
(cid:12) G2 G1 ··· G−r+3 (cid:12) |c1(0)|/|c2(0)| for γ <0 (BF channel).
σiyσiy+r =(cid:12)(cid:12)(cid:12) ... ... ... ... (cid:12)(cid:12)(cid:12)
(cid:10) (cid:11) (cid:12) (cid:12)
(cid:12) Gr Gr−1 ··· G1 (cid:12)
(cid:12) (cid:12)
σizσiz+r =h(cid:12)(cid:12)σzi2−GrG−r (cid:12)(cid:12) (18)
(cid:10) (cid:11) system and environment respectively. We assume that
where
the environment is represented by L independent reser-
1 π βω (1+λcosφ) voirs each of which interacts locally with a qubit consti-
φ
G = dφtanh cos(rφ)
r π Z0 (cid:18) 2 (cid:19) ωφ tuting the XY chain. The reduced density matrix ob-
tained from Eq. (20) can be written as
γλ π βω sinφ
φ
− dφtanh sin(rφ) (19)
π Z0 (cid:18) 2 (cid:19) ωφ ρr(0)=ρrs(0)⊗ρre(0) (21)
with r=|i−j| being the distance between the two spins where ρ and ρ represent the two-qubit reduced den-
rs re
atthesitesiandj (forthenearest-neighbour(n.n.) case, sity matrix of the transverse-fieldXY chainand the cor-
r =1). responding reduced density matrix of the two-reservoir
environmentrespectively. The quantumchanneldescrib-
Wenextconsidertheinteractionofthechainofqubits,
ing the interaction between a qubit and its environment
each qubit representing a S = 1 spin in transverse-field
2 can be of various types [18, 29]. We first assume the two
XY model, with an environment. We choose the initial
qubits to represent nearest-neighbour spins in the XY
state (time t = 0) of the whole system at time t = 0 to
chain. In this paper, we investigate the dynamics of the
be of the product form, i.e.,
two-qubitclassicalandquantumcorrelations(intheform
of the QD) under the influence of the bit-flip (BF), bit-
ρ(0)=ρ (0)⊗ρ (0) (20)
s e
phase-flip (BPF) and phase-flip (PF) channels. In the
where the density matrices ρ and ρ correspond to the Krausoperator representation,an initial state, ρ (0), of
s e rs
5
the qubits evolves as [20, 29] 0.8
(a)
5
ρ (t)= E ρ (0)E† (22)
rs µ,ν rs µ,ν
4
Xµ,ν 0.6
λ
where the Kraus operators Eµ,ν = Eµ ⊗Eν satisfy the dp/dsc 23 γγ==00..78
completeness relation E E† = I for all t. The
BF, BPF and PF chanPneµl,sνdeµs,tνroyµ,tνhe information con- psc 0.4 1
tained in the phase relations without involving an ex- 0
0.2 0.4 0.6 0.8 1 1.2
changeofenergy. TheKrausoperatorsforthesechannels λ
are 0.2 γ=0.7
γ=0.8
1 0
E0 = q′(cid:18)0 1 (cid:19); E1 = p/2σi (23)
p p 0
0.2 0.4 0.6 0.8 1 1.2
λ
where i = x for the BF, i = y for the BPF and i = z
0.3
for the PF channel with q′ = 1 − p/2. In the case of 0.4
Markovian time evolution, p is given by p = 1 − e−θt 0.3 λλ==00..87 (i) (b)
with θ denoting the decay rate. A fuller description of γp/dsc 0.2
the channels and the corresponding dynamics can be ob- d
0.1
0.2
tainedfromRefs. [20,24,29]. Usingthetwo-spinreduced
0
densitymatrixdescribedbyEqs. (7)and(15)-(19)asthe -1 -0.5 0 0.5 1
c γ
initial state ρrs(0), one can calculate the time-evolved ps 0.2 (ii)
stateρrs(t)foraspecificquantumchannelusingtheEqs. 0.15 λ=0.8
(22) and (23). Since ρrs(t) has the same form as Eq. 0.1 psc 0.1 λ=0.7
(7), the mutual information, I(ρAB), the QD Q(ρAB) 0.05
and the classical correlations C(ρAB) can be computed 0
at any time t with the help of the formulae in Eqs. (8)- -1 -0.5 γ 0 0.5 λλ==00..78
(14). The mainresults ofour calculationsforthe various 0
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1
quantum channels are described in the following section. γ
Figure 4. (Color online)(a) Variation of psc as a function
of λ with γ = 0.7 (solid line) and γ = 0.8 (dot-dashed line).
III. RESULTS (inset)Thefirstderivativeofpsc w.r.t. λdivergesastheQCP
λc = 1 is approached. (b) Variation of psc as a function of
γ with λ = 0.7 (solid line) and λ = 0.8 (dot-dashed line) for
A. Nearest-neighbour quantum correlations at zero the BPF channel. (inset) (i) The first derivative of psc w.r.t.
temperature γ exists when γc > 0 for the BPF channel. (ii) Variation of
psc as functions of γ with λ = 0.7 (solid line) and λ = 0.8
(dot-dashed line) for the BF channel.
1. Bit-flip and bit-phase-flip channels
We first consider the case of T = 0 so that the XY
BPF Channel:
spin chain is in its ground state. In the case of the BPF
channel,thedynamicalevolutionsofthemutualinforma- c1(p)=c1(0)(1−p)2
tion I(ρ ), the classical correlations C(ρ ), the QD
AB AB c2(p)=c2(0)
cQ2(aρsAaBf)uanncdtiothneoafbtshoeluptaeravmaleutersizoefdthtiemceoepffi(pci=ent1s−c1e−aθntd) c3(p)=c3(0)(1−p)2
are shown in Fig. 2 for λ = γ = 0.7. The dynamics are c4(p)=c4(0)(1−p) (24)
similar to type (ii) where sudden changes in the decay
BF Channel:
rates of the classical correlations and the QD occur at a
parametrizedtime instantp . Inset (a)shows the varia-
sc c1(p)=c1(0)
tgiiovnesnobfy|cp1(p.)|Iannsedt|(cb2()ps)h|ovwerssuthsep.tiTmhee-ecvroolsustiinognspooifntthies c2(p)=c2(0)(1−p)2
sc
mutualinformation,theclassicalcorrelationsandtheQD c3(p)=c3(0)(1−p)2
for λ = 0.7, γ = −0.7 which are described by type (iii) c4(p)=c4(0)(1−p) (25)
dynamics, i.e., both C(ρ ) and Q(ρ ) decay mono-
AB AB
tonically. The dynamics in the case of the BF channel From Eqs. (9) and (15), the coefficients c1(0) and c2(0)
have an interesting correspondencewith the dynamics of areidentifiedasthetwo-spincorrelationfunctionshσixσjxi
the BPF channel. Type (ii) and Type (iii) dynamics are andhσyσyirespectively(inthepresentstudy,r =|i−j|=
i j
obtained in the case of the BPF (BF) channel for +ve (- 1). Fig. 3(a) shows the variation of the absolute values
ve) and -ve (+ve) values of γ respectively. The following of the coefficients c1(0) and c2(0) with λ for γ > 0. The
analysisprovidesuswithanunderstandingofthisresult. generalobservation is that for γ >0, |c1(0)|≥|c2(0)| for
The rules of evolution for the coefficients c1 and c2 are λ∈[0,1]. Fig. 3(b) shows the plots of |c1(0)| and |c2(0)|
obtained fromEqs. (7), (9), (22) and(23)in the cases of versus γ for fixed λ values. One finds that when γ is
the BPF and BF channels. These are given by <0, |c1(0)|≤|c2(0)| for λ∈[0,1]. Since |c1(0)|≥|c2(0)|
6
0.2 tive of p w.r.t. λ, dpsc (inset), as functions of λ
sc dλ
(cid:16) (cid:17)γ
for the BPF channel (γ > 0). The first derivative of p
sc
w.r.t. λ diverges as the quantum critical point (QCP)
ns 0.15 QCI(((ρρρAAABBB))) pλscc=wi1thisthaepparnoiascohtreodp.yFpiga.ra4m(ebt)ershγowfosrtthheevBarPiaFticohnano-f
elatio 0.1 γ=+/- 0.7, λ=0.7 ninedl.ivAidsuaelxlypetcetnedd,tposcze6=ro0asfoγr→γ >0 f0o.rBλo∈th[0|c,11|]aanndd |tch2e|
orr
C ratio Rc(=|c2(0)|/|c1(0)|) → 1 as γ → 0 (Fig. 3). The
inset (i) shows the variation of the first derivative of p
sc
0.05
w.r.t. γ, dpsc , asa function ofγ for fixedvalues ofλ.
dγ
(cid:16) (cid:17)λ
The first derivative of p w.r.t. γ shows a discontinuity
psc at the anisotropy transistcion point, γ = 0, indicating a
0
0 0.2 0.4 0.6 0.8 1
QPT. Inset (ii) shows the variation of p as a function
p sc
of γ for the BF channel (p 6= 0 for γ < 0). In a recent
sc
Figure 5. (Color online) Phase-flip channel: Decay of mutual study [24], wehaveinvestigatedthe Markovianevolution
informationI(ρAB)(solidline),classicalcorrelationsC(ρAB) of the classical and quantum correlations for a number
(dashedline)andquantumdiscordQ(ρAB)(dot-dashedline) of quantum channels in the case of the transverse-field
as a function of the parametrized time p = 1−e−θt for λ = Ising model, a special case (γ =1) of the transverse-field
0.7,γ =±0.7. In this case, psc =0.173. XY model considered in this paper. For the BPF chan-
nel, we have pointed out that the first derivative of the
parametrized time instant at which the sudden changes
in the decay rates of classical and quantum correlations
for γ > 0, Eq. (24), describing the rules of evolution
occur w.r.t. λ diverges as the Ising model critical point
fortheBPFchannel,indicatesthatinthiscasethereisa
λ = 1 is approached. We have further shown that the
c
certainparametrizedtimeinstantatwhich|c1(p)|crosses BPF(BF)channelexhibits onlytype (ii) (type (iii)) dy-
|c2(p)|(=|c2(0)|). This specific value turns outto be the namics. Inthe case ofthe XY model, positive and nega-
same as p at which a sudden change in the decay rates
sc tive values of the anisotropy parameter γ allow for both
of the classical and quantum correlations occur (Fig. 2).
type (ii) (γ < 0) and (iii) (γ > 0) dynamics for the BF
From the condition |c1(p)| = |c2(p)| and Eq. (24), one channel. Similarly, unlike in the case of the transverse-
obtains the following expressionfor p in the case ofthe
sc fieldIsing model, type (iii) dynamicsare possiblefor the
BPF channel:
BPF channel (γ <0).
p =1− R (26)
sc c
p
where Rc = |c2(0)|/|c1(0)|. The QD (BPF channel, 2. Phase-flip channel
γ >0)isgivenbyQ2inEq. (14)forp∈[0,1]. Atp=psc,
(c1(p)−c2(p)) or (c1(p)+c2(p)) changes sign depending ThevariationsofthemutualinformationI(ρ ),clas-
AB
on whether c1(0) and c2(0) have the same or different sical correlations C(ρ ) and QD Q(ρ ) versus the
AB AB
signsrespectively. Noticingthatz(p)=(c1(p)+c2(p))/4 parametrized time p for (λ = 0.7,γ = ±0.7) have been
and f(p) = (c1(p)−c2(p))/4 (Eq. (9)), the magnitude plotted in Fig. 5 in the case of the PF channel. The dy-
of Γ which involves |z| and |f| (Eq. (14)) changes at
namics is of type (ii) and there is a specific time interval
the point p = p , while S(ρ ) and S(ρ ) remain un-
sc B AB during which the QD is greater in magnitude than that
changed, thus bringing about the sudden change in the
of the classical correlations. The classical correlations
decay rates of the quantum correlations. With γ < 0,
remainconstantat a value I(ρ ) , the mutual infor-
|c1(0)| is ≤ |c2(0)| so that c1(p) never crosses c2(p) and mation of the fully decohered sAtaBtep,=i1n the parametrized
one obtains type (iii) dynamics. We next compare the
time interval p < p < 1. In the interval 0 < p < p ,
sc sc
rules of evolution for the BF channel (Eq. (25)) with
the classical correlations decay with time. On the other
those of the BPF channel (Eq. (24)). One finds that
hand,the quantumcorrelations,asmeasuredbytheQD,
the rulesofevolutionforc3(p)andc4(p)remainidentical undergoes a sudden change in its decay rate at p = p
sc
whilethoseforc1(p)andc2(p)areinterchanged. Thisex- and goes to zero in the asymptotic limit p → 1. One
plains the interchange of dynamics type mentioned ear-
must note that the type of the dynamics in the case of
lier: for γ > 0, since |c1(0)| is ≥ |c2(0)|, c1(p) never the PF channel is unchanged under a change in sign of
crosses c2(p) for the BF channel so that type (iii) dy- the anisotropy parameter γ, the reason of which can be
namics hold true whereas for γ <0, the crossing point is
easily understood by examining the evolution rules for
given by Rc = |c1(0)|/|c2(0)|. The effective interchange the coefficients c ’s from Eq. (7), (9), (22) and (23):
i
in the identities of c1(0) and c2(0) across the γ = 0
(critical XX) line is understood by remembering that c1(p)=c1(0)(1−p)2
fcu1n(0c)ti=onhσhσixxσσix+x1iianisdicn2t(e0rc)h=anhgσeiydσiyw+i1tih. hTσhyeσycorireulantdioenr c2(p)=c2(0)(1−p)2
the transforimai+ti1on γ → −γ (see Eqs. (18i)ia+n1d (19)). c3(p)=c3(0)
The variations of Rc with λ and γ are shown in the in- c4(p)=c4(0) (27)
sets of Fig. 3.
Both the coefficients c1 and c2 decay with time identi-
Fig. 4(a) shows the plots of p and the first deriva- cally in the case of the PF channelwhereas the other co-
sc
7
0.8
4 (a)
3 γ=0.8 0.5
0.6 λd γ=0.7 0.4
dp/sc 2 0.3
cc
sc 0.4 1 ppss 0.2
p
0.1
0
0.2 0.4 0.6 0.8 1 1.2
λ 01
0.2 γ=0.8 0.8
γ=0.7 γ 0.6
0.4 1
0.2 0.8
0 0.6 λ
0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0.4
λ
(a)
0.5
1
(b)
0.4 γdp/dsc 0. 50 λλ==00..87 scsc 000...345
0.3 -0.5 pp 0.2
c 0.1
ps -1-1 -0.5 0 0.5 1 0
0.2 γ 10.8
0.6
0.4
γ 0.2
0 1
0.1 -0.-20.4 0.8 0.9
λλ==00..87 -0.-60.8-1 0.6 0.7 λ
0
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 (b)
γ
Figure 6. (Color online)(a) Variation of psc as functions of Figure 7. (Color online) (a) Variation of psc as a function
λ with γ = 0.7 (solid line) and γ = 0.8 (dot-dashed line) in of the system parameters λ and γ for the BPF channel (b)
the case of the PF channel. (inset) The first derivative of Variationofpsc asafunctionofthesystemparametersλand
psc w.r.t. λ diverges as the QCP λc = 1 is approached. (b) γ for thePF channel.
Variationofpsc asfunctionsofγ withλ=0.7(solidline)and
λ = 0.8 (dashed line) for the PF channel. (inset) The first
derivative of psc as a function of γ becomes discontinuous at anisotropyparameterγ divergesastheQPTpointλ =1
c
the anisotropy transition point γc =0. is approached (although the results for positive γ values
only have been shown, identical result can be obtained
fornegativeγ valuesaswell). Fig. 6(b)demonstratesthe
efficients remain independent of time. Since the change variationofp versusγ forfixedλvalues. Thebehaviour
sc
in sign of the anisotropy parameter γ only interchanges ofp issymmetricacrosstheanisotropytransitionpoint
sc
the values of c1 and c2, it can not affect the dynamics γ =0. Onapproachingγ =0,psc goestozero. Theinset
that the classical and quantum correlations undergo in of Fig. 6(b) shows the variation of the first derivative of
the PF channel. In this case, the sudden change in the p w.r.t. γ at the QCP γ = 0. The landscapes of p
sc sc
decay rates of Q(ρAB) and C(ρAB) at p = psc is un- versus the inverse of the transverse-field strength λ and
derstood by noting that in the interval 0 < p < p , the the anisotropy parameter γ are shown in Fig. 7.
sc
QD is determined by Q(ρAB) = Q2 (Eq. (14)) and for
psc < p < 1, Q(ρAB) = Q1 (Eq. (13)). In the regime
p <p<1, C(ρ ) is given by
sc AB B. Further-neighbour quantum correlations at zero
temperature
a b
C(ρ )=S(ρ )+alog +blog
AB A 2 a+b 2 a+b
Wehavesofarconsideredthespinsinapairtoben.n.s.
d b
+dlog +blog (28) ThedynamicsoftheQDwhenthespinsareseparatedby
2 d+b 2 d+b
furtherneighbourdistancescanbeinvestigatedfollowing
The matrix elements a, b and d are functions of only c3 the procedureoutlined in Sec. II. The two qubit reduced
and c4 which are independent of time (Eq. (27)). From density matrix ρsc(0) (Eq. (22)) now involves further-
Eq. (11), S(ρA), being a function of c4 only, is also neighbour spin-spin correlation functions (Eqs. (18) and
independent of p, thereby ensuring the constancy of the (19) with r > 1). Fig. 8 shows the decay of the mutual
classical correlations C(ρ ). The time instant p can informationI(ρ )(solidline),the classicalcorrelations
AB sc AB
be determined as a solution of the equation Q1 = Q2. C(ρAB) (dashed line) and the QD Q(ρAB) (dot-dashed
Fig. 6(a) shows the variation of p as a function of the line) as a function of the parametrizedtime p=1−e−θt
sc
inverse of the strength of the transverse field, λ. The for λ = 0.7, γ = 0.7 in the cases of the BPF (r = 2 (a)
first derivative of p w.r.t. λ for a fixed value of the and r = 3 (b)) and the PF (r = 2 (c) and r = 3 (d))
sc
8
0.05 (a) λ=γ=0.7 QCI(((ρρρAAABBB))) 00..001146 (b) λ=γ=0.7 QCI(((ρρρAAABBB))) 5 (a) BPF Channel, γ=0.7 5 (b) PF Channel, γ=0.7
Correlations 000...000234 Correlations 000 0...000.00011682 Correlations 0000 0....0000.0000000000 012468 QCI(((ρρρAAABBB))) λdp/dsc 234 rr==23 λdp/dsc 234 rr==23
0.4 0.5 0.6 0.7 0.8 0.9 1
0.01 0.004 p 1 1
0.002
0 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 λ 0.8 1 1.2 0 0.2 0.4 0.6 λ 0.8 1 1.2
p p
Correlations 0000....00002345 (c) λ=γ=0.7 QCI(((ρρρAAABBB))) Correlations 00000 0.....00000.000111168246 (d) Correlationsλ=γ 0000= 0....0000.00000000000. 7012468 QCI(((ρρρAAAQCBBBI((()))ρρρAAABBB))) λFancigd=uPre1F9i(n.b(t)Chceohlaconarsneoesnlsloinwfeir)th=Tthh2eeadanindviesrrogt=ernop3ceyaonpfdardafdpomλsrcettahetertγBheP=FQ0C.(7aP)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 C. Quantum correlations at finite temperatures
0.01 0.004 p
0.002
0 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 ThedynamicsoftheQDatfinitetemperatures(T 6=0)
p p
and for different values of r are investigated using the
procedure described in Sec. II. For infinite chain spin
Figure 8. (Color online) Decay of the mutual information
I(ρAB)(solidline),theclassicalcorrelationsC(ρAB)(dashed models likethe XXZ chain(inthe presenceandabsence
line) and the QD Q(ρAB) (dot-dashed line) as a function of of magnetic field) andthe transverse-fieldXY chain, the
the parametrized time p = 1−e−θt for λ = 0.7, γ = 0.7 in thermal quantum discord (TQD) has been shown to de-
the cases of the BPF (r = 2 (a) and r = 3 (b)) and the PF tect the critical points of QPTs at finite temperatures
(r=2(c)andr=3(d))channels. Insetsof(b)and(d)show [12, 32, 33]. In fact, the TQD is a better indicator of
on amagnified scalethesuddenchangesinthedecayratesof QPTs than thermodynamic quantities and different en-
the correlations
tanglement measures. Fig. 11 displays our results when
system-environment interactions, which give rise to the
dynamics of quantum correlations, are taken into ac-
count. The first (second) column in the figure pertains
channels with r = 2 and 3 corresponding respectively to to the BPF (PF) channel. The finite temperature calcu-
second and third neighbour distances between the spins lations are carried out for different two-qubit distances,
in a pair. ComparingFig. 8 with Figs. 2 and 5 one finds e.g., r = 1,2 and 3. An examination of Fig. 11 shows
that the magnitude of psc in the case of the PF channel that at finite temperature, the first derivative ddpλsc no
is larger than that in the case of the BPF channel. Our longer diverges at the QCP λc = 1 but exhibits a maxi-
computation shows that this feature is obtained when mum which is shifted from the QCP. With increasing T
r =1irrespectiveofwhetherλis lessorgreaterthanthe and r, the shift is greater in magnitude and the maxi-
QCPλ =1. Forr>1,however,themagnitudeofp in mum becomes more broadened. The last row of Fig. 11
c sc
thecaseoftheBPFchannelislessthanthatinthecaseof shows the estimated QCP at T 6= 0 and for r = 1,2,3,
the PF channel only for λ < λ . At λ > λ , the reverse given by the λ value at which the maximum of the dpsc
c c dλ
trend is observed. Also, for the same type of channel, versusλplotoccurs. Inthe caseoftheBPFchannel,the
BPF or PF, the magnitude of psc shifts towards larger estimatedQCP is alwayslowerthan λc =1 for allvalues
valuesasr,theseparationbetweenthetwoqubits(spins) ofr whereasinthe caseofthe PFchannel,theestimated
increases. Fig. 9showsthedivergenceof dpsc attheQCP QCP lies above the λc =1 value for r ≥2.
dλ
λ = 1 for r = 2 and 3 and the BPF (a) and the PF(b)
c
channels with γ = 0.7. Maziero et. al. [30, 31] have
shown that in the case of the transverse-field XY chain, IV. CONCLUDING REMARKS
the QD for spin-pairs with r>1 is able to signal a QPT
whereas pairwise entanglement is non-existent for such Thetransverse-fieldXY modelin1danditsderivative
distances. They have further demonstrated a noticeable model, the transverse-field Ising model, are well-known
change in the decay rate of the QD as a function of r as spin models which exhibit QPTs. The models have been
the system crosses the critical point λ = 1. The decay extensively studied in recent years from the perspective
rate is slower for λ > 1, i.e., in the ferromagnetic phase. of quantum information theory to quantify the different
In the presence of system-environment interactions, the types of quantum correlations like bipartite/multipartite
QD evolves as a function of time. We select three time entanglement and quantum discord (QD), present in the
points(threedifferentpvalues)atwhichwecomputethe ground and thermal states of the system, as a function
QD for r = 1,2,3 and 4. Figs. 10 (a) and (c) show ofdifferentmodelparameters. Onesignificantfactwhich
the results for the BPF and PF channels respectively for emerges from these studies is that the quantum correla-
γ = 0.7 and λ = 0.7, i.e., λ < λ = 1. Figs. 10 (b) and tion measures can provide signatures of QPTs. Another
c
(d) show the corresponding plots when λ is > λ with interestinglineofstudyinvolvesprobingthedynamicsof
c
λ=1.1. We find that the decay of the QD as a function differenttypeofcorrelations,bothclassicalandquantum,
of r becomes considerably slower as the QCP λ = 1 is due to the inevitable interaction of a quantum system
c
crossed in the case of both the BPF and PF channels. withitsenvironmentresultingindecoherence. Inthispa-
9
0.07 0.08 2.5 2.5
0.06 (a) λ=0.7, γ=0.7 p=p=p0=.00.070.551 0.07 (b) λ=1.1, γ=0.7 p=pp=0=.000.70.155 2 (a) krT=1=0.01 2 (b) krT=1=0.01
0.05 0.06 r=2 r=2
QD 00..0034 QD 0.05 λdp/dsc 1 .15 r=3 λdp/dsc 1 .15 r=3
0.04
0.02
0.01 0.03 0.5 0.5
0 0.02 0 0
1 2 3 4 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2
r r λ λ
0.07 0.09
0.06 (c) λ=0.7, γ=0.7 p=p=p0=.00.070.551 0.08 (d) λ=1.1, γ=0.7 p=pp=0=.000.70.155 1 .12 (c) kT=0.1 1 .12 (d) kT=0.1
0.05
QD 00..0034 QD 00..0067 λdp/dsc 00..68 rrr===123 λdp/dsc 00..68 rrr===123
0.02 0.4 0.4
0.05
0.01 0.2 0.2
0 0.04
1 2 3 4 1 2 3 4 0 0
0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2
r r λ λ
Figure 10. (Color online) Decay of the QD as a function of 0.8 0.8
(e) kT=0.2 (f) kT=0.2
r at three parametrized time points (cid:0)p=1−e−θt(cid:1) for the 0.7 0.7
(a),(b) BPF and (c),(d) PF channels. For each channel, the 0.6 r=1 0.6 r=1
r=2 r=2
dTehceaypslvoawlusedsocwonnsciodnerseidderaarbelypa=s th0e.0Q5C(sPolλidc =squ1airsecs)r,ospsed=. λdp/dsc 000...345 r=3 λdp/dsc 000...345 r=3
0.075 (empty circle) and p = 0.1 (solid circle). Also λ = 0.7
0.2 0.2
((a),(c)), 1.1 ((b),(d))and γ =0.7
0.1 0.1
0 0
0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2
λ λ
per, we study the Markoviantime evolutionoftwo-qubit
1.1
statesintheKrausoperatorformalismduetothelocalin- 1.05 rr==12 (g) 1.1 rr==12 (h)
r=3 r=3
teractions of the qubits with independent environments. CP 1 CP 1.05
Trehdeuctwedo-dqeunbsiittystmataetirsixdeosfctrhibeetdrabnystvheersnee-afireeldst-XneYigmhbooduerl mated Q 0 0.9.95 mated Q 1
ground state in 1d. The quantum channels, represent- Esti 0.85 Esti
0.95
ing system-environment interactions, are of three types: 0.8
bit-flip, bit-phase-flip and phase-flip. We investigate the 0.75 0.9
0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2
dynamics of the mutual information I(ρ ), a measure kT kT
AB
of the total correlations in the system, the classical cor-
relations C(ρ ) and the QD, Q(ρ ) as a function of
AB AB Figure 11. (Color online) The variation of dpsc versus λ for
the parametrized time p. An earlier study [21] identified dλ
r=1 (solid line), r=2 (dotted line) and r=3 (dashed line)
threedifferenttypesofdynamicsforadifferentsetofini-
at temperatures kT = 0.01,0.1 and 0.2. The first (second)
tial states of which only the type (ii) and (iii) dynamics
column refers to the BPF (PF) channel. In (g) and (h) the
are observed in the present study. A significant result
estimated QCPs are plotted against kT for the BPF (g) and
of our study is to identify quantities, ddpλsc and ddpγsc, as- PF (h) channels respectively and for r = 1 (solid squares),
sociated with the type (ii) dynamics which diverges and r=2 (empty circles) and r=3 (solid circles) with γ =1.
becomes discontinuous, respectively, as a quantum criti-
cal point is approached. The discontinuity occurs as the
critical point γ = 0 (λ ∈ [0,1]) of the anisotropy tran- Ising model are characterized by their specific type of
sition is approached and the divergence is obtained as dynamics [24] whereas in the case of the XY model, the
the quantum critical point λ = 1 is approached. The dynamics type changes across the γ = 0 line. We have
two different types of critical point transitions belong to further extended our study to finite temperatures and to
two different universality classes. We find an interest- r (the distance of separation between the two qubits de-
ing correspondence between the dynamics of bit-flip and scribed by the reduced density matrix) beyond the n.n.
bit-phase-flip channels, namely, the dynamics type is in- distance. In the latter case for T = 0, one finds that
terchanged across the γ = 0 line. We provide an expla- the derivatives dpsc diverges at the QCP λ = 1 as in
dλ c
nation for this feature and obtain an analytic expression the case of r = 1. We have further demonstrated that
for p , the parametrized time point at which a sudden at a specific time point, the decay of the QD as a func-
sc
change in the decay rates of the classical and quantum tion of r becomes significantly slower as the QCP λ =1
c
correlations occur. The transverse-field XY model has is crossed. We have obtained the results for the BPF
a phase diagram richer than that of the transverse-field and the PF channels but the same feature holds true for
Isingmodel. Thisisreflectedinthevarieddynamicspos- other channels like the amplitude damping channel. We
sible for a single quantum channel in the first case. The have checked that the decay rate decreases continuously
BPF and BF channels in the case of the transverse-field as the QCP is approached. At finite temperatures, the
10
TQD is able to provide estimates of critical points asso- however,obtained any evidence of the type (i) dynamics
ciated with QPTs, this is true for both n.n. as well as mentioned in Sec. I. In the present study, we have con-
further-neighbourqubitsthoughhighertemperaturesand sideredthetimeevolutionofthequantumcorrelationsto
larger values of r smear out the signatures. The general be Markovian in nature, an issue of significant interest
trendsobtainedaresimilartothosederivedinthecaseof would be to look for signatures of QPTs in the case of
isolated systems, i.e., when system-environment interac- non-Markoviantime evolution.
tions are ignored. In all our computations, we have not,
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