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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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022113-4

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