Artic le Quantum Information

Stabilized quantum coherence and remote state preparationin structured environments

Ping Zhang • Bo You • Li-Xiang Cen

Received: 9 May 2014 / Accepted: 21 May 2014 / Published online: 1 July 2014

� Science China Press and Springer-Verlag Berlin Heidelberg 2014

Abstract The radiation decay of a two-level atom could

be inhibited within structured environments even under

longtime evolution. We investigate the stabilized quantum

coherence of composite systems undergoing local dissipa-

tion and exploit it further as a resource for remote state

preparation. We focus on outputs of quantum states with

solely quantum discord (i.e., without entanglement) and

demonstrate that they could be resulted from various initial

states providing specific spectral structure of the reservoir.

In detail, we elaborate the behavior of stabilized quantum

discord and the corresponding fidelity for remote state

preparation in connection with structural spectra of Ohmic

class reservoir and of photonic band gap mediums.

Keywords Stabilized quantum correlation �Structured environment � Remote state preparation

1 Introduction

Quantum correlation is a characteristic trait of quantum

mechanics that is inaccessible to classical objects. Tradi-

tionally, entanglement was known to be of central impor-

tance in most applications of quantum information

processing, e.g., in quantum algorithms [1], quantum

teleportation [2] and quantum cryptography [3]. However,

the absence of entanglement does not remove all characters

of quantum behavior. A different notion of quantum cor-

relation, quantum discord, has been proposed to capture the

quantumness of correlations based on an information-

theoretic benchmark [4, 5]. Since it was found to exist in a

quantum algorithm without entanglement [6, 7], the prop-

erty of quantum discord and its operational interpretation

related to quantum information tasks have received a lot of

attention in the field.

In the real world, quantum coherence is rather fragile due

to the coupling of the quantum system with its surrounding

environments. This is one main obstacle to the practical

realization of quantum information processing. Since

quantum discord is more robust than entanglement (e.g.,

immune to sudden death under decoherence), one naturally

expects that whether the discord can be used as a resource in

certain quantum information tasks. In fact, recent studies

have found that quantum discord may play crucial roles for

remote state preparation (RSP) [8, 9] and quantum cryp-

tography [10, 11]. On the other hand, noticeable progress has

been achieved with respect to the issue of dissipative

dynamics within non-Markovian environments. It was

shown that the memory effect of the environment could

prolong quantum coherence and even lead to preservation of

quantum correlations [12–18]. Note that modifying the

property of the reservoir to reach the non-Markovian regime

was shown to be practicable in many physical systems, e.g.,

in photonic crystal materials [19–21] and optically confined

ultracold atomic gases [22]. These facts strengthen the

expectation to utilize quantum discord to accomplish quan-

tum information tasks in noisy environments.

In this paper, we propose to achieve steady coherent states

with solely quantum discord in the presence of noisy envi-

ronments. This provides a practicable scenario to verify

discord as a resource to implement quantum information

tasks. We show that by engineering properly the spectrum of

the reservoir, outputs of separable states with nonvanishing

discord could be resulted from various initial states of

composite systems undergoing longtime dissipation. As an

P. Zhang � B. You � L.-X. Cen (&)

Center of Theoretical Physics, College of Physical Science and

Technology, Sichuan University, Chengdu 610065, China

e-mail: lixiangcen@scu.edu.cn

123

Chin. Sci. Bull. (2014) 59(29–30):3841–3846 csb.scichina.com

DOI 10.1007/s11434-014-0497-x www.springer.com/scp

illustration, we demonstrate further that the stabilized

quantum coherence can be used as the resource for RSP.

2 Longtime behavior of two-level systems

within structured environments

Let us begin with the issue of typical spontaneous emission

of two-level systems but within some structured environ-

ments. In the rotating-wave approximation the Hamilton of

the model assumes the form

H ¼ x0rþr� þX

k

xkbykbk þ ðgkbkrþ þ g�kb

ykr�Þ

h i; ð1Þ

where x0 is the transition frequency between the upper

level j1i and the lower level j0i of the two-level atom (the

qubit) and r� denote the raising and lowering operators.

The field modes are described by the creation (annihilation)

operators byk (bk) with frequencies xk. Suppose that the

atom is initially in the excited state j1i and the field is in

the vacuum state jf0kgi. Depending on the coupling

coefficients gk of the atom with the reservoir, the sponta-

neous decay of the atom could be either complete or

incomplete [12, 13]. For the latter case, it is possible that

partial quantum coherence could be stabilized under the

longtime evolution of the dissipative process.

An efficient way to characterize the residual coherence

of the system is to invoke the functional relationship dis-

closed in Ref. [18]. In detail, in the subspace of the single-

excitation sector, the secular equation of the eigenvalue

problem of the Hamiltonian (1) is obtained as

x0 �Z 1

0

JðxÞx� E

dx ¼ E; ð2Þ

where a continuous spectrum of the reservoir has been

assumed and JðxÞ ¼P

k jgkj2dðx� xkÞ is the so-called

function of spectral density. It is possible that for particularly

structured environments Eq. (2) admits a unique solution

with a real root of E. Consequently, the eigenstate constitutes

an atom-photon bound state: jUBSi ¼ bj1; f0kgi þPk bkj0; 1ki with the coefficients bk ¼ gkb=ðE � xkÞ and

b ¼ 1þZ 1

0

JðxÞðx� EÞ2

dx

" #�1=2

: ð3Þ

On the other hand, as jUBSi is the only stationary solution,

the long-lived population of the asymptotic process hence

is given by [13]

jcðtÞj2 �!t!1 jc1j2 ¼ b4: ð4Þ

Equations (2)–(4) offer an implicit functional relationship

between the asymptotic population and the spectral density

function, i.e., jc1j2 ¼ b4½JðxÞ�. For given structured

environments with specified JðxÞ, it renders a straight way

to calculate the asymptotically stabilized population in

relation to the parameters of the reservoir spectra.

To describe the longtime behavior of quantum coher-

ence of composite systems under local dissipation, we

exploit two different measures: entanglement and quantum

discord. The amount of entanglement of a two-qubit state

qAB is expressed explicitly by the concurrence [23]:

CAB ¼ maxf0; k1=21 �

P4i¼1 k1=2

i g, where ki’s are eigen-

values of the matrix qABry � ryq�ABry � ry in decreasing

order. The discord of qAB is defined as [4]

QAB ¼ minfAkg

RkpkSðqkBjfAkgÞ þ SðqAÞ � SðqABÞ; ð5Þ

in which SðqÞ � �trðq log2 qÞ is the von Neumann entropy

and the minimization of the first term is taken over all

possible projective measurements on A. A natural question

arising here is that whether the residual coherence descri-

bed by these quantities will tend to be stationary in the

longtime limit. In the following we show that this is the

case as long as the system possesses a unique bound state

such that the relation (4) holds.

Specifically, we suppose that the qubit A is subject to

amplitude damping specified by Eq. (1) and the evolution

of qAB is then described by the Kraus representation

qABðtÞ ¼X2

i¼1

CiðtÞ � IBqABð0ÞCyi ðtÞ � IB; ð6Þ

where

C1ðtÞ ¼0 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� jcðtÞj2q

0

" #; C2ðtÞ ¼

cðtÞ 0

0 1

� �: ð7Þ

One may notice that as t!1; c1 will suffer a phase

uncertainty. However, by replacing cðtÞ with its module

jcðtÞj in C2, we observe that the yielded state will differ

from the original qABðtÞ only by a local phase shift uA � IB

with uA ¼ eiu 0

0 1

� �and u ¼ arg½cðtÞ�. Therefore the

yielded state qABðtÞ �!t!1

qstbAB will be determinate up to a

local unitary transformation on A and the amount of

entanglement and discord will be not affected. Both the

two latter quantities will be stabilized under the longtime

evolution as the module jc1j tends to a constant value.

3 Stabilized non-classical correlations

without entanglement

Given an initial state and the spectrum function of the

reservoir, the output under longtime dissipation can be

3842 Chin. Sci. Bull. (2014) 59(29–30):3841–3846

123

calculated straightforwardly via the above described

approach. For pure initial states, the asymptotically stabi-

lized quantum correlations have been characterized in Ref.

[18]. We mention that the derived state of Eq. (6) is of rank

two and it would be either entangled or of product form.

That is to say, from pure states it is not possible to obtain a

separable state with nonvanishing quantum discord. Nev-

ertheless, we show in the below that such outputs could be

achieved if we adopt mixed input states.

In detail, we consider the Werner state

qABð0Þ ¼ zjw�ihw�j þ1� z

4I4; ð8Þ

where jw�i ¼ ðj10i � j01iÞ=ffiffiffi2p

: It is known that this state

has nonzero discord as z [ 0 and nonzero entanglement as

z [ 1=3. To describe the property of the output state, we

obtain from Eq. (6) that

qstbAB ¼

d�4

0 0 0

0 dþ4

� z2

c1 0

0 � z2

c�112� d�

40

0 0 0 12� dþ

4

266664

377775; ð9Þ

where d� ¼ ð1� zÞjc1j2. It has nonvanishing discord as

long as the excited-state population c1 6¼ 0. This can be

seen from the fact that for any null-discord state rAB there

is ½rAB; rA � I� ¼ 0 [24]. In view of qstbA ¼

12(jc1j2 0

0 1þ �c21

), the commutator is worked out to be

g � ½qstbAB; q

stbA � ¼

z

2�c21½c1rþ � r� � c�1r� � rþ�; ð10Þ

where �c1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� jc1j2

q. Since g above will not vanish as

long as c1 6¼ 0, we can say that for initial states with

z [ 0, the outputs of the asymptotical process will possess

nonzero discord if the system allows a bound state.

To determine the region in which the output state pos-

sesses quantum discord but without entanglement, we note

that the concurrence of qstbAB is given as

CstbAB ¼ zjc1j �

jc1j2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� zÞð2� dþÞ

p: ð11Þ

Since c1 6¼ 0, the critical value of the parameter z

responsible for CstbAB ¼ 0 is obtained as

zc ¼2� jc1j2

4� jc1j2: ð12Þ

As z\zc, one achieves the expected discordant state with

vanishing entanglement. For specified reservoir spectra,

Eq. (12) together with jc1j2 ¼ b4½JðxÞ� (see Eqs. (2)–(4))

indicate that zc ¼ zc½JðxÞ�, namely, a direct connection

between zc and the spectral function is established.

Straightforward observation from Eq. (12) shows that zc

satisfies 1=3 zc 1=2.

We examine two kinds of reservoir spectra and report

the corresponding results in Fig. 1. One is the Ohmic class

spectra [25]

JoðxÞ ¼ gox1�sc xse�x=xc ; ð13Þ

where xc is the cutoff frequency and s is the parameter

whose range, s\1; s ¼ 1; s [ 1, corresponds to sub-Ohmic

reservoirs, Ohmic and super-Ohmic reservoirs, respec-

tively. The stabilized discord QstbAB and the critical curve zc

are depicted in Fig. 1a and b. We have taken the initial

z ¼ 0:454 (z\zc) to warrant that the output is a separable

state. Note that the spectral parameters here should satisfy

g�1ðxc=x0ÞCðsÞ (CðsÞ denotes the gamma function) so

that Eq. (3) allows a bound-state solution [18]. Our cal-

culation shows that as s ¼ 5:5 the maximal QstbAB ’ 0:0843

is achieved at go ¼ 0:082.

For the second reservoir we investigate the spectrum of

photonic band gap medium which is given by [13, 26]

JpðxÞ ¼gp

px3=2

effiffiffiffiffiffiffiffiffiffiffiffiffiffiffix� xep Hðx� xeÞ; ð14Þ

where xe is the band edge frequency, H is the Heaviside

step function. We take the initial parameter z ¼ 0:34 here

(z\zc in view of Fig. 1d). The peculiarity of QstbAB is

characterized as below (see Fig. 1c): (i) for the case that the

(a) (b)

(c) (d)

Fig. 1 (Color online) Stabilized quantum coherence of the initial

Werner state (8) undergoing asymptotical dissipation within struc-

tured environments. For the Ohmic class reservoirs the stabilized

discord QstbAB and the critical curve zc are depicted against go ((a) and

(b)) with the cut off frequency xc=x0 ¼ 0:3. As s ¼ 5:5, the maximal

QstbAB 0:0843 and the minimal zc 0:455 are achieved at

go ¼ 0:082. For the photonic band gap medium, QstbAB and zc are

depicted as functions of gp ((c) and (d)) with xe=x0 ¼ 0:8; 1:0 and

1.2, respectively. A constant QstbAB 0:67 and zc 0:44 are obtained

at xe ¼ x0

Chin. Sci. Bull. (2014) 59(29–30):3841–3846 3843

123

atomic level is inside the reservoir continuum

ðx0 [ xeÞ;QstbAB is an increasing monotone of gp; (ii) for

the atomic level outside the continuum ðx0\xeÞ;QstbAB is a

decreasing monotone of gp; (iii) for the exact resonance

case with x0 ¼ xe;QstbAB become constant since the popu-

lation jc1j is a constant at this point [13].

4 Creating quantum discord under asymptotic

dissipative processes

Since quantum discord could be generated through local

quantum channels [27, 28], it is of interest to ask whether

one can achieve, from initial states with null discord, some

nonzero stabilized discord through the described asymp-

totic dissipative process. In this section we present an

example and show that this could be the case. Specifically,

we consider the following initial state

qABð0Þ ¼ ð1� pÞjþihþj � je1ihe1j þ pj�ih�j � jehihehj;ð15Þ

where 0\p\1 and j�i ¼ ðj1i � j0iÞ=ffiffiffi2p

. The state

jehi ¼ cos hje1i þ sin hje2i and fje1i; je2ig are a set of

orthonormal bases of B. From Eq. (6), one obtains

qstbAB ¼

1� p

2j/þih/þj � je1ihe1j þ

p

2j/�ih/�j � jehihehj

þ �c21j0ih0j � ð

1� p

2je1ihe1j þ

p

2jehihehjÞ; ð16Þ

where j/�i ¼ c1j1i � j0i. To determine whether qstbAB

possesses nonzero discord, we examine the commutator

g ¼ ½qstbAB; q

stbA �. By setting the bases fje1i; je2ig as

fj1i; j0ig, one obtains the commutator as

g0 ¼ pðp� 1Þ sin h�c21ðc1rþ � c�1r�Þ �X � r; ð17Þ

where X ¼ fcos h; 0; sin hg. Subsequently for the state (16)

in which fje1i; je2ig relates to fj1i; j0ig via a local unitary

UB, the corresponding commutator is specified as

g ¼ UBg0UyB. Thus we can make assertion from above facts

that the state qstbAB will have nonzero discord when the

parameter of the input state (15) satisfies sin h 6¼ 0.

In Fig. 2, we figure out the generated discord in relation

to the spectral parameters of Ohmic class and of photonic

crystals, respectively. Since the stabilized discord is inde-

pendent of the choice of the bases fje1i; je2ig (the output

state (16) differs from each other by a local unitary trans-

formation), we only need to set the coefficients p and h of

the initial state. Note that the discord here is generally not a

monotone of jc1j (see the upper inset of Fig. 2a), their

relation together with the function jc1j2 ¼ b4½JðxÞ�

determine the behavior of QstbAB with respect to the spectral

parameters.

5 Remote state preparation with stabilized quantum

coherence in structured environments

As a measure of nonclassical correlations, the role of

quantum discord in quantum information tasks is still open

and its operational interpretation has been studied for some

particular situations [6, 29, 30]. Very recently, Dakic et al.

[9] reveal that quantum discord is a resource for RSP and

demonstrate that separable states with nonzero discord can

even outperform some entangled states for this special task.

In the below we demonstrate the efficiency that the stabi-

lized quantum coherence in dissipative systems could be

utilized as resources for RSP.

For the ideal channel of RSP, i.e., the two partners A and B

share a maximally entangled two-qubit state, A could prepare

(a)

(b)

Fig. 2 The generated quantum discord of the initial state (15)

undergoing asymptotic dissipation within structured environments.

(a) Stabilized discord QstbAB against go of the super-Ohmic spectrum

with s ¼ 5:5 and xc ¼ 0:3x0; (b) QstbAB against gp of the photonic band

gap medium with xe ¼ 1:2x0. The coefficients of the initial state (15)

are set as p ¼ 3=4 and h ¼ p=4. The upper inset in (a) shows the

connection of QstbAB with the long-lived population jc1j of the yielded

state (16) and the shadow area denotes the scope of jc1j and QstbAB

achievable within the specified reservoir

3844 Chin. Sci. Bull. (2014) 59(29–30):3841–3846

123

deterministically any state in the equatorial plane of Bob’s

Bloch sphere [8]. When the channel is not perfectly entan-

gled, the state will be prepared with a reduced fidelity. In

detail, for a general channel of the two-qubit state

qAB ¼1

4ðI � I þ x � r� I þ I � y � rþ

X

i;j

tijri � rjÞ;

ð18Þ

the fidelity of RSP is shown to be [9]

F ¼ 1

2ðE2

2 þ E23Þ; ð19Þ

where E22 and E2

3 are the two lowest eigenvalues of the

symmetric tensor T ¼ tTt, and I and ri ði ¼ 1; 2; 3Þ in

Eq. (18) denote the identity matrix and the Pauli matrices,

respectively.

Let’s employ the state (9) as a channel for RSP which is

resulted from the Werner state under the asymptotic dis-

sipative process. To calculate the fidelity of RSP, we note

that the correlation tensor of the state (9) reads

tstb ¼�z Re c1 z Im c1 0

�z Im c1 �z Re c1 0

0 0 �zjc1j2

264

375: ð20Þ

The fidelity of RSP is then obtained from Eq. (19) as

F ¼ ðjc1j2 þ jc1j4Þz2: ð21Þ

In Fig. 3a we depict the fidelity as well as the discord as

functions of the initial parameter z of the Werner state. It is

shown that the RSP fidelity and discord manifest similar

behavior in this case. Particularly, we mention that as

z zc, the output state of the asymptotic process has van-

ishing entanglement, however, both the discord and the

fidelity in this region have nonzero values.

Let us look into the RSP fidelity via the channel of Eq. (16)

in which the quantum coherence is locally created from null-

discord state. Similarly, the fidelity is independent of the

choice of the bases fje1i; je2ig of the qubit B. This can be

seen since the singular values of the correlation tensor tstb (or

eigenvalues of Tstb) of the output state are invariant under the

local unitary UB on B. Explicitly, by setting the bases as

fj1i; j0ig, one obtains the tensor Tstb ¼ tTstbtstb with non-

vanishing elements specified as

T11stb ¼ p2 sin2 2hðjc1j2 þ �c4

1Þ; T33stb ¼ c2

�jc1j2 þ c2

þ�c41;

T13stb ¼ T31

stb ¼ p sin 2hðc�jc1j2 � cþ�c4

1Þ; ð22Þ

where c� ¼ 1� p� p cos 2h. Since Tstb here is of rank

two, the RSP fidelity is worked out to be

F ¼ 1

4½T11

stb þ T33stb �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðT11

stb � T33stbÞ

2 þ 4T13stb

q�: ð23Þ

Both the RSP fidelity and discord against the parameter hare depicted in Fig. 3b with p ¼ 3=4. We assume jc1j ¼0:577 which corresponds approximately to the maximal

value achieved in Ohmic class reservoir (see Fig. 2a). Our

calculation reveals the different behavior of the two

quantities depending on the parameter h. Finally, we

mention that the efficiency of locally created coherence for

RSP has also been displayed in Ref. [31].

6 Conclusion

In summary, we have studied the stabilized quantum

coherence of dissipative composite systems within struc-

tured environments. We have shown that outputs of dis-

cordant states without entanglement could be achieved for

various initial states by engineering the reservoir spectra

properly. The behavior of stabilized discord in connection

with spectral parameters has been elaborated for Ohmic

class reservoirs and photonic band gap mediums. Further-

more, we have shown that the stabilized quantum coher-

ence resulted from such asymptotic dissipative processes

could be exploited as a resource for RSP.

Acknowledgments This work was supported by the National Nat-

ural Science Foundation of China (10874254).

Conflict of interest The authors declare that they have no conflict

of interest.

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