Role of mixedness in the evolution of classical and quantum correlations under Markovian decoherence

Abstract

Quantum correlation lies at the very heart of almost all the non-classical
phenomena exhibited by quantum systems composed of more than one subsystem. In
the recent days it has been pointed out that there exists quantum correlation,
namely discord which is more general than entanglement. Some authors have
investigated that for certain initial states the quantum correlations as well
as classical correlation exhibit sudden change under simple Markovian noise. We
show that, this dy- namical behavior of the both types of correlations can be
explained using the idea of complementary correlations introduced in
[arXiv:1408.6851]. We also show that though certain class of mixed en- tangled
states can resist the monotonic decay of quantum correlations,it is not true
for all mixed states. Moreover, pure entangled states of two qubits will never
exhibit such sudden change.

Full text

Role of mixedness in the evolution of classical and quantum correlations under
Markovian decoherence
Prasenjit Deb1, ∗and Manik Banik2 , †
1Department of Physics and Center for Astroparticle Physics and Space Science,
Bose Institute, Bidhan Nagar Kolkata - 700091, India.
2Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India.
Quantum correlation lies at the very heart of almost all the non-classical phenomena exhibited
by quantum systems composed of more than one subsystem. In the recent days it has been pointed
out that there exists quantum correlation, namely discord which is more general than entanglement.
Some authors have investigated that for certain initial states the quantum correlations as well as
classical correlation exhibit sudden change under simple Markovian noise. We show that, this dy-
namical behavior of the both types of correlations can be explained using the idea of complementary
correlations introduced in [arXiv:1408.6851]. We also show that though certain class of mixed en-
tangled states can resist the monotonic decay of quantum correlations,it is not true for all mixed
states. Moreover, pure entangled states of two qubits will never exhibit such sudden change.
PACS numbers:
I. INTRODUCTION
Quantum information processing protocols [1,2] re-
quire resources which are quantum in nature. In the
last few years it has been proved that different kind of
non-classical correlations are necessary resources for per-
forming various quantum-information-processing tasks.
One of the most important non-classical correlations is
quantum entanglement [3–5], which is a central area of
research in quantum information science for a long pe-
riod of time. Various information theoretic tasks such
as quantum teleportation [1], quantum dense coding [2],
quantum key distribution,state merging [6] can be per-
formed in presence of entanglement. However, there ex-
ists another kind of quantum correlation called discord
[7] which is more general than quantum entanglement.
Unlike entanglement, quantum discord has received at-
tention during the last few years and it has been proved
to be an useful non classical resource. Researchers have
shown that discord provides speedup in performing some
tasks in a non-universal model of quantum computation
[8]. Some other operational significance of discord have
been also pointed out by others [9–14]. On the other
hand, quantum mutual information is the information-
theoretic measure of the total correlation in a bipartite
quantum state. Groisman [15], Schumacher and West-
ermoreland [16] showed the significance of quantum mu-
tual information, which can be thought as the sum of
quantum and classical correlation [17]. In most of the
ideal cases it is assumed that quantum systems are iso-
lated from environment and one can use the unitary evo-
lution to illustrate the dynamics of such systems. Un-
fortunately, during the practical applications the quan-
tum systems interact with environment resulting in loss
∗Electronic address: devprasen@gmail.com
†Electronic address: manik11ju@gmail.com
of quantum coherence which in turn destroys the quan-
tum correlations. This destruction of quantum proper-
ties by the inevitable interaction with the environment
is perhaps the major hindrance to the development of
quantum technologies till date. Recently several studies
revealed the dynamics of quantum and classical correla-
tions under both Markovian [18–21] and non-Markovian
[22] decoherence. Interestingly, contrary to the entangle-
ment dynamics where sudden death may occur [23,24],
quantum correlation measured by quantum discord dose
not exhibit this behavior. However sudden change may
occur in the decoherence process. More specifically, for
certain class of states e.g Bell diagonal states, discord re-
mains constant for a particular period of time and then
decays, while classical correlation decays first and then
becomes constant [20].
In this paper we have focused mainly on two questions:
(1) What are the underlying physical mechanisms for
which classical and quantum correlations (mea-
sured in terms of discord) suffer sudden change in
the decoherence process as studied by Mazzola et.
al [20]?
(2) Does mixedness provide some advantage to prevent
the loss of quantum coherence, and hence quantum
correlations?
To answer the 1st question we have used the idea of
complementary correlations introduced in [25,26] and we
have succeed to describe the physical mechanism going
under the phenomena observed in [20]. More precisely,
we have shown that under some restrictions for certain
class of bipartite qubit states the amount of classical cor-
relation and quantum discord are exactly equal to the
correlations between the complementary observables of
two sides. We have also found out the initial class of
states exhibiting such behavior. With a lot of surprise,
we have found that the two qubit pure entangled states
will never show the said behavior. Rather, some par-
ticular mixture of pure states exhibit sudden change in
arXiv:1411.3160v1 [quant-ph] 12 Nov 2014

2
the classical and quantum correlations in the decoherence
process which answers the 2nd question affirmatively.
The organization of the paper is as follows: Section (II)
contains a brief overview on quantum discord and mutual
infomation; in section (III) we give a brief description on
complementary correlations and discuss about correla-
tion tensor matrix; Markovian decoherence and quantum
channels are discussed briefly in section (IV); we present
our results in section (V) and Section (VI) contains dis-
cussions and concluding remarks.
II. DISCORD AND MUTUAL INFORMATION
Entanglement is perhaps the most familiar non-
classical correlation observed in quantum systems com-
posed with more than one subsystem. But, it is impor-
tant to note that some states with zero entanglement
can perform tasks which are not possible in the classi-
cal regime. It is due to the fact that those states have
nonclassical correlation even though they are unentan-
gled. Apart from entanglement, the measure of quantum
correlation that has received a great deal of attention
is Quantum Discord (D), originally proposed by Ollivier
and Zurek [7]. The quantum discord is defined as:
D(ρAB )≡ I(ρAB )− C(ρAB ),(1)
where, C(ρAB ) and I(ρAB ) are respectively the classical
correlations and quantum mutual information of the bi-
partite state ρAB . Quantum mutual information I(ρAB )
measures the total correlation present in the state ρAB,
and it is defined as:
I(ρAB ) = S(ρA) + S(ρB)−S(ρAB ),(2)
where, ρAand ρBare the reduced density matrix of
the subsystems Aand Brespectively and S(ρ) =
−Tr{ρlog2ρ}is the von Neumann entropy. From the
above definition of mutual information it is clear that one
can straightforwardly calculate it for a given state ρAB .
The maximum value of I(ρAB ) is 2 log2d, achieved by
two-qudit maximally entangled states and it’s minimum
value is zero, achieved by product states.
On the other hand, classical correlations, C(ρAB ) of
a composite quantum state can be quantified via the
measure proposed by Henderson and Vedral [17] which
is given by:
C(ρAB )≡max
{Πj}[S(ρA)−S{Πj}(ρA|B)],(3)
where, the maximum is taken over the set of projective
measurements {Πj}on subsystem B.S{Πj}(ρA|B) =
ΣjpjS(ρj
A) is the entropy of subsystem Aconditioned
on B,ρj
A=TrB(ΠjρAB Πj)/pjis the density matrix of
subsystem Adepending on the measurement outcome of
Band pj= TrAB (ρAB Πj) is the probability of jth out-
come. Note that, unlike I(ρAB ), the classical correlation
is asymmetric with respect to the subsystems involved
and so is D(ρAB ).
As the definition of classical correlation requires opti-
mization over all possible projective measurements (more
generally over the positive operator valued measurements
[27]) that can be performed at one part of the composite
system, it is in general very hard to find the amount of
quantum discord for arbitrary bipartite state. To avoid
this complexity different computably easy measures of
quantum discord have been proposed recently [28–30].
Few important properties of quantum discord are the fol-
lowing:
(a) D(ρAB )≥0,∀ρAB .
The proof is straight forward. Putting the explicit
form of I(ρAB ) and C(ρAB ) in Eq.(1), we get
D(ρAB ) = S{Πj}(ρA|B)−S(ρA|B),(4)
where, S(ρA|B) = S(ρAB )−S(ρB) is the quan-
tum conditional entropy. Finding the value
of S{Πj}(ρA|B) involves measurement on B side
which typically increases the entropy and hence
S{Πj}(ρA|B)≥S(ρAB )−S(ρB), which completes
the proof. If we put our concern on quantum con-
ditional density matrix ρA|B, then from ref. [31] it
can be said that this conditional density matrix re-
tains the quantum phases and coherence. So phys-
ically quantum discord is a measure of information
which cannot be extracted without joint measure-
ments [7].
(b) D(ρAB ) = 0 for quantum-classical (QC) states
which are of the form:
ρQC = Σipiρi
A⊗ |iBihiB|,(5)
where {|iBi} is an orthonormal basis for subsys-
tem B, and ρi
A’s are density matrices of subsystem
Aand {pi}is a probability distribution. For two-
qudit maximally entangled states D(ρAB) = log2d.
(c) Quantum discord is non-increasing under com-
pletely positive trace preserving (CPTP) maps on
unmeasured party A[32], i.e,
D(ρAB )≥ D([ΛA⊗IB]ρAB ),(6)
where, ΛAbeing the CPTP map on A[33].
III. COMPLEMENTARY CORRELATIONS AND
CORRELATION TENSOR MATRIX
Complementary correlations: The concept of com-
plementary correlations is recently introduced in ref.[25].
Consider a quantum mechanical system described by
d-dimensional Hilbert space. Let Mand Nare two
observables acting on the system with {|mii}d
i=1 and

3
FIG. 1: (Color on-line) System 1 and System 2 are the sub-
systems of some composite quantum system. Aand Care the
complementary observables for system 1, while Band Dare
the same complementary observables for system 2
{|nji}d
j=1 denoting the non-degenerate eigenstates, re-
spectively. Mand Nare called complementary observ-
ables, if |hmi|nji|2=1
d, for all i, j. It means that if one
knows the value of one of the complementary observables
i.e., if the system is prepared in one of the eigenstates of
one of the complementary observables then all the possi-
ble values of the other observable are equal probable.
The authors of [25] have shown that, for a composite
quantum system the correlations in the measurement of
such complementary observables is a good signature of
quantum correlations present in the state. If two quan-
tum systems of finite dimension are considered and two
observables A ⊗ B and C ⊗ D are taken into account,
where Aand Care complementary observables on one
subsystem and Band Don the other, then the quantity
|χAB|+|χCD|denotes the value of complementary corre-
lations with |χAB|and |χCD|denoting the absolute value
of correlations on complementary observables. The sum
not only tells about the quantum correlations present in
a composite quantum system but it also represents the
overall correlations of the composite system.
Consider that χAB =IAB and χAB =ICD, where I
is the mutual information defined earlier and having an
alternate definition as:
IAB ≡H(A)−H(A|B),(7)
where, H(A) is the Shannon entropy of the outcome
probabilities of the measurement Aperformed on the
first system and H(A|B) denotes the conditional entropy,
conditioning being done on second system. Therefore in
terms of mutual information the complementary correla-
tions reads as: IAB +ICD. It can be easily shown that:
(i) If IAB +ICD = 2 log2d, then the bipartite quan-
tum system is maximally entangled and there exists
two complementary measurement bases or in other
words mutually unbiased bases (MUBs).
(ii) If IAB +ICD >log2d, then there is entanglement
in the composite system.
(iii) If IAB +ICD = log2d, the bipartite state is a clas-
sically correlated (CC) state which belongs to the
set of separable states and the quantum correla-
tions for such states is zero.
Correlation tensor matrix: Here we concentrate on
two-qubit quantum system defined on the Hilbert space
H=C2⊗C2. The collection of Hermitian operators act-
ing on Hconstitute a inner product space with Hilbert-
Schmidt inner product defined as hα, βi=Tr(α†β), where
αand βare Hermitian operators acting on H. In such
a Hilbert-Schmidt space any generic state of the system
can be expressed as [34]:
ρ=1
4(12⊗12+a·σ⊗12+12⊗b·σ+
3
X
m,n=1
cnmσn⊗σm)
(8)
where, 12is the identity operator, aand bdenote the lo-
cal Bloch vectors for each subsystem and {σn}3
n=1 are the
standard Pauli spinors σx,σyand σz. The 3 ×3 matrix
Tformed by the coefficients cnm is called the correlation
tensor matrix as it is responsible for the correlations:
E(a,b)≡Tr(ρa·σ⊗b·σ) = (a,Tb).(9)
Note that cnm = Tr(ρσn⊗σm) are the expectation values
of the observables σn⊗σm. The state ρ, as expressed in
Eq.(8) can always be transformed to a state eρfor which
the matrix Tbecomes diagonal by acting local unitaries
U1and U2. Though the unitaries transform the state ρ
to ˜ρ, the inseparability (separability) remains invariant.
The transformed state can be represented as
eρ=U1⊗U2ρU†
1⊗U†
2.(10)
The transformation of the state ρis possible due to
the fact that for any unitary transformation Uthere is
always a unique rotation Osuch that Uˆn·σU†= (Oˆn)·σ
and the parameters a,band Ttransform themselves as
a0=O1a,b0=O1band T0=O1TO†
2respectively, where
a0,b0and T0are the new parameters for the state eρ.
As the unitaries U1and U2diagonalize the correlation
tensor matrix, hence we have,
T0=
c10 0
0c20
0 0 c3

,(11)
where, c1= Tr(ρσx⊗σx), c2= Tr(ρσy⊗σy), c3=
Tr(ρσz⊗σz), are the expectation values of the observ-
ables σx⊗σx,σy⊗σyand σz⊗σz, respectively.
IV. MARKOVIAN DECOHERENCE AND
QUANTUM CHANNELS
Markovian decoherence: Decoherence is a phys-
ical process which describe the gradual loss of coher-
ence present in any quantum system [35]. The pro-
cess can be represented by some family of linear maps

4
{Λ(t2,t1), t2≥t1≥t0}, where, t0,t1and t2denote time
[36]. If the linear map Λ(t2,t1)satisfies following three
properties i.e.,
(i) trace-preserving i.e., Tr(ρ) = Tr(Λ[ρ]),
(ii) completely positive i.e., Λ ⊗1Cnis positive for all
n, where 1Cndenote identity map acting on n-
dimensional Hilbert space and
(iii) Λ(t3,t1)= Λ(t3,t2)Λ(t2,t1),
then the decoherence process is called Markovian deco-
herence [36]. The linear map Λ(.) basically describes the
time evolution of the quantum system that interacts with
its environment. The above mentioned conditions, which
a linear map should follow to represent a Markovian pro-
cess comes from a very useful theorem, namely the Kraus
representation theorem [37] which provides the following
operator sum representation for any CPTP map,
Λ(A) =
r
X
j=1
ΓjAΓ†
j(12)
where, Γjare the Kraus operators and ris the Kraus
rank which determines the number of Kraus operators in
the operator - sum representation of the linear map Λ(.).
The normalization principle leads to the fact that Λ(.) is
trace preserving iff PjÆ
jΓj= 1.
Quantum Channels: In any communication proto-
col one has to send information through channels. If the
information is quantum in nature, then the time evo-
lution of the quantum system carrying the information
can be modelled by quantum channels. Mathematically,
quantum channels are some superoperators or CPTP lin-
ear maps having the operator-sum representation. The
three important classes of quantum channels are depo-
larizing channel, amplitude damping channel and phase
damping channel.
(I) Depolarizing channel: The Kraus operators
which represent the depolarizing channel are:
Γ0=p1−p I, Γ1=rp
3σ1,
Γ2=rp
3σ2,Γ3=rp
3σ3,
where, prepresents probability. Under depolarizing
channel any general density matrix ρevolves as:
ρ→ρ0= (1 −p)ρ+p
3(σ1ρσ1
+σ2ρσ2+σ3ρσ3).
(II) Amplitude damping channel: Kraus opera-
tors representing amplitude damping channel as a
schematic model of the decay of an excited state of
a two state quantum system are as follows:
Γ0=1 0
0√1−p,Γ1=0√p
0 0 .
The evolution a density matrix ρis given by:
ρ→Λ(ρ)=Γ0ρΓ†
0+ Γ1ρΓ†
1.
(III) Phase damping channel: Caricaturing of deco-
herence process in realistic physical situations is
possible by phase damping channel. Though there
are phenomenological models leading to decoher-
ence, but none of the models represent the real
physical situations. Nevertheless, using operator-
sum representation the evolution of a quantum
state under phase damping channel can easily be
understood. The Kraus operators required to rep-
resent phase-damping channel are
Γ0=r1−p
2I, Γ1=rp
2σ3.
The most general single-qubit density matrix can
be written as:
ρ=ρ00 ρ01
ρ10 ρ11
where, the diagonal real elements represent the
probabilities of finding the qubit in the state |0i
or |1i, respectively, if measurement is done in σz
basis. The off-diagonal elements (quantum coher-
ences) have no classical analogue and the phase-
damping channel induces a decay of those elements
resulting in decoherence. The single-qubit density
matrix evolves as
Λ(ρ)=Γ0ρΓ†
0+ Γ1ρΓ†
1
= (1 −p
2)ρ+p
2ρ
=ρ00 ρ01(1 −p)
ρ01(1 −p)ρ11 .(13)
It is clear from the above equation that the off-
diagonal terms will gradually decay as the time
elapse and the initial coherent superposition will
turn into incoherent superposition or mixture, i.e,
ρ→ρ0=|ρ00|2|0ih0|+|ρ11 |2|1ih1|.
The phase-damping channel plays the central role
in the transition from the quantum to the classical
world. The decay of the off-diagonal terms and
hence decoherence can be well understood if we
consider the interaction of the qubit with the envi-
ronment as a rotation (phase-kick) about z-axis of
the Bloch sphere through an angle θ, due to which
the axes transform as x0=e−λx,y0=e−λyand
z0=z, where λis the damping parameter. In other
words it can be said that the channel picks out a
prefered basis for the qubit, which is {|0i,|1i}, as
z-basis is the only one in which bit flip never occurs.

5
V. RESULTS
A. Complementary correlations and decoherence
Consider a generic two qubit bipartite state ρAB as
expressed in Eq.(8). Applying local unitaries let us diag-
onalize the correlation tensor matrix and transform the
state ρAB to ρ0
AB so that
ρ0
AB =1
4(12⊗12+a·σ⊗12+12⊗b·σ+
3
X
n=1
cnσn⊗σn).
(14)
For simplicity we consider the states with maximally
mixed marginals, i.e, a=0and b=0. Thus we have:
ρ0
AB =1
4(12⊗12+
3
X
n=1
cnσn⊗σn),(15)
here, cn’s are the diagonal elements of correlation tensor
matrix Tand 0 ≤ |cn| ≤ 1. The class of states rep-
resented by ρ0
AB in Eq.(15) are called the Bell diagonal
states, which includes pure Bell states (|c1|=|c2|=|c3|=
1) and Werner class of states (|c1|=|c2|=|c3|= c) [20].
When both the subsystems of the composite state of
Eq.(15) are subjected to local Markovian noise, the time
evolution of the composite state is given by:
ρ0
AB (t) = λ+
Ψ(t)|Ψ+ihΨ+|+λ+
Φ(t)|Φ+ihΦ+|
+λ−
Φ(t)|Φ−ihΦ−|+λ−
Ψ(t)|Ψ−ihΨ−|,(16)
where,
λ±
Ψ(t) = 1
4[1 ±c1(t)∓c2(t) + c3(t)],(17)
λ±
Φ(t) = 1
4[1 ±c1(t)±c2(t)−c3(t)],(18)
and |Φ±i=1
√2(|00i±|11i),|Ψ±i=1
√2(|01i ± |10i) are
Bell states.
If we consider phase damping channel as the local
Markovian noise, then the co-efficients in Eq.(17) and
Eq.(18) will be:
c1(t) = c1(0)e−2γt ,
c2(t) = c2(0)e−2γt ,
c3(t) = c3(0) ≡c3,(19)
where γis the phase damping rate. For our analysis
we consider the initial states as c1(0) = ±1 and c2(0) =
∓c3(0) with the condition |c3|≤ 1 , as considered by other
authors [20] also. Thus the states read as:
ρAB =(1 + c3)
2|Ψ±ihΨ±|+(1 −c3)
2|Φ±ihΦ±|.(20)
The subsystems of the above state are qubit and σxand
σzare complementary observables for a qubit quantum
system. From the definition of complementary correla-
tions, the total complementary correlations is therefore:
Ic[ρAB (t)] = I(σA
x:σB
x) + I(σA
z:σB
z),(21)
where, the superscript “c” signifies complementarity.
The first term on the right hand side of the Eq.(21) de-
notes the correlation between the outcomes of σxmea-
surement performed on both sides, similarly second term
denotes the same, but for σz. Using Eq.(7) we have:
I(σA
x:σB
x) = P[c1(t)] + P[−c1(t)],(22)
I(σA
z:σB
z) = P[c3] + P[−c3],(23)
where P[α] = 1+α
2log2(1 + α). Inserting Eq.(22)-(23) in
Eq.(21) we get:
Ic[ρAB (t)] = P[c1(t)] + P[−c1(t)] + P[c3]
+P[−c3].(24)
Interestingly, for the concerned class of states Ic[ρAB(t)]
is exactly equal to the mutual information (I) of the state
ρAB (t), i.e.,
Ic[ρAB (t)] = I(ρAB (t)).(25)
On the other hand, the classical correlation C(ρAB (t))
in this case turns out to be:
C(ρAB (t)) = P[K(t)] + P[−K(t)],(26)
where K(t) = max{|c1(t)|,|c2(t)|,|c3(t)|}. It is to be
noted that the coefficients c1(t), c2(t) and c3(t) are the
expectation values of σ1⊗σ1,σ2⊗σ2and σ3⊗σ3respec-
tively. So, during the calculation of classical correlations
of the specific states considered, the conditional entropy
in the Eq.(3) reaches the minimum when the projective
measurements are performed on eigenstate of that com-
plementary observable σ(B)
nfor which Tr(ρABσ(A)
n⊗σ(B)
n)
is maximum. Hence, we conclude that, for the class
of states taken into consideration, σx,σy,σzform a set
of complementary observables and classical correlation
C(ρAB (t)) is
C(ρAB (t)) = max
l∈x,y,z[I(σA
l:σB
l)].(27)
For our purpose we assume,
I(ρAB ) = Q(ρAB ) + C(ρAB ),(28)
which in turn yields
Ic[ρAB (t)] = Q(ρAB ) + C(ρAB ).(29)
We are now in a position to explain the sudden transi-
tion in classical and quantum decoherence for the states
considered in Eq.(20).
(i) At time t= 0, Tr(ρAB σ(A)
1⊗σ(B)
1) = c1(0) = 1.
So projective measurements on eigen-states of σxwill
yield minimum conditional entropy S(A|B) and hence

6
the amount of classical correlations, which is equal to the
correlation between the measurement outcomes of σxon
both sides of the bipartite state reads as
C(ρAB (t0)) = I(σA
x:σB
x)
= 2P[c1(t0)].(30)
From Eqs. (1), (23) and (28), the value of quantum cor-
relations or discord comes out to be
D(ρAB (t0)) = I(σA
z:σB
z)
= 2P[c3].(31)
(ii) In the time interval 0 < t < t0=−ln(|c3|)/2γ,
c1(t)> c3. So due to the same reasons as described
above, the classical correlations(C) and discord(D) of the
initial state will be
C(ρAB (t)) = I(σA
x:σB
x)
=P[c1(t)] + P[−c1(t)],(32)
D(ρAB (t)) = I(σA
z:σB
z)
=P[c3] + P[−c3].(33)
However, in this time interval the classical correlation
C(ρAB (t)) decays and discord D(ρAB(t)) remains con-
stant. In other words we can say that the correlations
I(σA
x:σB
x) decay and correlations I(σA
z:σB
z) remain
constant. The physical origin of this fact is that at t > 0
the phase-damping channel induces a decay in the quan-
tum coherence(phase) of the state ρAB (t), which results a
decay in the expectation value c1(t) =Tr[ρAB(t)σA
1⊗σB
1],
whereas the expectation value c3(t) =Tr[ρAB(t)σA
3⊗σB
3]
remains constant. The phase-damping channel picks the
{|0i,|1i} basis as preferred basis for each qubit and de-
stroys all other superpositions of |0iand |1i, resulting in
the decay of c1(t) and hence C[ρAB(t)]. Here the discord
remains constant.
(iii) For t > t0,c1(t)< c3. So, in this case the con-
ditional entropy S(A|B) will be minimum if projective
measurements are performed on eigenstates of σz. Thus
the classical correlation will be:
C(ρAB (t)) = I(σA
z:σB
z)
=P[c3] + P[−c3].(34)
From Eq.(1), (22) and (28), discord of ρAB (t) will be:
D(ρAB (t)) = I(σA
x:σB
x)
=P[c1(t)] + P[−c1(t)].(35)
Hence, for t>t0, the classical correlation is constant
in time, whereas, discord starts to decay. To under-
stand this sudden change in the evolution of classical
and quantum correlations for t>t0, we focus on the
definition of classical correlations and notice that dur-
ing this time projective measurement on the eigenstates
of σzwill yield the classical correlations, i.e., only the
term I(σA
z:σB
z) contributes in C(ρAB(t)). Now, as the
channel is phase damping channel, the expectation value
c3does not change with time and as a result classical
correlation becomes constant after time t0.
On the other hand, during this time the correlation
between measurement outcomes of another complemen-
tary observable σxrepresents the amount of D(ρAB(t)),
present in the state ρAB(t). Whereas, it is mentioned
before that the expectation value txdecays due to the ef-
fect of phase daming noise. Therefore, D(ρAB (t)) decays
after time t0.
B. Mixedness and decoherence
We now focus on two questions.(i) Will the pure entan-
gled two qubit states exhibit sudden change in the evo-
lution of C(ρAB (t)) and D(ρAB (t)) ? (ii) Does mixedness
will always ensure sudden change in decoherence process?
To answer the first question, we consider a pure two qubit
entangled state
|ΨAB i=a|00i+b|11i(36)
where, |a|2and |b|2are probabilities. This state belongs
to the class of states for which the correlation tensor ma-
trix (T) is diagonal. The three diagonal elements or the
expectation values of the observables σ1⊗σ1,σ2⊗σ2and
σ3⊗σ3are found to be,
c1= (|a|2+|b|2)=1,
c2=−(|a|2+|b|2) = −1,
c3= (|a|2+|b|2)=1.(37)
If both the qubits of the state are subjected to local phase
damping channel, then all the ci’s change as Eq.(19). The
classical correlation C(ρAB (t)) and discord D(ρAB (T)) of
the state for t > 0 read as:
C(ρAB (t)) = I(σA
z:σB
z)
=P[c3] + P[−c3]=1,(38)
D(ρAB (t)) = I(σA
x:σB
x)
=P[c1(t)] + P[−c1(t)]
=P[e−2γt ] + P[−e−2γt ].(39)
From the Eq.(38) and (39) it is clear that C(ρAB (t)) will
always remain constant, while D(ρAB (t)) will decay from
the beginning, which means there will be no such sudden
transition in the evolution of classical correlations and
discord. So, we can conclude that pure two-qubit entan-
gled states will never show sudden transition.
To address our 2nd question, we first consider two qubit
Werner class of states, which is represented as:
ρAB =β|ψ−ihψ−|+(1 −β)14
4,(40)
where, βrepresents the singlet fraction and (1 −β) rep-
resents random fraction. Simple calculations show that
for such states the diagonal elements of the correla-
tion tensor matrix are all equal, i.e, |c1|=|c2|=|c3|=

7
constant = k(say). Therefore, for such states C(ρAB(t))
and D(ρAB (t)) are found to be:
C(ρAB (t)) = I(σA
z:σB
z)
=P[c3] + P[−c3]
=P[k] + P[−k],(41)
D(ρAB (t)) = 1
2(P[k+ 2k.e−2γt ]
+P[k−2k.e−2γt ]−2P[k]).(42)
Hence, for phase damping noise classical correlations for
FIG. 2: (Color on-line) Evolution of Dunder phase-damping
channel for Werner class of states.
this class of states will remain constant while the discord
will gradually decay without showing any kind of sudden
transition. It is thus confirmed that mixedness is not the
only factor for sudden transition in classical and quantum
decoherence. Along with mixed nature of the bipartite
state ρAB , it is also important that the state should have
asymmetry in the correlations between the measurement
outcomes of different complementary observables.To say
more specifically, I(σA
1:σB
1)>I(σA
2:σB
2) = I(σA
3:
σB
3).
VI. CONCLUSIONS
We have shown that in case of certain Bell diagonal
states, the sudden transition in the evolution of classical
and quantum correlations under Markovian noise(phase
damping channel) can be well understood in terms of
comlementary correlations.For two qubit Bell-diagonal
states σxand σzare complementary observables and
for those specific Bell-diagonal states the overall com-
plementary correlations are exactly equal to the mutual
information(I) of the states.
We have also proved that two qubit pure entangled
states will never show such type of sharp transition in
the classical and quantum decoherence. Taking the ex-
ample of Werner class of states we have shown that the
freezing property of discord (D), which is in contrast with
entanglement sudden death, is not inherent for all mixed
entangled states of Bell-diagonal class. Our analysis put
forward an interesting question. Using complementary
correlations, is it possible to find the general class of
two qubit composite states exhibiting sudden change in
the evolution of classical and quantum correlations? We
hope that our findings will provide better insight to un-
derstand the evolution of classical and quantum correla-
tions of composite quantum systems, when subjected to
noises.
This work is funded by DST, Govt. of India. We are
grateful to Dr. Guru Prasad Kar for useful discussions.
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