â2[cos rb cos rc|0ãA|0ãBI |0ãCI + cos rb sinrc|0ãA|0ãBI |1ãCI ... ÏABI CI. = 1. â2[cos r2 b cos r2 c |000ãã000| + cos r2 b sinr2 ...

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arXiv:1205.3133v1 [quant-ph] 14 May 2012

(Dated: January 23, 2018) Quantum discord is an optimal resource for the quantification of classical and non-classical correlations as compared to other related measures. Geometric measure of quantum discord is another measure of quantum correlations. Recently, the geometric quantum discord for multipartite states has been introduced by Jianwei Xu [arxiv:quant/ph.1205.0330]. Motivated from the recent study [Ann. Phys. 327 (2012) 851] for the bipartite systems, I have investigated global quantum discord (QD) and geometric quantum discord (GQD) under the influence of external environments for different multipartite states. Werner-GHZ type three-qubit and six-qubit states are considered in inertial and non-inertial settings. The dynamics of QD and GQD is investigated under amplitude damping, phase damping, depolarizing and flipping channels. It is seen that the quantum discord vanishes for p > 0.75 in case of three-qubit GHZ states and for p > 0.5 for six qubit GHZ states. This implies that multipartite states are more fragile to decoherence for higher values of N . Surprisingly, a rapid sudden death of discord occurs in case of phase flip channel. However, for bit flip channel, no sudden death happens for the six-qubit states. On the other hand, depolarizing channel heavily influences the QD and GQD as compared to the amplitude damping channel. It means that the depolarizing channel has the most destructive influence on the discords for multipartite states. From the perspective of accelerated observers, it is seen that effect of environment on QD and GQD is much stronger than that of the acceleration of non-inertial frames. The degradation of QD and GQD happens due to Unruh effect. Furthermore, QD exhibits more robustness than GQD when the multipartite systems are exposed to environment.

Keywords: Multipartite correlations; GQD; Werner-GHZ states; decoherence

2 I.

INTRODUCTION

During recent years, quantum discord [1-14] has become the main focus of fundamental research in the discipline of quantum information theory. It quantifies the total non-classical correlations in a quantum state. Recently, quantum discord for multipartite system has been investigated [15-17]. Its extension to N -partite GHZ state is given by Jianwei Xu [18]. The dynamics of quantum discord has been extensively studied in different contexts by various authors [19-45]. It has been shown that quantum discord is more robust than entanglement for bipartite systems. Furthermore, the quantum discord for a qubit-qutrit [46] and qubit-qudit [47] systems has been proposed. Quantum discord has been used in studies of quantum phase transition [48, 49] and to measure the quantum correlation between relatively accelerated observers [50]. The geometric interpretation of the geometric discord has been discussed by Yao et al. [51]. The lower and upper bounds of quantum discord have also been investigated [52, 53]. Its experimental evidences can be seen in references [54-57]. Recently, the geometric measure of quantum discord (GMQD) has been proposed [58, 59], which quantifies the amount of non-classical correlations of a state in terms of its minimal distance from the set of genuinely classical states. However, most of the studies in this connection deal with bipartite quantum states. It has also been studied over two-sided projective measurements by [60]. Recently, the geometric quantum discord (GQD) for multipartite quantum states have been introduced by Jianwei Xu [61] and its lower bound is given. Motivated from the previous studies regarding the influence of environment on the bipartite quantum discord [62] and its geometric measure [63], I have investigated the global quantum discord (QD) and its geometric measure (GQD) under the influence of external environments for different types of multipartite quantum states. Quantum systems can never be isolated from their environments completely and the interactions with the environment deteriorate the purity of the quantum states. This gives rise to the phenomenon of decoherence [64], which appears when a system interacts with its environment in a irreversible way. It plays a fundamental role in the description of the quantum-to-classical transition [65] and has been successfully applied in the cavity QED [66]. Another familiar aspect, the degradation of entanglement has also been investigated recently by several authors [67-70], with special attention from the non-inertial perspective. Entanglement in noninertial frames was first time introduced by Alsing et al. [71]. The subject have attracted much attention during recent years [72-80]. It has also been investigated under decoherence for bipartite [81-85], qubit-qutrit [86] and multipartite [87] systems. The entanglement dynamics for noninertial observers in a correlated

3 environment is considered in Ref. [88], where it is shown that correlated noise compensates the loss of entanglement caused by the Unruh effect. Recently, I have studied the decoherece dynamics of GMQD and MIN at finite temperature for non-inertial observers [89]. In this paper, I have investigated the decoherence dynamics of multipartite quantum discord and geometric quantum discord for GHZ-type initial states in inertial and non-inertial frames. Different decoherence channels are considered parameterized by decoherence parameter p such that p ∈ [0, 1]. The lower and upper limits of decoherence parameter represent the fully coherent and fully decohered system, respectively. It is seen that the quantum discord is more robust than geometric quantum discord under environmental effects. However, Werner-GHZ type states are found to be more fragile to decoherence for higher values of N . The depolarizing channel heavily influences the QD and GQD as compared to the amplitude damping channel. It is also seen that the effect of environment on QD and GQD is much stronger than that of the acceleration of the accelerated observer. The degradation of QD and GQD happens due to Unruh effect.

II.

EVOLUTION OF MULTIPARTITE QUANTUM STATES

The quantum discord, a measure of the minimal loss of correlation in the sense of quantum mutual information, can be defined for a bipartite quantum state ρAB composed of subsystems A and B, as [1] QD(ρAB ) = I(ρAB ) − C(ρAB )

(1)

I(ρAB ) = S(ρA ) + S(ρB ) − S(ρAB )

(2)

where

is the quantum mutual information and measures. Here ρA,B =TrB,A ρAB are the reduced density matrices and S(ρ) = −Tr(ρ log ρ)

(3)

is the von-Neumann entropy. C(ρAB ) = max{Π} [S(ρA ) − S(ρAB |{Πk })] is the measure of classical correlations between the two subsystems. It is defined as the maximum information about one system that can be obtained by performing a set of projective measurements on the other subsystem and the maximum is taken over the set of projective measurements {Πk } as S(ρ|ΠA ) :=

X k

pk S(ρk )

(4)

4 and ρk =

1 A (Π ⊗ IB )ρ(ΠA k ⊗ IB ) pk k

(5)

with A pk = Tr[(ΠA k ⊗ IB )ρ(Πk ⊗ IB )], k = 1, 2

(6)

Recently, Rulli et al. [15] have proposed global quantum discord (QD) for multipartite quantum 1 states. QD for an arbitrary N -partite state ρA1 A2 .......AN under the set of local measurement {ΠA k ⊗ N ...... ⊗ ΠA k } can be defined as

D(ρA1 A2 .......AN ) = min[S(ρA1 A2 .......AN ||Φ(ρA1 A2 .......AN )) − {Π}

where Φj (ρAj ) =

P

Aj Aj i Πi ρAj Πi

and Φ(ρA1 A2 .......AN ) =

P

k

N X

S(ρAj ||Φj (ρAj ))]

j=1

(7)

1 Πk ρA1 A2 .......AN Πk with Πk = ΠA j1 ⊗

N ...... ⊗ ΠA jN . One can select the set of von-Neumann measurements as

A

Π1 j =

A

θ

θ

θ

θ

sin2 ( 2j )

θ

θ

sin2 ( 2j ) θ

θ

eiφj cos( 2j ) sin( 2j )

(8)

θ

e−iφj cos( 2j ) sin( 2j )

Π2 j =

θ

eiφj cos( 2j ) sin( 2j )

cos2 ( 2j )

θ

−e−iφj cos( 2j ) sin( 2j ) θ

cos2 ( 2j )

(9)

One must find the measurement bases that minimize the QD by varying the angles θj and φj , which is achieved by adopting local measurements in the σ z eigenbases for GHZ-type initial states. The geometric quantum discord for bipartite X-state system has been proposed by Dakic et al [58] as DG (ρ) := min kρ − χk2

(10)

ΠA

where the minimum is over the set of zero-discord states χ. The square of Hilbert-Schmidt norm of Hermitian operators, kρ − χk2 =Tr[(ρ − χ)2 ]. Whereas the geometric quantum discord (GQD) for N -partite state ρρA

1 A2 .......AN

is defined as

D G (ρA1 A2 .......AN ) =

N X j=1

S(ρAj ) − S(ρA1 A2 .......AN )

− max Π

hX

S(ΠAj (ρAi )) − S(Π(ρρA

1 A2 .......AN

i ))

(11)

5 where Π = ΠρA

1 .......AN

is a locally projective measurement on A1 A2 .......AN . Here in this paper, we

consider three different types of initial states as part of the following general N -qubit Werner-GHZ initial state of the form

ρ = (1 − µ)

I ⊗N + µ|ψi hψ| 2N

(12)

where I is 2 × 2 identity operator, µ ∈ [0, 1] and |ψi is the N -qiubit GHZ state |ψi = (|00....0i + √ |11.....1i)/ 2. (i) Let the three partners share a three-qubit Werner-GHZ initial state as given by ρ = (1 − µ)

I ⊗3 + µ|ψi hψ| 8

(13)

√ where |ψi = (|000i + |111i)/ 2. (ii) A six-qubit Werner-GHZ initial state as given by ρ = (1 − µ)

I ⊗6 + µ|ψi hψ| 64

(14)

√ where |ψi = (|000000i + |111111i)/ 2. (iii) Let the three observers: Alice, an inertial observer, Bob and Charlie, the accelerated observers moving with uniform acceleration, share the following maximally entangled GHZ-type initial state 1 |ΨiABC = √ (|0ω a iA |0ω b iB |0ω c iC + |1ωa iA |1ω b iB |1ω c iC ) 2

(15)

where |0ω a(bc) iA(BC) and |1ω a(bc) iA(BC) are vacuum states and the first excited states from the perspective of an inertial observer respectively. Let the Dirac fields, as shown in Refs. [90], from the perspective of the uniformly accelerated observers, are described as an entangled state of two modes monochromatic with frequency ω i , ∀i |0ω i iM = cos ri |0ω i iI |0ω i iII + sin ri |1ω i iI |1ω i iII

(16)

and the only excited state is |1ω i iM = |1ω i iI |0ω i iII

(17)

where cos ri = (e−2πωc/ai + 1)−1/2 , ai is the acceleration of ith observer. The subscripts I and II of the kets represent the Rindler modes in region I and II, respectively, in the Rindler spacetime diagram (see Ref. [82], Fig. (1)). Considering that an accelerated observer in Rindler region I has no access to the field modes in the causally disconnected region II and by taking the trace over

6 the inaccessible modes, one obtains the following tripartite state in Rindler spacetime as given by [78] 1 |ΨiABI CI = √ [cos rb cos rc |0iA |0iBI |0iCI + cos rb sin rc |0iA |0iBI |1iCI 2 + sin rb cos rc |0iA |1iBI |0iCI + sin rb sin rc |0iA |1iBI |1iCI +|1iA |1iBI |1iCI ]

(18)

For the sake of simplicity, the frequency subscripts are dropped and in density matrix formalism, the above state can be written as 1 ρABI CI = √ [cos rb2 cos rc2 |000i h000| + cos rb2 sin rc2 |001i h001| 2 + sin rb2 cos rc2 |010i h010| + sin rb2 sin rc2 |011i h011| + cos rb cos rc (|000i h111| + |111i h000|) + |111i h111|]

(19)

In order to simplify our calculations, it is assumed that Bob and Charlie move with the same acceleration, i.e. rb = rc = r. The interaction between the system and its environment introduces the decoherence to the system, which is a process of the undesired correlation between the system and the environment. The evolution of a state of a quantum system in a noisy environment can be described by the super-operator Φ in the Kraus operator representation as [91]

ρf = Φ(ρini ) =

X

Ek ρi Ek†

(20)

k

where the Kraus operators Ek satisfy the following completeness relation

P

k

Ek† Ek = I. The Kraus

operators for the evolution of N -partite system can be constructed from the single qubit Kraus operators by taking their tensor product over all nN combinations of π (i) indices as Ek = ⊗eπ(i) , π

where n corresponds to the number of Kraus operators for a single qubit channel. The single qubit Kraus operators for different channels are given in table 1. Using equations (7-11) along with the initial density matrices as given in equations (13, 14 and 19), the QD and GQD for the multipartite system under different environments can be found. The analytical relations for GQD for all the three types of initial states are given in tables 2, 3 and 4 respectively. However, the analytical expressions for QD are too lengthy, therefore, these are not presented in the text and have been explained in figures.

7 III.

DISCUSSIONS

Analytical expressions for the quantum discord and geometric quantum discord are calculated for different situations. Influence of different decoherence channels such as amplitude damping, phase damping, depolarizing and phase flip, bit flip and bit-phase flip channels is investigated for GHZ type initial states in inertial and non-inertial frames. The results consist of three parts (i) the effect of decoherence parameter p on the quantum discord (QD) and geometric quantum discord (GQD) for three-qubit Werner-GHZ states (ii) the effect of decoherence parameter p on QD and GQD for six-qubit Werner-GHZ states (iii) the dynamics of quantum discord and geometric quantum discord for three-qubit GHZ state in non-inertial frames influenced by different decoherence channels. In figure 1, the quantum discord is plotted as a function of decoherence parameter p for µ = 0.5 for different noisy channels for three-qubit and six-qubit GHZ states. It is seen that the quantum discord vanishes for p > 0.75 in case of three-qubit GHZ states and for p > 0.5 for six qubit GHZ states. The depolarizing channel has dominant effect on the discord as compared to the amplitude damping channel. Whereas the behaviour of flipping channels such as bit flip, phase flip and bit-phase flip channels is symmetrical around 50% decoherence as expected. It is seen that rapid sudden death of discord occurs in case of phase flip channel. However, for bit flip channel, no sudden death happens in case of six-qubit states. The rise and fall of quantum discord is seen for all the flipping channels. It is shown that states with higher number of qubits (N = 6) are more prone to decoherence as compared to the states with less number of qubits (N = 3). In figure 2, the geometric quantum discord is plotted as a function of decoherence parameter p for µ = 0.5 for different noisy channels, panel (a) three-qubit GHZ states and panel (b) six-qubit GHZ states. It can be seen from the figure that the depolarizing channel heavily influences the geometric quantum discord if compared with amplitude damping channel. It is seen that quantum discord is more robust than geometric quantum discord. Furthermore, it is seen that the vanishing of geometric quantum discord happens for both cases of the qubit states for bit-flip channel. In figure 3, the geometric quantum discord is plotted as a function of decoherence parameter p and acceleration r (a) amplitude damping channel and (b) bit-phase flip channel, for different values of acceleration r lower panel, respectively. It is seen that the behaviour of bit-phase flip channel is symmetrical at p = 0.5. It is also seen that effect of environment on multipartite geometric quantum discord is much stronger than that of the acceleration of non-inertial frames. In figure 4, the quantum discord and geometric quantum discord are plotted as a function of

8 acceleration r for different decoherence channels at p = 0.5. To illustrate the environmental effects, a comparison for different values of decoherence parameter p is given in sub figure (d) for amplitude damping channel only. It is seen that the degradation of quantum discord and geometric quantum discord occurs due to Unruh effect. The depolarizing channel influences the discords more heavily as compared to the other channels in non-inertial frames. It means that the depolarizing channel has the most destructive influence on the discords for multipartite states.

IV.

CONCLUSIONS

Quantum discord (QD) and geometric quantum discord (GQD) for multipartite quantum states is investigated under the influence of external environments. Different types of initial states such as, Werner-GHZ type three-qubit and six-qubit states are considered in inertial and non-inertial frames. Dynamics of QD and GQD is investigated under amplitude damping, phase damping, depolarizing and flipping channels. It is seen that the quantum discord is more robust than geometric quantum discord under environmental effects. However, Werner-GHZ type states are found to be more fragile to decoherence for higher values of N . Sudden death and birth of discords occur in case of flipping channels. However, for bit flip channel, no sudden death happens for six-qubit GHZ states. Whereas depolarizing channel heavily influences the QD and GQD as compared to the amplitude damping channel. This implies that the depolarizing channel has the most destructive influence on the discords for multipartite states. For the case of accelerated observers, it is seen that the effect of environment on QD and GQD is much stronger than that of the acceleration of the accelerated observer. However, the degradation of QD and GQD happens due to Unruh effect. In general, QD exhibits more robustness than GQD when multipartite systems are exposed to environment.

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(a) N=3

(b) N=6

0.4

0.5 AD Dep

0.3

12

AD Dep

0.4

QD

QD

0.3 0.2

0.2 0.1

0

0.1

0

0.5

0

1

0

0.5

p

1

p

0.35

0.25

PF BPF

0.4

0.2

QD

QD

0.5

BF PF BPF

0.3

0.15

0.3 0.2

0.1 0.1

0.05 0

0

0.5

0

1

0

0.5

p

1

p

FIG. 1: (Color online). Quantum discord (QD) is plotted as a function of decoherence parameter p for µ = 0.5 for different noisy channels, panel (a) three-qubit GHZ states and panel (b) six-qubit GHZ states.

TABLE I: Single qubit Kraus operators for amplitude damping, depolarizing, phase damping, bit-phase flip, bit flip and phase flip channels where p represents the decoherence parameter.

Depolarizing channel

0

0

√ p

, A1 = √ 1−p 0 0 0 0 1 0 , E1 = E0 = √ √ 0 p 0 1−p q p A1 = p4 σ x A0 = 1 − 3p 4 I, p p A2 = p4 σ y , A3 = p4 σ z

Amplitude damping channel A0 = Phase damping channel

1 0

Bit-phase flip channel

A0 =

√ 1 − pI,

A1 =

√ pσ y

Bit flip channel

A0 =

√ 1 − pI,

A1 =

√ pσ x

Phase flip channel

A0 =

√ 1 − pI,

A1 =

√ pσ z

(a) N=3

(b) N=6

0.2

0.2 AD Dep 0.15

GQD

GQD

0.15

0.1

0.05

0.1

0.05

0

0

0.5

0

1

0

0.5

p

1

p

0.2

0.2 BF PF BPF

PF BPF 0.15

GQD

0.15

GQD

13

AD Dep

0.1

0.05

0.1

0.05

0

0

0.5

0

1

0

0.5

p

1

p

FIG. 2: (Color online). Geometric quantum discord (GQD) is plotted as a function of decoherence parameter p for µ = 0.5 for different noisy channels, panel (a) three-qubit GHZ states and panel (b) six-qubit GHZ states.

(b) BPF

(a) AD

0.6 GQD

GQD

0.4 0.2 0

0.4 0.2 0

1

0.5

1

0.5

0.5 r

0 0

0.5 r

p

0.5

r=0 r=π/6 r=π/4

0.4

0.3

GQD

GQD

p

0.5 r=0 r=π/6 r=π/4

0.4

0.2 0.1 0

0 0

0.3 0.2 0.1

0

0.5

p

1

0

0

0.5

1

p

FIG. 3: (Color online). Geometric quantum discord (GQD) is plotted as a function of decoherence parameter p for panel (a) amplitude damping channel, for different values of acceleration r panel (b) bit-phase flip channel respectively.

(a) AD

(b) Dep

0.25

0.035 GQD QD

0.2

14

GQD QD

0.03 0.025

0.15

0.02

0.1

0.015 0.01

0.05 0

0.005 0

0.2

0.4

0.6

0

0.8

0

0.2

0.4

r (c) BPF

0.8

(d) AD

0.35

0.7 GQD QD

0.3

0.4

QD

0.5

0.2 0.15

0.3

0.1

0.2

0.05

0.1 0

0.2

0.4

0.6

p=0.3 p=0.7

0.6

0.25

0

0.6

r

0.8

0

0

0.2

r

0.4

0.6

0.8

r

FIG. 4: (Color online). Quantum discord and Geometric quantum discord are plotted as a function of acceleration r for different decoherence channels at p = 0.5. A comparison for different values of decoherence parameter p is given in fig (d) for the case of amplitude damping channel.

TABLE II: Analytical expressions of GQD for Werner-GHZ three-qubit state under different environments. Channel Description GQD (3-qubit GHZ state) Amplitude damping

Depolarizing

Phase damping

Bit flip

Phase flip

Bit-phase flip

1 2

3

(1 − p) µ2

1 512

4

2

(4 − 3 p) (1 − p) µ2

1 2 + 512 p (4 + p (7 − 3 p))2 µ2

1 2

3

(1 − p) µ2

3 2 2p

1 − 3p + 2 p2

2

µ2

+ 21 1 − p 3 − 3p + 2p2 1 2

2

µ2

2

µ2

(1 − 2p)6 µ2

3 2 2p

1 − 3p + 2 p2

2

µ2

+ 21 1 − p 3 − 3p + 2p2

15

TABLE III: Analytical expressions of GQD for Werner-GHZ six-qubit state under different environments. Channel Description GQD (6-qubit GHZ state) Amplitude damping

1 2

(1 − p)6 µ2

Depolarizing

1 2

(1 − p)

Phase damping

1 2

(1 − p) µ2

12

µ2

6

6

Bit flip

Phase flip

4

4 20 (1 − p) p6 µ2 + 15 2 (1 − p) p × 2 2 1 − 2p + 2p2 µ2 + 21 (1 − p)6 + p6 µ2 2 2 +3 (1 − p) p2 1 − 4p + 6p2 − 4p3 + 2p4 µ2

1 2

12

(1 − 2p)

µ2

6

Bit-phase flip

4

4 20 (1 − p) p6 µ2 + 15 2 (1 − p) p × 2 2 1 − 2p + 2p2 µ2 + 21 (1 − p)6 + p6 µ2 2 2 +3 (1 − p) p2 1 − 4p + 6p2 − 4p3 + 2p4 µ2

TABLE IV: Analytical expressions of GQD for GHZ state in non-inertial frames (Eq. 19) under different environments. Channel Description GQD (3-qubit state Eq. (19)) Amplitude damping

Depolarizing

1 2

3

(1 − p) cos4 (r)

1 512

4

2

(4 − 3p) (1 − p) cos4 (r)

1 2 p (4 − 7p + 3p2 ) cos4 (r) + 512 3

Phase damping

1 2

(1 − p) cos4 (r)

Phase flip

1 2

(1 − 2p) cos4 (r)

Bit-phase flip

3 2 2p

+ 21

6

1 − 3p + 2p2

2

cos4 (r) 2 1 − 3p + 3p2 − 2p3 cos4 (r)