# Entanglement of Tripartite States with Decoherence in Noninertial frames

Salman Khan<EMAIL_ADDRESS>Department of Physics, Quaid-i-Azam
University, Islamabad 45320, Pakistan

(Jun 8 2011)

###### Abstract

The one-tangle and $\pi$-tangle are used to quantify the entanglement of a
tripartite GHZ state in noninertial frames when the system interacts with a
noisy environment in the form of phase damping, phase flip and bit flip
channel. It is shown that the two-tangles behave as a closed system. The one-
tangle and $\pi$-tangle have different behaviors in the three channel. In the
case of phase damping channel, depending on the kind of coupling, the sudden
death of both one-tangle and $\pi$-tangle may or may not happen. Whereas in
the case of phase flip channel the sudden death cannot be avoided. The effect
of decoherence may be ignored in the limit of infinite acceleration when the
system interacts with a bit flip channel. Furthermore, a sudden rebirth of the
one-tangle and $\pi$-tangle occur in the case of phase flip channel that may
be delayed when collective coupling is switched on.

PACS: 03.65.Ud; 03.65.Yz; 03.67.Mn;04.70.Dy

Keywords: Entanglement; Tripartite GHZ state; Decoherence; Noninertial frames

Entanglement; Tripartite GHZ state; Decoherence; Noninertial frames

###### pacs:

03.65.Ud; 03.65.Yz; 03.67.Mn;04.70.Dy

††preprint:

##  I Introduction

Entanglement is not only one of the most striking properties of quantum
mechanics but also the core concept of quantum information and quantum
computation springer . The structure of all the subfields of quantum
information theory, such as teleportation of unknown states Bennett , quantum
key distribution Ekert , quantum cryptography Bennett2  and quantum
computation Grover ; Vincenzo , stand on quantum entanglement. The dynamics of
entanglement in various bipartite qubit and qutrit states have been
extensively studied under various circumstances. The study of entanglement of
various fields in the accelerated frames has taken into account recently and
valuable results about the behavior of entanglement have been obtained Alsing
; Ling ; Gingrich ; Pan ; Schuller ; Terashima . The effect of decoherence on
the behavior of entanglement under various quantum channels in noninertial
frames have been studied in Refs. Wang ; Salman . However, these studies are
limited only to bipartite qubit systems in the accelerated frames. Recently,
the dynamics of entanglement in tripartite qubit systems in noninertial frames
are studied in Refs. Hwang ; Wang2 ; Shamirzai . These studies show that the
degree of entanglement is degraded by the acceleration of the frames and, like
the two-tangles in inertial frames, the two-tangles are zero when one or two
observers are in the accelerated frames.

In this paper, I investigate the effect of decoherence on the tripartite
entanglement of Dirac field in accelerated frames by using a phase damping
channel, a phase flip channel and a bit flip channel. The effect of amplitude
damping channel and depolarizing channel on tripartite entanglement of Dirac
field in a noninertial frames is recently studied in Ref. Winpin . I consider
three observers Alice, Bob and Charlie that initially share a GHZ tripartite
state. One of the observers, say Alice, stays stationary and the other two
observers move with a constant acceleration. I work out the effect of
acceleration and of decoherence by using the three channels on the initial
entanglement of the shared state between the observers. I consider different
kinds of coupling of each channel with the system. For example, in one case
each qubit interacts locally with the noisy environment. In the second case,
all the three qubits are influenced collectively by the same environment. I
show that the entanglement sudden death (ESD) Yu1 ; Yu2  can either be
completely avoided or can be slowed down depending on the various coupling of
the system and a particular channel. Furthermore, I also show that the ESD can
happen faster and becomes independent of the acceleration when the system
interacts with a phase flip channel.

Table 1: A single qubit Kraus operators for phase damping channel, phase flip channel and bit flip channel.  phase damping channel | $E_{o}=\left(\begin{array}[c]{cc}1&0\\\ 0&\sqrt{1-p}\end{array}\right),\qquad E_{1}=\left(\begin{array}[c]{cc}0&0\\\ 0&\sqrt{p}\end{array}\right)$
---|---
phase flip channel | $E_{o}=\sqrt{1-p}\left(\begin{array}[c]{cc}1&0\\\ 0&1\end{array}\right),\qquad E_{1}=\sqrt{p}\left(\begin{array}[c]{cc}1&0\\\ 0&-1\end{array}\right)$
bit flip channel | $E_{o}=\sqrt{1-p}\left(\begin{array}[c]{cc}1&0\\\ 0&1\end{array}\right),\qquad E_{1}=\sqrt{p}\left(\begin{array}[c]{cc}0&1\\\ 1&0\end{array}\right)$

##  II The system in noisy environment

The evolution of a density matrix of a system in a noisy environment is
described in terms of Kraus operators formalism. The Kraus operators for a
single qubit system of the three channels that I use in this paper are given
in Table $1$. Each channel is parameterized by the decoherence parameter $p$
that has values between $0$ and $1$. For the lower limit of $p$, a channel has
no effect on the system and is said to be undecohered whereas for the upper
limit it introduces maximum error and is said to be fully decohered. The final
density matrix of a composite system when it evolves in a noisy environment is
given by the following equation

$\rho_{f}=\sum_{i,j,k,...}K_{i}K_{j}K_{k}...\rho...K_{k}^{{\dagger}}K_{j}^{{\dagger}}K_{i}^{{\dagger}},$
(1)

where $\rho$ is the initial density matrix of the system and $K_{n}$ are the
Kraus operators that satisfy the completeness relation
$\sum_{n}K_{n}^{{\dagger}}K_{n}=I$. I consider that Alice, Bob and Charlie
initially share the following maximally entangled GHZ state

$|\psi\rangle_{ABC}=\frac{1}{\sqrt{2}}\left(|000\rangle_{A,B,C}+|111\rangle_{A,B,C}\right),$
(2)

where the three alphabets represent the three observers and the entries in
each ket correspond to the three observers in the order of alphabet. From the
perspective of an inertial observer, the Dirac fields can be expressed as a
superposition of Minkowski monochromatic modes
$|0\rangle_{M}=\bigoplus_{i}|0_{\omega_{i}}\rangle_{M}$ and
$|1\rangle_{M}=\bigoplus_{i}|1_{\omega_{i}}\rangle_{M}$ $\forall i$, with
AsPach ; Martin

$\displaystyle|0_{\omega_{i}}\rangle_{M}$ $\displaystyle=\cos
r_{i}|0_{\omega_{i}}\rangle_{I}|0_{\omega_{i}}\rangle_{II}+\sin
r_{i}|0_{\omega_{i}}\rangle_{I}|0_{\omega_{i}}\rangle_{II},$
$\displaystyle|1_{\omega_{i}}\rangle_{M}$
$\displaystyle=|1_{\omega_{i}}\rangle_{I}|1_{\omega_{i}}\rangle_{II},$ (3)

where $\cos r_{i}=\left(e^{-2\pi\omega_{i}c/a_{i}}+1\right)^{-1/2}$. The
parameters $\omega_{i}$, $c$ and $a_{i}$, in the exponential stand,
respectively, for Dirac particle’s frequency, speed of light in vacuum and
acceleration of the $i$th observer. In Eq. (3) the subscripts $I$ and $II$ of
the kets represent the Rindler modes in region $I$ and $II$, respectively, in
the Rindler spacetime diagram. The Minkowski mode that defines the Minkowski
vacuum is related to a highly nonmonochromatic Rindler mode rather than a
single mode with the same frequency Martin ; Bruschi . Let the angular
frequencies $\omega_{i}$ for the three observers be, respectively, given by
$\omega_{a}$, $\omega_{b}$ and $\omega_{c}$ then with respect to an inertial
observer $|0_{\omega_{a(b,c)}}\rangle_{A(B,C)}$ and
$|1_{\omega_{a(b,c)}}\rangle_{A(B,C)}$ are the vacuum and the first excited
states. Furthermore, let us suppose that each observer is equipped with a
monochromatic device that are sensitive only to their respective modes. In
order to save space and present the relation in simple form, I drop the
frequencies in subsript of each entry of the kets. Then substituting the
Rindler modes from Eq. (3) for the Minkowski modes in Eq. (2) for the two
observers in the accelerated frames gives

$\displaystyle|\psi\rangle_{ABC}$ $\displaystyle=\frac{1}{\sqrt{2}}(\cos
r_{b}\cos r_{c}|00000\rangle_{A,BI,BII,CI,CII}$ $\displaystyle+\cos r_{b}\sin
r_{c}|00011\rangle_{A,BI,BII,CI,CII}$ $\displaystyle+\sin r_{b}\cos
r_{c}|01100\rangle_{A,BI,BII,CI,CII}$ $\displaystyle+\sin r_{b}\sin
r_{c}|01111\rangle_{A,BI,BII,CI,CII}$
$\displaystyle+|11010\rangle_{A,BI,BII,CI,CII}).$ (4)

Let the noninertial frames of Bob and Charlie be moving with same
acceleration, then $r_{b}=r_{c}=r$. Since the Rindler modes in region $II$ are
inaccessible, tracing out over those modes leave the following initial mixed
density matrix

$\displaystyle\rho$
$\displaystyle=\frac{1}{2}[\cos^{2}r_{b}\cos^{2}r_{c}|000\rangle\langle
000|+\cos^{2}r_{b}\sin^{2}r_{c}|001\rangle\langle 001|$
$\displaystyle+\sin^{2}r_{b}\cos^{2}r_{c}|010\rangle\langle
010|+\sin^{2}r_{b}\sin^{2}r_{c}|011\rangle\langle 011|$ $\displaystyle+\cos
r_{b}\cos r_{c}(|000\rangle\langle 111|+|111\rangle\langle
000|)+|111\rangle\langle 111|.$ (5)

Note that I have dropped the subsript $I$ that indicates the Rindler modes in
region $I$. In the rest of the paper, all calculations correspond to the
Rindler modes in region $I$.

The degree of entanglement in bipartite qubit states is quantified by using
the concept of negativity Peres ; Horodecki1 . It is given by

$\mathcal{N}_{AB}=\left\|\rho_{AB}^{T_{B}}\right\|-1,$ (6)

where $T_{B}$ is the partial transpose over the second qubit $B$ and
$\left\|.\right\|$ gives the trace norm of a matrix. For a bipartite density
matrix $\rho_{m\nu,n\mu}$, the partial transpose over the second qubit $B$ is
given by $\rho_{m\mu,n\nu}^{T_{B}}=\rho_{m\nu,n\mu}$ and for the first qubit,
it can similarly be defined. On the other hand, the entanglement of a
tripartite state $|\psi\rangle_{ABC}$ is quantified by using the concept of
$\pi$-tangle, which is given by

$\pi_{ABC}=\frac{1}{3}(\pi_{A}+\pi_{B}+\pi_{C}),$ (7)

where $\pi_{A(BC)}$ is the residual entanglement and is given by

$\pi_{A}=\mathcal{N}_{A(BC)}^{2}-\mathcal{N}_{AB}^{2}-\mathcal{N}_{AC}^{2}.$
(8)

In Eq. (8), $\mathcal{N}_{AB(AC)}$ is a two-tangle and is given as the
negativity of the mixed density matrix
$\rho_{AB(AC)}=Tr_{C(B)}|\psi\rangle_{ABC}\langle\psi|$ whereas
$\mathcal{N}_{A(BC)}$ is a one-tangle and is defined as
$\mathcal{N}_{A(BC)}=\left\|\rho_{ABC}^{T_{A}}\right\|-1$.

Now, in the following I use the three channels to investigate the behavior of
entanglement of the tripartite initially entangled density matrix of Eq. (8)
in a noisy environment.

###  II.1 Phase damping channel

When all the three qubits are locally coupled to a phase damping channel, then
Eq. (1) for the evolution of a tripartite state can be written as

$\rho_{f}=\sum_{i,j,k}(K_{i}K_{j}K_{k})\rho\left(K_{k}^{{\dagger}}K_{j}^{{\dagger}}K_{i}^{{\dagger}}\right).$
(9)

The Kraus operators in Eq. (9) with subscripts $i$, $j$, $k$ act,
respectively, on Alice’s, Bob’s and Charlie’s qubits. These operators are
formed from the single qubit Kraus operators of a phase damping channel and
are, respectively, given by $K_{i}=(E_{i}\otimes I\otimes I)$,
$K_{j}=(I\otimes E_{j}\otimes I)$, $K_{k}=(I\otimes I\otimes E_{k})$ with
$i=j=k=0$, $1$. Using the initial density matrix of Eq. (5) in Eq. (9), the
final density matrix of the system is given as

$\displaystyle\rho_{f}$ $\displaystyle=\frac{1}{2}\cos^{4}r|000\rangle\langle
000|+\frac{1}{8}\sin^{2}2r(|001\rangle\langle 001|+|010\rangle\langle 010|)$
$\displaystyle+\frac{1}{2}\sin^{4}r|011\rangle\langle
011|+\frac{1}{2}|111\rangle\langle 111|$
$\displaystyle+\frac{1}{2}\sqrt{(1-p_{0})(1-p_{1})(1-p_{2})}\cos^{2}r(|000\rangle\langle
111|+|111\rangle\langle 000|),$ (10)

where $p_{0}$, $p_{1}$ and $p_{2}$ are the decoherence parameters of the
locally coupled channels with the qubits of Alice, Bob and Charlie,
respectively. The three one-tangles that can straightforwardly be calculated
by using its definition are given as

$\displaystyle\mathcal{N}_{A(BC)}$
$\displaystyle=\frac{1}{4}(-2+2\cos^{4}r+2\sqrt{(1-p_{0})(1-p_{1})(1-p_{2})\cos^{4}r}$
$\displaystyle+2\sqrt{(1-p_{0})(1-p_{1})(1-p_{2})\cos^{4}r+\sin^{8}r}+\sin^{2}2r),$
(11) $\displaystyle\mathcal{N}_{B(AC)}$
$\displaystyle=\mathcal{N}_{C(AB)}=\frac{1}{16}(-1+8\sqrt{(1-p_{0})(1-p_{1})(1-p_{2})\cos^{4}r}$
$\displaystyle+\cos
4r+2\sqrt{16(1-p_{0})(1-p_{1})(1-p_{2})\cos^{4}r+\sin^{4}2r}).$ (12)

(a) |
---|---

(b) |
---|---

Figure 1: (color online) (a) The one-tangles, for $r=\pi/6$, are plotted
against the decoherence parameter $p$ when only one qubit is coupled to the
phase damping channel. (b) The one tangles, for $r=\pi/6$, are plotted against
$p$ when all the three qubit are influenced collectively by the same phase
damping channel.

Note that $\mathcal{N}_{B(AC)}=\mathcal{N}_{C(AB)}$ shows that the two
accelerated subsystems are symmetrical for any value of the acceleration. It
can easily be checked that all the one-tangles reduce to
$[(1-p_{0})(1-p_{1})(1-p_{2})]^{1/2}$ when $r=0$ and for $p_{i}=r=0,$ the
result is $1$, which is the result obtained in the rest frames both for Dirac
and scalar fields Hwang ; Wang2 . It is also clear that for $r=0$ case, the
one-tangles goes to zero when either of the three qubits is coupled to a fully
decohered channel. Moreover, for $r=\pi/4$ all the three one-tangles become
indistinguishable, that is,
$\mathcal{N}_{A(BC)}=\mathcal{N}_{B(AC)}=\mathcal{N}_{C(AB)}$ irrespective of
the value of decoherence parameter and of which qubit is coupled to the
channel. Also, in this case the one-tangle is lost only when either of the
three local channels is fully decohered. However, the initial entanglement is
small enough as compared to the case for $r=0$. For example, when only one
qubit is locally coupled to the channel, the initial value of the one-tangle
is $0.4$ for $p_{i}=0$. This shows the strong dependence of the one tangle
over the acceleration. However, the on-tangles are different for other values
of the acceleration. To see the behavior of the one-tangles for other values
of the acceleration, I plot it against the decoherence parameters for
$r=\pi/6$ in Fig. $1$($a,b$). Fig. $1a$ shows the behavior of the one-tangles
when only one qubit is locally influenced by the channel. This behavior does
not change by switching the coupling of the channel from one qubit to another,
that is, whether it’s plotted against $p_{0}$ or $p_{1}$($p_{2}$), which means
that the one-tangles are symmetrical in the phase damping environment. It can
be seen from the figure that both $\mathcal{N}_{A(BC)}$ and
$\mathcal{N}_{B(AC)}$($\mathcal{N}_{C(AB)}$) varies identically with the
increasing value of decoherence parameter, however, the acceleration effect is
different on both of them. Like for the case of $r=0$ and $r=\pi/4$, the one-
tangles goes to zero only when the channel is fully decohered, which shows
that no sudden death occurs. The behavior of one-tangles when all the three
qubits are under the influence of collective environment
($p_{0}=p_{1}=p_{2}=p$) is shown in Fig. $1b$ for $r=\pi/6$. In this case, the
one-tangles are heavily damped by the decoherence parameter as compared to Fig
$1a$ and become indistinguishable at $p=0.85$.

Next, I find the two-tangles according to the definition given above. Taking
partial trace of the final density matrix of Eq. (10) over the Bob’s qubit or
Charlie’s qubit leads to the following mixed state

$\rho_{AB(AC)}=\rho_{ABC}^{T_{C(B)}}=\frac{1}{2}(\cos^{2}r|00\rangle\langle
00|+\sin^{2}r|01\rangle\langle 01|+|11\rangle\langle 11|).$ (13)

Since non of the elements of the density matrix depends on the decoherence
parameter, the two-tangles are unaffected by the noisy environment. Similarly,
one can straightforwardly prove that tracing over Alice’s qubit leads to a
density matrix whose every element is independent of the decoherence
parameter. From these matrices it can easily be verified that
$\mathcal{N}_{AB}=\mathcal{N}_{AC}=\mathcal{N}_{BC}=0$. The zero value of all
the two-tangles verifies that no entanglement exists between any two
subsystems of the tripartite state.

\put(-320.0,220.0){} |  |
---|---|---
Figure 2: (color online) The $\pi$-tangle is plotted against the decoherence
parameter for $r=\pi/6$ and $r=\pi/4$ both for one qubit coupled with the
phase damping channel and for all the three qubit coupled collectively with
the same phase damping channel.

I am now in position to find the $\pi$-tangle by using its defining Eq. (7).
Since all the two-tangles are zero, the $\pi$-tangle simply becomes

$\displaystyle\pi_{ABC}$
$\displaystyle=\frac{1}{3}\left(\mathcal{N}_{A(BC)}^{2}+\mathcal{N}_{B(AC)}^{2}+\mathcal{N}_{C(AB)}^{2}\right)$
$\displaystyle=\frac{1}{384}[8(-2+2\cos^{4}r+2\sqrt{(1-p_{0})(1-p_{1})(1-p_{2})\cos^{4}r}$
$\displaystyle+2\sqrt{(1-p_{0})(1-p_{1})(1-p_{2})\cos^{4}r+\sin^{8}r}+\sin^{2}2r)^{2}$
$\displaystyle+(-1+8\sqrt{(1-p_{0})(1-p_{1})(1-p_{2})\cos^{4}r}+\cos 4r$
$\displaystyle+2\sqrt{16(1-p_{0})(1-p_{1})(1-p_{2})\cos^{4}r+\sin^{4}2r})^{2}].$
(14)

From Eq. (14), it can easily be verified that for the case of inertial frames
the $\pi$-tangle reduces to $(1-p_{i})$ when only one qubit is locally coupled
to the noisy environment. This is easy to check that this result goes to zero
only when the decoherence parameter is $1$. Moreover, for any acceleration,
the $\pi$-tangle does not depend on whether the stationary or the accelerated
qubit is coupled to the environment. In Fig. $2$, I plot the $\pi$-tangle for
different accelerations when only one qubit is coupled to the environment. It
can be seen that the $\pi$-tangle is lost only when the channel is fully
decohered and thus no sudden death occurs in this case as well. I have also
plotted the $\pi$-tangle for the case of collective environment in the same
figure. The graphs show that the sudden death can happen when the system is
influenced by collective environment.

###  II.2 Phase flip channel

When the three qubits are locally coupled to a phase flip channel, the final
density matrix of the system is again given by Eq. (9). The subscripts and the
number of Kraus operators are the same as described for the previous case,
however, these are now made from single qubit phase flip Kraus operators. The
final density matrix of the system in this case becomes

$\displaystyle\rho_{f}$ $\displaystyle=\frac{1}{2}\cos^{4}r|000\rangle\langle
000|+\frac{1}{8}\sin^{2}2r(|001\rangle\langle 001|+|010\rangle\langle 010|)$
$\displaystyle+\frac{1}{2}(1-2p_{0})(1-2p_{1})(1-2p_{2})\cos^{2}r(|000\rangle\langle
111|+|111\rangle\langle 000|)$
$\displaystyle+\frac{1}{2}\sin^{4}r|011\rangle\langle
011|+\frac{1}{2}|111\rangle\langle 111|.$ (15)

Using the definition of one-tangle, the three one-tangles in this case become

$\displaystyle\mathcal{N}_{A(BC)}$
$\displaystyle=1/4(-2+2\cos^{2}r(\left|(1-2p_{0})(1-2p_{1})(1-2p_{2})\right|+\cos^{2}r)$
$\displaystyle+2\sqrt{(1-2p_{0})^{2}(1-2p_{1})^{2}(1-2p_{2})^{2}\cos^{4}r+\sin^{8}r}+\sin^{2}2r)$
$\displaystyle\mathcal{N}_{B(AC)}$
$\displaystyle=\mathcal{N}_{C(AB)}=\frac{1}{8}(-4+4\left|(1-2p_{0})(1-2p_{1})(1-2p_{2})\right|\cos^{2}r+4\cos^{4}r$
$\displaystyle+4\sin^{4}r+\sin^{2}2r+\sqrt{16(1-2p_{0})^{2}(1-2p_{1})^{2}(1-2p_{2})^{2}\cos^{4}r+\sin^{4}2r})$
(16)

(a) |
---|---

(b) |
---|---

Figure 3: (color online) (a) The one-tangles, for $r=\pi/6$, are plotted
against the decoherence parameter $p$ when only one qubit is coupled to the
phase flip channel. (b) The one tangles, for $r=\pi/6$, are plotted against
$p$ when all the three qubit are influenced collectively by the phase flip
channel.

where, $\left|.\right|$ represents the absolute value. Once again, the
equality between the one-tangles for the accelerated frames shows the
symmetrical nature of the problem. Also, for $r=0$, the three one-tangles are
indistinguishable, as expected, and are given by
$\left|(1-2p_{0})(1-2p_{1})(1-2p_{2})\right|$. This shows that the one-tangles
vanishes at $p_{i}=0.5$ irrespective of which qubit is locally coupled to the
noisy environment. For undecohered case, this leads to the result of static
frames. Again, like in the case of phase damping channel, all the one-tangles
becomes indistinguishable at $r=\pi/4$, however, unlike the phase damping
channel it goes to zero at $p_{i}=0.5$. This means that the phase flip channel
destroys all the one-tangles earlier, in other words, the tripartite
entanglement is caused to sudden death. The one-tangles are shown in Fig. $3a$
for $r=\pi/6$ when only one qubit is locally influenced by the phase flip
channel. It is important to mention here that the channel effect is locally
symmetrical for both inertial and noninertial qubits. One can see that
initially the one-tangles are acceleration dependent and this dependence on
acceleration gradually decreases as the decoherence parameter increases and
the difference vanishes near $p_{i}=0.5$. This shows that the one-tangles are
heavily damped by the decoherence parameter in comparison to the damping
caused by the acceleration. Hence one can ignore the acceleration effect near
a $50\%$ decoherence level. All the one-tangles become zero at $p_{i}=0.5$,
which in fact happens for any acceleration. Interestingly enough, the one-
tangles’ sudden rebirth takes place for values of $p_{i}>0.5$ irrespective of
the acceleration of the frames. However, the regrowing rate is different for
different values of acceleration. This means that at $50\%$ decoherence level
all the tripartite entanglement first shifts to the environment and then part
of it shifts back to the system at values of decoherence parameter higher than
this value. In Fig. $3b$, the effect of collective environment on all the one-
tangles is shown for $r=\pi/6$ and $r=\pi/4$. It can be seen that the damping
caused by the collective environment is heavier than the damping in one qubit
local coupling case and a sudden death prior to a $50\%$ decoherence level may
happen. Moreover, the sudden rebirth of the one-tangles is also delayed.

One can further verify that by taking partial trace of Eq. (15) over Bob’s or
Charlie’s qubit leads to the same result as given in Eq. (13). This means that
no bipartite entanglement exits for this case as well, because all the two-
tangles are zero as before. The $\pi$-tangle can be found straightforwardly as
done in the previous case, which is one-third of the sum of the squares of the
three one-tangles. It is given by

$\displaystyle\pi_{ABC}$
$\displaystyle=\frac{1}{96}[2(-2+2\cos^{2}r(\left|(1-2p_{0})(1-2p_{1})(1-2p_{2})\right|+\cos^{2}r)$
$\displaystyle+2\sqrt{(1-2p_{0})^{2}(1-2p_{1})^{2}(1-2p_{2})^{2}\cos^{4}r+\sin^{8}r}+\sin^{2}2r)^{2}$
$\displaystyle+(-4+4\left|1-2p_{0}\right|\left|1-2p_{1}\right|\left|1-2p_{2}\right|\cos^{2}r+4\cos^{4}r+4\sin^{4}r$
$\displaystyle+\sin^{2}2r+\sqrt{16(1-2p_{0})^{2}(1-2p_{1})^{2}(1-2p_{2})^{2}\cos^{4}r+\sin^{4}2r})^{2}].$
(17)

\put(-320.0,220.0){} |  |
---|---|---
Figure 4: (color online) The $\pi$-tangle is plotted against the decoherence
parameter for $r=\pi/6$ and $r=\pi/4$ both for one qubit coupled with the
phase flip channel and for all the three qubit coupled collectively with the
same phase flip channel.

I plot it for one qubit local coupling case for $r=\pi/6$, $\pi/4$ in Fig.
$4$. It can be seen that the $\pi$-tangle is strongly acceleration dependent
for the lower limit of the decoherence parameter. However, as the decoherence
parameter increases, the acceleration dependence decreases and it goes to
zero, irrespective of the value of acceleration, at $p_{i}=0.5$. The sudden
rebirth of the $\pi$-tangle is not as quick as in the case of one-tangles.
This behavior of the $\pi$-tangle does not depend on which one qubit is
coupled to the environment. I have also plotted the $\pi$-tangle for the case
of collective environment for $r=\pi/6$, $\pi/4$. The figure shows that the
sudden death is faster when the system evolves under collective environment
and the rebirth is quite delayed.

(a) |
---|---

(b) |
---|---

Figure 5: (color online) (a) The one-tangles, for $r=0$ and $r=\pi/4$, are
plotted against the decoherence parameter $p$ when only Alice’s qubit is
coupled to the bit flip channel. (b) For the same two values of acceleration,
the one-tangles are plotted against the decoherence parameter when the
collective environment is switched on.

###  II.3 Bit flip channel

Finally, I investigate the effect of bit flip channel on the entanglement of
the tripartite system. When each qubit is locally coupled to the channel, the
final density matrix of the system is again given by Eq. (9) and the Kraus
operators have the same meaning as before. However, these are now made from
the single qubit bit flip channel in the same way as mentioned earlier.
Instead of writing the mathematical relations for one-tangles, which are quite
lengthy, I prefer to see the effect of decoherence only for special cases.
Unlike the two previous cases, the one-tangles for the accelerated observers
are not always equal. Similarly, for $r=0$ case, the equality of one-tangles
depends on which qubit is locally coupled to the noisy environment. For
example, when only Alice’s qubit is locally coupled to the channel, then
$\mathcal{N}_{B(AC)}=\mathcal{N}_{C(AB)}=1$ and
$\mathcal{N}_{A(BC)}=-1+2\sqrt{1-2p_{0}+2p_{0}^{2}}$ and if Bob’s qubit is
under the influence of the channel then
$\mathcal{N}_{A(BC)}=\mathcal{N}_{C(AB)}=1$ and
$\mathcal{N}_{B(AC)}=-1+2\sqrt{1-2p_{1}+2p_{1}^{2}}$. A similar relation
exists when only Charlie’s qubit is coupled to the environment. On the other
hand, all the three one-tangles are equal when the system is coupled to
collective environment and are given by
$-1+2\sqrt{2}\sqrt{(-1+p)^{2}p^{2}}+2\sqrt{1-6p+16p^{2}-20p^{3}+10p^{4}}$. A
similar situation arises for $r=\pi/4$ as well. That is, two one-tangles are
always equal and like for $r=0$ case, the equality between two one-tangles
depends on which qubit is under the action of the channel. However, unlike
$r=0$ case, non of the one-tangles is always equal to $1$, that is, every
tangle is decoherence parameter dependent. Fig. $5a$ shows the dynamics of the
one-tangles for $r=0$, $\pi/4$ when only Alice’s qubit is locally coupled to
the channel. The acceleration dependence of the one-tangles can easily be seen
as compared to $r=0$ case. Also, $\mathcal{N}_{B(AC)}=\mathcal{N}_{C(AB)}$ for
the whole range of decoherence parameter. Even though the behavior of each
one-tangle is symmetrical around $50\%$ decoherence level, the damping is more
for $\mathcal{N}_{A(BC)}$. The interesting features are the symmetrical
increase beyond $50\%$ decoherence level of each one-tangle back to its
initial value for a fully decohered channel and the occurrence of no sudden
death even in the limit of infinite acceleration. The behavior of one-tangles
for the case of collective environment is shown in Fig.$5b$. Even in this case
of strongly decohered system, there is no sudden death of any one-tangle. The
second remarkable feature is the three regions where the subsystems become
indistinguishable, that is, for lower limit, intermediate range and upper
limit of the decoherence parameter. In Fig. $6$, the results of one-tangles
are plotted for $r=\pi/6$ against the decoherence parameter when Alice’s qubit
interacts with the channel. Again, each one-tangle is symmetrical around
$p_{1}=0.5$, however, they are affected differently. It can be seen that there
are different particular couple of values of the decoherence parameter at
which all the three one-tangles are same and the subsystems become
indistinguishable. Interestingly, in between these points, the one-tangles of
the accelerated observers are less effected than the one-tangle of the static
observer.

\put(-320.0,220.0){} |  |
---|---|---
Figure 6: (color online) The one-tangles are plotted against the decoherence
parameter for $r=\pi/6$ when only Alice’s qubit is locally coupled to the bit
flip channel.

Like in the previous two cases, all the two-tangles are zero in this case as
well. The $\pi$-tangle can be found similar to the previous two cases. To see
how it is effected by the noisy environment, I plot it in Fig. $7$ against the
decoherence parameter for different values of the acceleration. The figure
shows the behavior of $\pi$-tangle for the case when only one qubit is
influenced by the channel. This behavior of the $\pi$-tangle does not change
by switching the coupling of the channel from one qubit to another, which
shows that the $\pi$-tangle is invariant with respect to the local coupling of
the channel with a single qubit. The acceleration dependence of the
$\pi$-tangle is quite clear from the figure. It can also be seen that the
damping caused by the decoherence parameter is heavy for small values of the
acceleration as compared to the case of large values of the acceleration. In
other words, the effect of decoherence diminishes as the acceleration
increases. This means that in the limit of infinite acceleration the
decoherence effect may be ignored. The behavior of $\pi$-tangle when the
system is under the action of collective environment has also been plotted in
Fig. $7$ for two different values of the acceleration. In comparison to single
qubit coupling, the damping is heavier, however, no sudden death occurs.

\put(-320.0,220.0){} |  |
---|---|---
Figure 7: (color online) The $\pi$-tangle is plotted against the decoherence
parameter for $r=\pi/6$ and $r=\pi/4$ both for one (Alice) qubit coupled with
the bit flip channel and for all the three qubit coupled collectively with the
same bit flip channel.

##  III Summary

In conclusion, the effects of different channels on the tripartite
entanglement of GHZ state in noninertial frames by using one-tangle and
$\pi$-tangle as the entanglement quantifier are investigated. It is shown that
non of the reduced density matrix of any two subsystems depends on the
decoherence parameter irrespective of the channel used. That is, the reduced
density matrices of the subsystems behave as closed systems for which the two-
tangles are always zero. This means that no entanglement exists between two
subsystems. In other words, the entanglement resource cannot be utilized by
any two observers without the cooperation of the third one. Under the
influence of phase damping and phase flip channels, the one-tangles of the
accelerated observers are indistinguishable for the whole range of
acceleration and decoherence parameters. Similarly, in the limit of infinite
acceleration all the three one-tangles are equal which shows that all the
subsystems are equally entangled. On the other hand, the situation is
different when the system is coupled to a bit flip channel. That is, the
equality between any two one-tangles depends on which qubit is coupled to the
channel. In other words, the share of entanglement between subsystems is
dependent on the coupling of a particular qubit to the channel. This asymmetry
of the one tangles can be used to identify the frame of the observer coupled
to the noisy environment. Also, under no circumstances the three one-tangles
become equal for the whole range of decoherence parameter, except for a couple
of values, when the system is coupled to a bit flip channel. In the case of
phase damping channel, no sudden death of any one-tangle happens under local
coupling of any qubit with the channel. However, it goes to zero when the
channel is fully decohered. The sudden death of one-tangles may happen when
the system interacts with collective phase damping channel. Under the action
of phase flip channel the one-tangle sudden death occurs for every
acceleration and then a sudden rebirth of the one-tangle that increases as the
decoherence parameter increases. In the case of bit flip channel, the one-
tangle always survives and the effect of decoherence may be ignored in the
range of large acceleration. The sudden death of $\pi$-tangles may or may not
happens under the action of phase damping channel. Whereas it’s sudden death
cannot be avoided when the system is influenced by phase flip channel. The
$\pi$-tangle is never lost when the system is coupled to a bit flip channel.
Which means that the entanglement of tripartite GHZ state is robust against
bit flip noise. The fact that both one-tangles and $\pi$-tangle are never lost
in bit flip channel may be useful for faithful communication in noninertial
frames. Finally, it needs to be pointed out that the presence of a noisy
environment does not violate the CKW inequality Coffman
$\mathcal{N}_{AB}^{2}+\mathcal{N}_{AC}^{2}\leq\mathcal{N}_{A(BC)}^{2}$ for GHZ
initial state in the noninertial frames.

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