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PHYSICAL REVIEW A 84, 022113 (2011)

Necessary and sufficient condition for Markovian-dissipative-dynamics-induced quantum discord
Xueyuan Hu,1 Ying Gu,1,* Qihuang Gong,1 and Guangcan Guo2
1

State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100871, China
Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
(Received 4 May 2011; published 26 August 2011)

2

A Markovian dissipative quantum channel can generate quantum discord from some bipartite product states
if and only if it cannot be reduced to individual decoherence channels independently acting on each qudit. As a
byproduct, we also prove that proper individual decoherence can transform the classical correlation into quantum
discord. Our result builds a tight connection between quantum discord and collective decoherence.
DOI: 10.1103/PhysRevA.84.022113

PACS number(s): 03.65.Ud, 03.65.Yz, 03.65.Ta, 03.67.Mn

Quantum discord, a measure of quantum correlation, has recently attracted much interest since the discovery that quantum
discord is presented in some quantum computing model without entanglement [1,2]. It is defined as the difference between
mutual information and classical correlation. Quantum discord
is closely associated to quantum channels. The necessary and
sufficient condition for a completely positive map is vanishing
quantum discord contained in the initial state of the system and
its environment [3]; besides, the dynamics of quantum discord
in an open composite system has also been widely studied
[4–12]. In contrast to entanglement sudden death, quantum
discord does not suddenly vanish within a finite time either in
a Markovian [7] or in a non-Markovian [8] regime. Thus,
quantum discord is more robust against decoherence than
entanglement. This is because quantum discord includes but is
not limited to entanglement [13,14]. Evidence has been given
that when two noninteracting qubits are coupled to a common
reservoir, entanglement can be built between the two qubits,
under the condition that the effect from correlations caused
by the collective decoherence surpasses that for individual
decoherence [15,16]. This lead to the expectation that similar
dissipative dynamics can also induce quantum discord on less
stringent conditions.
In this article, we will prove that the necessary and
sufficient condition on which Markovian decoherence can
create quantum discord between two qudits initially in a
product state is that it cannot be reduced to individual
decoherence channels independently acting on each qudit. Our
result establishes a tight connection between quantum discord
and collective decoherence. It is interesting that any collective
decoherence can create quantum correlation, no matter how
badly the individual qudits are degraded. This fact impacts on
practical quantum information processes. On the one hand, it
provides a method for generating states with strictly positive
quantum discord but vanishing entanglement; on the other, it
can also serve as a criterion for checking whether a dissipative
dynamics is collective.
Moreover, we will show that the proper individual decoherence can turn classical correlations into quantum discord.
Counter-intuitively, local noise can create quantum correlation
between two qudits.

*

ygu@pku.edu.cn

1050-2947/2011/84(2)/022113(4)

The total correlation in a bipartite state AB is quantified
by the mutual information between qudits A and B: I( AB ) =
S( A ) + S( B ) − S( AB ) with A(B) = TrB(A) AB the reduced
density matrix of qudit A (B) and S( ) = −Tr( log2 )
the von Neumann entropy. Mutual information can be interpreted as the amount of information that qudits A and
B have in common [17]. The maximum amount of mutual
information that can be revealed by local von Neumann
measurements { m
B } on qudit B is called
the classical correlation [18]: C( AB ) = max{ mB } [S( A ) − m pm S( A| mB ), with
pm = TrB ( AB m
B ) being the probability of the outcome m
m
and A| mB = TrB ( m
B AB B )/pm the corresponding state of
qudit A after the measurements. The quantum discord is then
defined as
D(ρAB ) = I(ρAB ) − C(ρAB ),

(1)

measuring the “quantumness” of the correlation between
qudits A and B. According to Ref.
[19], a zero-discord
A
⊗ m
quantum state can be written as AB = m pm m
B and,
consequently,
[ AB ,IA ⊗ B ] = 0,

(2)

where I is the identity operator. In the following, we will use
Eq. (2) as a necessary criterion for vanishing quantum discord.
Consider that two noninteracting qudits, initially in a
product state, are weakly coupled to a common bath via some
noisy mechanism. The two qudits, A and B, are defined in
the Hilbert spaces HA and HB of dimensions dA and dB ,
respectively. The total Hamiltonian of the two qudits and the
bath R is HABR = HA ⊗ IB ⊗ IR + IA ⊗ HB ⊗ IR + IA ⊗
IB ⊗ HR + HI , where HA , HB , and HR are the individual
Hamiltonians of the two qudits and the bath, respectively,
whereas
dA2 −1

HI =


i=1

dA2 +dB2 −2

(Fi ⊗ I) ⊗ Ri +



I ⊗ Gi−dA2 +1 ⊗ Ri ,

i=dA2

(3)
dA2 −1

is the interaction Hamiltonian. Here, {Fi }i=0 (F0 = I)
d 2 −1

B
(G0 = I)] constitutes a complete basis for the vector
[{Gi }i=0
space of bounded operators acting on the Hilbert space HA
(HB ). Notice that the two qudits are coupled to the bath through
different operators Ri , but do not interact with each other
directly.

022113-1

©2011 American Physical Society

XUEYUAN HU, YING GU, QIHUANG GONG, AND GUANGCAN GUO

Assuming Markovian dynamics and an initial decoupling
between the system and bath, semigroup theory is employed
to describe the dynamics of the two qudits. The density matrix
ρ(t) of qudits A and B then evolves according to the following
master equation [20]:
∂t ρ(t) = −i[H,ρ(t)] + L[ρ(t)].

(4)

Here, H is the effective Hamiltonian of the two qudits, and
its contribution will be ignored in the following discussion,
since we are interested in the possibility of inducing quantum
discord by pure dissipative dynamics. According to semigroup
theory, the general form of the dissipative term L[ρ(t)] is


N

1
Dij Ki ρ(t)Kj − {Kj Ki ,ρ} ,
(5)
L[ρ(t)] =
2
i,j =1
where N d 2 − 1, d = dA dB is the dimension of the system
Hilbert space H = HA ⊗ HB , and {Ki }N
i=0 (K0 = I) constitute
a basis for the vector space of bounded operators acting
on H. When {Ki }N
i=0 is given, the positive N × N matrix
D = D † completely describes the properties of the dissipative
dynamics.
For the situation we consider here, Ki = Fi ⊗ I for i =
1,2, . . . ,dA2 − 1, Ki = I ⊗ Gi for i = dA2 ,dA2 + 1, . . . ,dA2 +
dB2 − 2. The dissipative contribution L[ρ(t)] can then be
written as
L[ρ(t)] = LA [ρ(t)] + LB [ρ(t)] + LAB [ρ(t)],
(6)
dA2 −1
where LA [ρ(t)] = i,j =1 Aij [(Fi ⊗ I)ρ(t)(Fj ⊗ I) − {(Fj Fi ⊗
d 2 −1
I),ρ}/2] and LB [ρ(t)] = i,jB =1 Cij [(I ⊗ Gi )ρ(t)(I ⊗ Gj ) −
{(I ⊗ Gj Gi ),ρ}/2] are dissipative terms individually affecting
qudits A and B, whereas
dA2 −1 dB2 −1



LAB [ρ(t)] =
Bij (Fi ⊗ I)ρ(t)(I ⊗ Gj )

time t. Therefore, the dissipative dynamics L[ρ(t)] with
B = 0 does not drive any product state into a positive-discord
state.
Now we will concentrate on proving the “if” part. More
specifically, we will prove an equivalent statement that if
the dissipative dynamics L[ρ(t)] in Eq. (6) cannot drive
any product state into a positive-discord state, then we have
B = 0. In order to witness quantum discord, we employ
Eq. (2) as the necessary criterion for zero quantum discord.
Since Eq. (2) holds for t = 0, the condition that there exists
an initial product state ρ ≡ ρ(0) = ρA ⊗ ρB such that ≡
∂t [ρAB (t),IA ⊗ ρB (t)]t=0 = 0 is sufficient for environmentinduced quantum discord. Therefore, we only need to prove
that = 0 for any choice of ρA and ρB only if B = 0. Ignoring
the contribution of the Hamiltonian in Eq. (4), we have
= [L(ρ),IA ⊗ ρB ] + [ρ,IA ⊗ TrA [L(ρ)]].

= [LAB (ρ),IA ⊗ ρB ] + [ρ,IA ⊗ TrA [LAB (ρ)]]
dA2 −1 dB2 −1

=



Re(Bij )[ρA ,Fi ] ⊗ [[Gj ,ρB ],ρB ]

i=1 j =1

˜
+ iIm(B
ij ){Tr(ρA Fi )ρA ⊗ [ρB ,Gj ]
+ [Fi ⊗ ρB ,ρA ⊗ [Gj ,ρB ]]},

describes the correlation effect caused by the collective
decoherence. Correspondingly, the coefficient matrix in Eq.
(5) takes the form


A B
,
(8)
D=
B† C
with the (dA2 − 1) × (dA2 − 1) matrix A = A† and the (dB2 −
1) × (dB2 − 1) matrix C = C † describing the individual decoherence acting on qudits A and B, respectively, and the
(dA2 − 1) × (dB2 − 1) matrix B corresponding to the correlation
effect caused by the collective decoherence.
Now we are ready to prove the following statement, which is
the main result of this paper: the dissipative dynamics L[ρ(t)]
can drive some product states into positive-discord states if
and only if B = 0.
Proof: The proof of the “only if” part is obvious. If B = 0,
the decoherence dynamics described by Eq. (6) reduces to
independent decoherence acting on qudits A and B, respectively; hence, we have ρAB (t) = ρA (t) ⊗ ρB (t) for arbitrary

(10)

where i˜ stands for the imaginary unit. By choosing
ρA =

(7)

(9)

It is obvious that the term LA [ρ(t)] of Eq. (6) does
not contribute to , whereas the contribution of the
term LB [ρ(t)] also vanishes for the product state:
[LB (ρ), IA ⊗ ρB ] + [ρ, IA ⊗ TrA [LB (ρ)]] = [ρA ⊗ L˜ B (ρB ),
IA ⊗ ρB ] + [ρA ⊗ ρB ,IA ⊗ L˜ B (ρB )] = 0, where L˜ B (ρB ) =
dB2 −1
i,j =1 Cij [Gi ρ(t)Gj − {Gj Gi ,ρ}/2]. Therefore, we have

i=1 j =1



1
− {(Fi ⊗ Gj ),ρ} + H.c.
2

PHYSICAL REVIEW A 84, 022113 (2011)

I + Gj
I + Fi
, ρB =
dA
dB

(11)

with i = 1,2, . . . ,dA2 − 1 and j = 1,2, . . . ,dB2 − 1 in turn
and setting = 0, we finally obtain B = 0. Consequently,
a nonzero matrix B is sufficient for environment-induced
quantum discord, completing the proof.
Here, we provide an example to explore in more detail
this environment-induced quantum discord. Consider a twoqubit initial state ρ = (|+ A +|) ⊗ (|+ B +|) evolving in a
collective dephasing channel with A = C = diag{0,0,γ }, B =
x,y,z
x,y,z
in Eq. (6).
diag{0,0,γ1 } and F1,2,3 = σA ,G1,2,3 = σB
Here, γ is the coupling strength between the qubits and the
reservoir, and the coefficient γ1 ∈ (0,γ ]. For the situation
where qubits A and B are placed close enough, we have γ1 =
γ . Since the correlation effect caused by collective dephasing
is no larger than the individual decoherence (γ1 γ ), such
dissipative dynamics cannot induce entanglement between
qubits A and B [15]. The resulting state at time t is

022113-2



1

1⎜ q
ρ(t) = ⎜
4⎝ q
q 2 q12

q
1
(q/q1 )2
q

q
(q/q1 )2
1
q


q 2 q12
q ⎟

⎟,
q ⎠
1

(12)

NECESSARY AND SUFFICIENT CONDITION FOR . . .

PHYSICAL REVIEW A 84, 022113 (2011)

where q = exp(−2γ t) and q = exp(−2γ1 t). Direct calculation leads to
˜ −6γ t sinh(4γ1 t)σ y ⊗ σ z . (13)
[ρ(t),IA ⊗ ρB (t)] = −2ie
A

d
B −1

pi ρiA ⊗ |φi B φi |,

(14)

i=0

where {|φi B } are to orthogonal states of qudit B, and ρiA
are different density matrices of qudit A. For state ρAB as
in Eq. (14), the classical correlation is equal to the mutual
information when qudit B is measured on basis {|φi B }; thus,
no quantum discord is shared between qudit A and qudit B.

(i)†
Now a quantum decoherence channel εB (·) ≡ i E(i)
B (·)EB ,
with Kraus operators E(i)
B , is acting on qudit B, causing the
initial two-qubit state to become

ρAB

=

d
B −1

pi ρiA ⊗ ξiB ,

B B
ξi ,ξj = 0

B

At finite time t ∈ (0, + ∞), an arbitrarily small but positive
γ1 can cause Eq. (13) to be nonzero. It means that collective
dephasing, which cannot induce entanglement between any
initially separable qubits, can produce quantum discord. In
the steady-state limit (t → ∞), we calculate the quantum
discord for γ = γ1 . Using the method in Ref. [21], we obtain
for the total and classical correlations and quantum discord
the respective values I = 0.5, C = 34 log2 3 − 1 ≈ 0.189, and
D = 1.5 − 34 log2 3 ≈ 0.311. Clearly, quantum discord is induced by collective dephasing and preserved in the limit of
t → ∞. It is worth noting that, for the state ρ(t → ∞), the
classical correlation is smaller than quantum discord.
It is worth mentioning that the condition for creating
quantum discord through collective decoherence does not
depend on matrices A and C. The significance is that no matter
how badly L[ρ(t)] affects the qudits individually, quantum
discord can be created by purely dissipative dynamics as long
as collective decoherence can cause correlation effects. This
result builds a tight connection between collective decoherence
and quantum discord: any collective decoherence can induce
quantum discord in some initially product state of two qudits.
On the other hand, our result provides a method for checking
whether a given dissipative channel is collective. We only
need to set the input state of the two qudits as in Eq. (11)
subsequently, and then detect the quantum discord of the
output state. Whenever the output discord is nonzero for some
input state, the channel is a collective decoherence channel;
otherwise, it can only cause individual decoherence.
Next, we will show that quantum discord can be induced
by local decoherence between a pair of initially classically
correlated qubits.
Consider an initial two-qudit state,
ρAB =

channel εB (·) acting on the individual qudit B induces quantum
discord between qudit A and qudit B once the inequality

(15)

i=0

where ξiB = εB (|φi B φi |). Here, we employ Eq. (2) to check
whether the discord has been created. Direct calculation
provides





(16)
[ρAB
,IA ⊗ ρB ] =
pi pj ρiA − ρjA ξiB ,ξjB .
i<j

Therefore, we arrive at the sufficient condition for quantum discord induced by individual decoherence: the local decoherence

(17)

holds for at least one pair of indices i and j .
Now we show that this condition can be satisfied by
providing an example. Suppose A and B are a pair of
qubits, |φi B in Eq. (14) are the two eigenstates√ of the
Pauli matrix σBx : |φ0,1 B = |± B ≡ (|0 a ± |1 b )/ 2, and
the local decoherence channel is the amplitude damping channel εBAD (·) for which the Kraus operators are


[17] E(0)
1 − p|1 B 1| and E(1)
p|0 B 1|.
B = |0 B 0| +
B =
±
Then we have √ξB ≡ εBAD (|± B ±|) = [(1 + p)|0 B 0| +
(1 − p)|1 B 1| ± 1 − p(|0 B 1| + |1 B 0|)]/2 and, consequently,

[ξB+ ,ξB− ] = p 1 − pσBx ,

(18)

which does not vanish for 0 < p < 1. Thus, we conclude that
amplitude damping acting locally on qubit B of the classically
correlated state ρAB can induce quantum discord between qubit
A and qubit B and that the induced discord does not vanish in
finite time.
That individual dissipative procedure can induce quantum
discord between two qudits that are initially in a state with
strictly classical correlation is quite interesting. It has been
stated that the set of all states with zero quantum discord
is negligible in the whole Hilbert space: an arbitrarily small
perturbation can drive a given zero-discord state to a positivediscord state [19]. Our result provides an explicit example of
the above statement: the zero-discord state ρAB in Eq. (14) can
be driven to a positive-discord state by the amplitude damping
channel εBAD (·) for arbitrarily small p. Notice this procedure
can only turn classical correlations into quantum discord; the
individual dissipative dynamics cannot drive a product state to
a positive-discord state.
In conclusion, we have proved that the necessary and
sufficient condition for inducing quantum discord between two
initially uncorrelated qudits is that the dissipative dynamics
cannot be reduced to individual decoherence. This fact
establishes a tight connection between quantum discord and
collective decoherence. We have also shown that when two
qudits are initially classically correlated, even the individual
decoherence acting on one of the qudits can create quantum
discord between the qudit pair. Our result has practical
significance in that it provides a method for generating states
with strictly positive discord and vanishing entanglement;
moreover, it can also serve as a criterion for checking whether
dissipative dynamics is collective.
ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant Nos. 10874004 and
10821062 and National Key Basic Research Program Grant
Nos. 2007CB307001 and 2006CB921601.

022113-3

XUEYUAN HU, YING GU, QIHUANG GONG, AND GUANGCAN GUO
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