]https://www.peterrohde.org

]https://sites.google.com/view/chandrashekar

# Accessible and inaccessible quantum coherence in relativistic quantum
systems

Saveetha Harikrishnan<EMAIL_ADDRESS>Centre for Quantum Science &
Technology, Chennai Institute of Technology, Chennai 600069, India  Segar
Jambulingam<EMAIL_ADDRESS>Department of Physics, Ramakrishna Mission
Vivekananda College, Mylapore, Chennai 600 004, India  Peter P. Rohde
<EMAIL_ADDRESS>[ Centre for Quantum Software & Information (UTS:QSI),
University of Technology Sydney, Ultimo, NSW 2007, Australia  Hearne Institute
for Theoretical Physics, Department of Physics & Astronomy, Louisiana State
University, Baton Rouge LA, United States  Chandrashekar Radhakrishnan
<EMAIL_ADDRESS>Centre for Quantum Information, Communication and
Computing, Indian Institute of Technology Madras, Chennai 600036, India
Centre for Quantum Science & Technology, Chennai Institute of Technology,
Chennai 600069, India [

###### Abstract

The quantum coherence of a multipartite system is investigated when some of
the parties are moving with uniform acceleration and the analysis is carried
out using the single mode approximation. Due to acceleration the quantum
coherence is divided into two parts as accessible and inaccessible coherence
and the entire analysis has been carried out in the single-mode approximation.
First we investigate tripartite systems, considering both GHZ and W-states. We
find that the quantum coherence of these states does not vanish in the limit
of infinite acceleration, rather asymptoting to a non-zero value. These
results hold for both single- and two-qubit acceleration. In the GHZ and
W-states the coherence is distributed as correlations between the qubits and
is known as global coherence. But quantum coherence can also exist due to the
superposition within a qubit, the local coherence. To study the properties of
local coherence we investigate separable state. The GHZ state, W-state and
separable states contain only one type of coherence. Next we consider the
$W\bar{W}$ and star states in which both local and global coherences coexist.
We find that under uniform acceleration both local and global coherence show
similar qualitative behaviour. Finally we derive analytic expressions for the
quantum coherence of $N$-partite GHZ and W-states for $n<N$ accelerating
qubits. We find that the quantum coherence of a multipartite GHZ state falls
exponentially with the number of accelerated qubits, whereas for multipartite
W-states the quantum coherence decreases only polynomially. We conclude that
W-states are more robust to Unruh decoherence and discuss some potential
applications in satellite-based quantum communication and black hole physics.

Quantum coherence, non-inertial frame, single mode approximation, coherence
distribution

###### pacs:

03.67.-a, 03.30+p, 03.67.Hk

##  I Introduction

Entanglement is a widely studied quantum resource with applications in quantum
computing [1], quantum algorithms [2, 3, 4], metrology [5], teleportation [6,
7], and cryptography [8, 9]. In recent times, it has become more apparent that
there are several other quantum properties such as non-locality [10], steering
[11], discord [12] and coherence [13], which can also be used as a resource.
Based on their relative presence, it is known that there is a hierarchy among
these different resources. Of all these resources, quantum coherence is more
extensively present, even when the quantum system does not have steering,
entanglement or discord. Hence an investigation of quantum coherence can
provide more complete information about the ‘quantumness’ of physical systems.
Consequently, several studies have been performed into quantum coherence with
the aim to understand the fundamental quantum behaviour of systems. Some
important investigations carried out so far are on the measurement of quantum
coherence [13, 14], the formulation of resource theory of quantum coherence
[15, 16, 17, 18], the effect of external environments on coherence [19, 20]
and finally applications of quantum coherence [21, 22, 23].

Investigations in quantum information theory are generally carried out in
inertial reference frames. An extension to non-inertial reference frames has
been discussed through several works [24, 25, 26, 27]. In this context
entanglement in non-inertial frames of reference has been an important area of
research [28, 29, 30]. Initial works [31] in this direction considered a
family of peaked Minkowski wave packets which admits only a single Unruh mode.
Later on the entanglement of a general set of states has been discussed in a
multimode setting [32, 33]. Though the multimode approximation is more
generally valid, several studies are still carried out in the single mode
approximation [34], due to the ease of obtaining a clear analytic expression.
Another approach is also available in non-inertial scenario by considering
Unruh-Dewitt detector model[35, 36]. Several interesting results have been
obtained in the fields of black hole physics [31, 37, 38, 39, 40, 41], quantum
error correction [42], relativistic quantum metrology [43, 44, 45, 46],
relativistic teleportation [47, 48, 49, 50] and communication [51, 52, 53,
54]. Similar studies on relativistic effects on quantum discord [55, 56] have
also been performed. Recent experimental advances have increased the scope of
quantum technologies from applications in terrestrial situations [57, 58, 59]
to satellite-based space level technologies [60, 61, 62, 63, 64, 65, 66].
Hence, we investigate quantum coherence under uniform acceleration since
compared to entanglement or other resources it is more commonly found and can
be measured more easily.

The effect of relativistic motion on quantum coherence has been studied in the
following works [67, 68, 69, 70]. In Ref. [67], the relativistic effects on
the quantum coherence between a pair of Unruh-Dewitt detectors is studied.
Here they find that compared to entanglement, quantum coherence is more robust
to Unruh decoherence. A generalisation of this result to tripartite system was
performed by [69]. Both these works consider a massless scalar field. The
relativistic coherence of a system of Dirac fields was considered in [70]. In
our work we consider the modes of a massless scalar field and measure the
coherence between different modes as well as the coherence within a mode. Here
we consider a uniformly accelerating system which can be described using a
family of peaked Minkowski wave packets and consequently we work in the single
mode setting. We estimate the information-theoretic change in the coherence of
the system using the $\ell_{1}$-norm of coherence. For a complete study of
tripartite systems, we investigate both the SLOCC (Stochastic Local Operations
& Classical Communication) class of states, namely the GHZ (Greenberger-Horne-
Zeilinger) and W class of states. But these two states have only global
coherence which arises due to correlation between qubits. For completeness
sake, we investigate the relativistic effects on separable states with only
local coherence, a type of quantum coherence arising due to superposition
within qubits. We also look into two different tripartite states, namely the
$W\bar{W}$ and star states in which both global and local coherence coexist.

The structure of the manuscript is as follows: In Sec. II we discuss the
notion of relativity in the field of quantum information and also introduce
the $\ell_{1}$-norm of coherence which is used to measure coherence in our
work. The SLOCC class of states is studied in detail in Sec. III with the GHZ
class and W class forming two different subsections. The separable state and
an introduction to the notions of local coherence and global coherence is
described in Sec. IV. The tripartite systems with both local and global
coherence are analysed in Sec. V. In Sec. VI we discuss certain applications
related to satellite-based communication as well as black holes. Finally in
Sec. VII we discuss our results.

##  II Relativity & quantum coherence

To describe events independent of the inertial frame of reference, the
combination of three space and one time dimension known as Minkowski space are
used. For non-inertial reference frames we need to use the Rindler co-
ordinates. The Minkowski and Rindler co-ordinates are related to each other
via a Rindler transformation. For a situation with one spatial and one time
dimension $(z,t)$, the Minkowski space-time is divided into four wedges as
shown in Fig. 1. The regions $F$ and $P$ represent the future and the past
light cones and the regions I and II are the two causally disconnected Rindler
regions. The equations $|z|=t$ and $|z|=-t$ describe the future and past event
horizons. For situations with one spatial and one time dimension $(z,t)$, the
world lines of uniformly accelerated observers in Minkowski co-ordinates
correspond to a hyperbola. The two branches of the hyperbola constitute the
regions I and II of the Rindler co-ordinates. The co-ordinates of the two
regions are,

$\displaystyle t=a^{-1}e^{a\xi}\sinh a\tau,z=a^{-1}e^{a\xi}\cosh a\tau;|z|<t,$
$\displaystyle t=-a^{-1}e^{a\xi}\sinh a\tau,z=a^{-1}e^{a\xi}\cosh
a\tau;|z|>t.$ (1)

where $\xi$ is the space-like co-ordinate, $\tau$ is proper time, and $a$ is
acceleration.

Figure 1: The $(z,t)$ plane of the Minkowski co-ordinates is divided into four
regions of the Rindler co-ordinates. The solid lines are the future ($F$) and
past ($P$) event horizons and the dashed lines are the trajectories of the
uniformly accelerated observers. Here $H$ represents the horizon.

Alice, Bob and Charlie each having monochromatic detector with their
corresponding frequencies, $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$ of a
free massless scalar field in a Minkowski space time. A maximally entangled
GHZ state in the framework reads,

$\displaystyle\frac{|0_{\omega_{1}}\rangle^{\mathcal{M}}|0_{\omega_{2}}\rangle^{\mathcal{M}}|0_{\omega_{3}}\rangle^{\mathcal{M}}+|1_{\omega_{1}}\rangle^{\mathcal{M}}|1_{\omega_{2}}\rangle^{\mathcal{M}}|1_{\omega_{3}}\rangle^{\mathcal{M}}}{\sqrt{2}},$
(2)

where $|0_{\omega_{i}}\rangle^{\mathcal{M}}$
($|1_{\omega_{i}}\rangle^{\mathcal{M}}$) is the vacuum (single particle
excitation) state of frequency $\omega_{i}$ in Minkowski space. Initially when
all three parties are in inertial frames then
$|0_{\omega_{1}}\rangle^{\mathcal{M}}$, $|0_{\omega_{2}}\rangle^{\mathcal{M}}$
and $|0_{\omega_{3}}\rangle^{\mathcal{M}}$ are their respective ground states.
Their excited states can be obtained by applying their corresponding Minkowski
creation operators as follows,

$\displaystyle|n_{\omega_{i}}\rangle^{\mathcal{M}}$
$\displaystyle=\frac{(\hat{a}_{\omega_{i}}^{{\dagger}})^{n}}{\sqrt{n!}}|0_{\omega_{i}}\rangle^{\mathcal{M}}.$
(3)

If Charlie starts moving with uniform acceleration the wavefunction becomes
higly delocalized in space and the quantum state corresponding to the
frequency $\omega_{3}$ can be specified using Rindler or Unruh co-ordinates
[32]. In both the Rindler and Unruh basis, the initial Minkowski space is
divided into two regions which are casually disconnected from each other. Let
$\Omega$ be the dimensionless Rindler frequency and the Unruh modes are also
sharply peaked at the same frequency. Using an analytic continuation arguement
it was shown [32] that Unruh mode is a purely positive frequency linear
combinations of the Minkowski modes. But the definition of positive frequency
in the Rindler mode differs from the positive frequency definition on the
Minkowski mode. So we can relate the Minkowski and the Rindler modes via the
Unruh modes. The field in each of these three bases are expanded as follows:

$\displaystyle\phi$
$\displaystyle=\int_{0}^{\infty}(a_{\omega,\mathcal{M}}u_{\omega,\mathcal{M}}+a^{\dagger}_{\omega,\mathcal{M}}u^{*}_{\omega,\mathcal{M}})\,d\omega$
$\displaystyle=\int_{0}^{\infty}(A_{\Omega,R}u_{\Omega,R}+A^{\dagger}_{\Omega,R}u^{*}_{\Omega,R}$
$\displaystyle+A_{\Omega,L}u_{\Omega,L}+A^{\dagger}_{\Omega,L}u^{*}_{\Omega,L})\,d\Omega$
$\displaystyle=\int_{0}^{\infty}(b_{\Omega,I}u_{\Omega,I}+b^{\dagger}_{\Omega,I}u^{*}_{\Omega,I}$
$\displaystyle+b_{\Omega,II}u_{\Omega,II}+b^{\dagger}_{\Omega,II}u^{*}_{\Omega,II})\,d\Omega,$

where $a_{\omega,\mathcal{M}}$ is the Minkowski annihilation operator and
$A_{\Omega,R}\&A_{\Omega,L}$ are the Unruh annihilation operator for the right
and left regions and $b_{\Omega,I}\&b_{\Omega,II}$ are the Rindler
annihilation operators in $I\&II$ regions. The operators obey the bosonic
commutation relations
$[a_{\omega_{1},\mathcal{M}},a^{\dagger}_{\omega_{2},\mathcal{M}}]=\delta_{\omega_{1}\omega_{2}}$,
$[A_{\Omega_{1},R},A^{\dagger}_{\Omega_{2},R}]$ =
$[A_{\Omega_{1},L},A^{\dagger}_{\Omega_{2},L}]=\delta_{\Omega_{1}\Omega_{2}}$
and
$[b_{\Omega_{1},I},b^{\dagger}_{\Omega_{2},I}]=[b_{\Omega_{1},II},b^{\dagger}_{\Omega_{2},II}]=\delta_{\Omega_{1}\Omega_{2}}$.
For the Unruh mode, the commutator between operators in the R and L region
vanish. Similarly the commutator between the operators in region $I$ and $II$
vanish. The creation and annihilation operators of the Minkowski and Unruh
bases do not mix and hence we have
$|0\rangle_{M}=|0\rangle_{U}=\prod_{\Omega}|0{{}_{\Omega}}\rangle_{U}$. But
the state $|0\rangle_{U}$ does not coincide with the Rindler vacuum and we
have

$\displaystyle|0_{\Omega}\rangle_{U}=\sum_{N}\frac{\tanh^{n}r_{\Omega}}{\cosh{r}_{\Omega}}|n_{\Omega}\rangle_{I}|n_{\Omega}\rangle_{II}.$
(5)

Here $|n_{\Omega}\rangle_{I}$ is the $n^{th}$ excited state of the Rindler $I$
vacuum state $|0_{\Omega}\rangle_{I}$.

We consider a wave packet which is narrowly peaked in $\Omega$ and for this
the Unruh and Rindler commutators read $[A_{\Omega,R},A^{\dagger}_{\Omega,R}]$
= $[A_{\Omega,L},A^{\dagger}_{\Omega,L}]=1$ and
$[b_{\Omega,I},b^{\dagger}_{\Omega,I}]=[b_{\Omega,II},b^{\dagger}_{\Omega,II}]=1$.
Under these ideal conditions, the most general creation operator of a purely
positive Minkowski frequency can be written as,

$\displaystyle
a^{\dagger}_{\Omega,U}=q_{L}A^{\dagger}_{\Omega,L}+q_{R}A^{\dagger}_{\Omega,R}.$
(6)

Here the factors $q_{R}$ and $q_{L}$ are complex numbers with
$|q_{R}|^{2}+|q_{L}|^{2}=1$. Under these conditions we have

$\displaystyle
a^{\dagger}_{\Omega,U}|0_{\Omega}\rangle_{U}=\sum_{n=0}^{\infty}\frac{\tanh^{n}r_{\Omega}}{\cosh
r_{\Omega}}(\frac{\sqrt{n+1}}{\cosh r_{\Omega}})|\Phi^{n}_{\Omega}\rangle,$
$\displaystyle|\Phi^{n}_{\Omega}\rangle=q_{L}|n_{\Omega}\rangle_{I}|(n+1)_{\Omega}\rangle_{II}+q_{R}|(n+1)_{\Omega}\rangle_{I}|n_{\Omega}\rangle_{II},$
(7)

where, in general we consider $q_{R}=1$ and $q_{L}=0$. Let us consider a
Minkowski smearing function which is a Gaussian in $\ln(\omega l)$,

$\displaystyle
f(\omega)=\left(\frac{\lambda}{\pi\omega^{2}}\right)^{1/4}\exp\Big{\\{}\frac{-1}{2}\lambda[\ln(\omega/\omega_{0})]^{2}\Big{\\}}(\omega/\omega_{0})^{-i\mu},$
(8)

When the uniformly accelerated particle has the above smearing function and
has negligible overlap with the other states, then it is well approximated by
a single Unruh frequency. Thus we use this monochromatic wave approximation in
our investigation. Under this condition, the Minkowski and Rindler modes can
be connected via the relations:

$\displaystyle\hat{a}_{\omega_{3}}^{{\dagger}}$
$\displaystyle=\hat{b}^{{\dagger}}_{\Omega_{3}I}\cosh
r-\hat{b}_{\Omega_{3}II}\sinh
r=\hat{S}_{\Omega_{3}}\hat{b}^{{\dagger}}_{\Omega_{3}I}\hat{S}^{{\dagger}}_{\Omega_{3}}$
$\displaystyle\hat{a}_{\omega_{3}}$ $\displaystyle=\hat{b}_{\Omega_{3}I}\cosh
r-\hat{b}^{{\dagger}}_{\Omega_{3}II}\sinh
r=\hat{S}_{\Omega_{3}}\hat{b}_{\Omega_{3}I}\hat{S}^{{\dagger}}_{\Omega_{3}},$
(9)

where,

$\displaystyle\hat{S}_{\Omega}(r)=\exp[r(\hat{b}^{{\dagger}}_{\Omega
I}\hat{b}^{{\dagger}}_{\Omega II}-\hat{b}_{\Omega I}\hat{b}_{\Omega II})].$
(10)

Here the operator $(\hat{S})$ effecting the transformation from Minkowski co-
ordinates to Rindler co-ordinates is structurally identical to the two mode
squeezing operator,

$\displaystyle\hat{S}(\zeta)=\exp(\zeta^{*}ab-\zeta
a^{{\dagger}}b^{{\dagger}}),$ (11)

in the quantum optics context. Hence for a non-inertial observer, the single
mode Minkowski vacuum becomes a two mode squeezed state in the Rindler vaccum,

$\displaystyle|0_{\omega}\rangle^{\mathcal{M}}$
$\displaystyle=\hat{S}_{\Omega_{3}}(r)(|0\rangle_{I}\otimes|0\rangle_{II})$
$\displaystyle=\frac{1}{\cosh
r}\sum_{n=0}^{\infty}\tanh^{n}r|n_{\Omega}\rangle_{I}|n_{\Omega}\rangle_{II},$
(12)

where $\cosh r=(1-e^{-2\pi\Omega})^{-1/2}$ and $\Omega=|\omega|c/a$. The
factors $\omega$ and $c$ are the wave vector and velocity of light
respectively. Here $|n_{\Omega}\rangle_{I}$ and $|n_{\Omega}\rangle_{II}$ are
the mode decompositions in Rindler regions I and region II respectively. For
the single particle excitation state we have,

$\displaystyle|1_{\omega}\rangle^{\mathcal{M}}$
$\displaystyle=\hat{a}_{\omega_{3}}^{{\dagger}}|0_{\omega_{3}}\rangle^{\mathcal{M}}=\hat{S}_{\Omega_{3}}(r)\hat{b}^{{\dagger}}_{\Omega_{3}I}(|0\rangle_{I}\otimes|0\rangle_{II})$
$\displaystyle=\frac{1}{\cosh^{2}r}\sum_{n=0}^{\infty}\sqrt{n+1}\tanh^{n}r|(n+1)_{\Omega}\rangle_{I}|n_{\Omega}\rangle_{II}.$

We observe that due to the squeezing behaviour of a non-inertial observer, a
single Minkowski mode can be written as a superposition of two Rindler modes.
Consequently there exists coherence between the two Rindler modes. Since these
two modes are not causally connected, this coherence cannot be experimentally
observed. Actually there is no new coherence created in the system.

To illustrate this let us consider a tripartite system with Alice and Bob at
rest, and Charlie moving with constant acceleration. The coherence shared by
Charlie with Alice and Bob is split into two parts. One part remains in
Rindler mode I and can be experimentally observed, which we call the
accessible coherence. The other part of the coherence which is shared with
Rindler mode II cannot be measured, which we refer to as inaccessible
coherence. This inaccessible coherence quantifies the decrease in coherence
due to relativistic effects. A schematic diagram explaining the accessible and
inaccessible coherence is shown in Fig. 2.

Figure 2: A schematic diagram of coherence in tripartite is shown where the
circles labelled $A$, $B$ and $C$ represent the qubit and the blue arrow
between them representing the coherence. Fig (a) contains the coherence in a
tripartite system when all the qubits are in inertial frame. The part (b)
contains the coherence in the tripartite system when qubit $C$ is under
acceleration. Here qubit $C$ enclosed by the Red ellipse with dashed boundary
line, is split into two Rindler modes I and II. Consequently the coherence
shared by qubit $C$ also is split into two parts. The coherence shared with
$C$ of Rindler mode I is the accessible coherence and the coherence shared
with $C$ of Rindler mode II is the inaccessible coherence. The lighter shades
of blue for the accessible and inaccessible coherence represents their
relative strength to the coherence initially present with $C$ and shown in
part (a).

In a tripartite state when two qubits are accelerated the vacuum
($|000\rangle$) and excited ($|111\rangle$) states can be expressed as,

$\displaystyle|000\rangle_{ABC}$
$\displaystyle=|0\rangle\otimes|0\rangle\otimes|0\rangle$
$\displaystyle\overset{accl}{\longrightarrow}|0_{\omega_{1}}\rangle\otimes[\hat{S}_{\Omega_{2}}(r_{2})(|0_{\Omega_{2}}\rangle_{I}\otimes|0_{\Omega_{2}}\rangle_{II})]\otimes[\hat{S}_{\Omega_{3}}(r_{1})(|0_{\Omega_{3}}\rangle_{I}\otimes|0_{\Omega_{3}}\rangle_{II})]$
$\displaystyle=|0\rangle\otimes\frac{1}{\cosh
r_{2}}\left[\sum_{m=0}^{\infty}\tanh^{m}r_{2}|m\rangle_{I}|m\rangle_{II}\right]\otimes\left[\frac{1}{\cosh
r_{1}}\sum_{n=0}^{\infty}\tanh^{n}r_{1}|n\rangle_{I}|n\rangle_{II}\right],$
(13) $\displaystyle|111\rangle_{ABC}$
$\displaystyle=|1\rangle\otimes|1\rangle\otimes|1\rangle$
$\displaystyle\overset{accl}{\longrightarrow}|1_{\omega_{1}}\rangle\otimes[\hat{S}_{\Omega_{2}}(r_{2})\hat{b}^{\dagger}_{{\Omega_{2}}{I}}(|0_{\Omega_{2}}\rangle_{I}\otimes|0_{\Omega_{2}}\rangle_{II})]\otimes[\hat{S}_{\Omega_{3}}(r_{1})\hat{b}^{\dagger}_{{\Omega_{3}}{I}}(|0_{\Omega_{3}}\rangle_{I}\otimes|0_{\Omega_{3}}\rangle_{II})]$
$\displaystyle=|1\rangle\otimes\left[\frac{1}{\cosh^{2}r_{2}}\sum_{m=0}^{\infty}\sqrt{m+1}\tanh^{m}r_{2}|m+1\rangle_{I}|m\rangle_{II}\right]\otimes\left[\frac{1}{\cosh^{2}r_{1}}\sum_{n=0}^{\infty}\sqrt{n+1}\tanh^{n}r_{1}|n+1\rangle_{I}|n\rangle_{II}\right].$
(14)

The modes $I$ and $II$ correspond to the two causally disconnected regions in
Rindler co-ordinates of Minkowski space. Hence mode $II$ is physically
inaccessible to Alice, Bob and Charlie and is partially traced out. For an
inertial observer, a quantum system in Minkowski space-time is independent of
the nature of the observer. But when the observer is moving with constant
acceleration, they perceive a thermal bath with temperature proportional to
acceleration. This thermal bath is made up of Rindler particles which are
associated with the vacuum state of Minkowski space. Due to this thermal bath,
the observer perceives decoherence of the quantum system, a phenomenon known
as Unruh decoherence.

Quantum coherence is in general measured as the distance between the quantum
state under consideration and the closest incoherent state in the same basis.
Several measures of quantum coherence [13, 71, 72, 14] have been presented,
but in the first work on quantum coherence by Baumgratz, Cramer and Plenio
[13] two measures were introduced corresponding to the entropic class and
geometric class of measures. The relative entropy based measure of quantum
coherence belongs to the entropic class, and the $\ell_{1}$ norm of coherence
to the geometric class. Here we use the $\ell_{1}$-norm of quantum coherence,
defined as,

$\displaystyle
C_{l_{1}}(\hat{\rho},\hat{\rho}_{d})=\|\hat{\rho}-\hat{\rho}_{d}\|_{l_{1}}=\sum_{i\neq
j}|\hat{\rho}_{i,j}|,$ (15)

where $\hat{\rho}$ is a given density matrix and $\hat{\rho}_{d}$ is the
associated decohered density matrix defined as,

$\displaystyle\hat{\rho}_{d}=\sum_{i}\hat{\rho}_{i,i}|i\rangle\langle i|.$
(16)

We can observe that this is equivalent to the sum of the off-diagonal elements
of the density matrix. Using interference fringes, the quantum coherence of a
physical system can be measured using the robustness of coherence [73], a
measure introduced in Ref. [74]. In Ref. [75] it was shown that for a pure
state the robustness of coherence measure is equal to the $\ell_{1}$-norm of
coherence. Hence the $\ell_{1}$-norm of coherence can be directly obtained
from the interference fringes, providing a method to compare theoretical
results with experimental data.

##  III Relativistic effects in the SLOCC class of states

Based on the local operations and classical communication (LOCC), tripartite
entangled quantum states can be divided into two distinct classes, namely the
GHZ class and the W class. The entanglement distribution is different in these
two class of states. In a GHZ state all the entanglement vanishes even with
the loss of a single qubit whereas in a W-state a finite amount of
entanglement is always present even when we loose a single qubit. The
coherence of these class of states in inertial frames has been discussed in
detail in Ref. [20]. We extend upon this by investigating the coherence when
some of the qubits of the tripartite system are in a non-inertial reference
frame. In particular, we probe the loss of coherence due to the acceleration
of qubits.

###  III.1 GHZ class

A GHZ state is maximally entangled and so the loss of just a single qubit
results in the complete loss of entanglement and coherence. The general form
of a tripartite Greenberger-Horne-Zeilinger (GHZ) state is,

$\displaystyle|\mathrm{GHZ}\rangle_{ABC}=\cos\theta|000\rangle_{ABC}+\sin\theta|111\rangle_{ABC},$
(17)

where $\theta\in[0,2\pi)$ is a parameter generalizing the GHZ state via the
bias between the two basis states. On accelerating qubit $C$ its Minkowski
modes are replaced with their corresponding Rindler modes and the tripartite
states becomes,

$\displaystyle|\mathrm{GHZ}\rangle_{ABC}$ $\displaystyle=\frac{1}{\cosh
r}\sum_{n=0}^{\infty}\tanh^{n}r\bigg{[}\cos\theta|00\rangle|n\rangle_{I}|n\rangle_{II}+\frac{\sqrt{n+1}}{\cosh
r}\;\sin\theta|11\rangle|n+1\rangle_{I}|n\rangle_{II}\bigg{]}.$ (18)

Here, $|00\rangle|n\rangle_{I}|n\rangle_{II}$ and
$|11\rangle|n+1\rangle_{I}|n\rangle_{II}$ refer to the quantum state in which
the Minkowski modes of the first two qubits are $|ab\rangle$ and the Rindler
modes corresponding to the third qubit is as $|c\rangle_{I}|c\rangle_{II}$. We
know that the modes from the Rindler I and II regions are not causally
connected, so we can trace out the modes corresponding to Rindler region II
and the density matrix of the state with Alice, Bob and Charlie reduces to,

$\displaystyle\hat{\rho}_{\mathrm{GHZ}}$
$\displaystyle=\frac{1}{\cosh^{2}r}\sum_{n=0}^{\infty}\tanh^{2n}r\bigg{[}\cos^{2}\theta|00n\rangle\langle
00n|+\cos\theta\sin\theta\frac{\sqrt{n+1}}{\cosh r}\bigg{(}|00n\rangle\langle
11n+1|+|11n+1\rangle\langle 00n|\bigg{)}$
$\displaystyle+\frac{n+1}{\cosh^{2}r}\sin^{2}\theta|11n+1\rangle\langle
11n+1|\bigg{]}.$ (19)

The total coherence in the generalised GHZ state is calculated using the
$\ell_{1}$-norm measure of coherence. The sum of the off-diagonal elements of
Eq. (19) is

$\displaystyle
C(\hat{\rho})=\frac{2\sin\theta\cos\theta}{\cosh^{3}r}\sum_{n=0}^{\infty}\sqrt{n+1}\tanh^{2n}r.$
(20)

Using the trigonometric identities,

$\displaystyle\sum_{n=0}^{\infty}\tanh^{2n}r$ $\displaystyle=\cosh^{2}r,$
$\displaystyle\sum_{n=0}^{\infty}(n+1)\tanh^{2n}r$ $\displaystyle=\cosh^{4}r,$
(21)

and the polylogarithm function

$\displaystyle{\rm
Li}_{-1/2}(z)=\sum_{n=0}^{\infty}\sqrt{n+1}(\tanh^{2}r)^{n+1},$ (22)

where the polylogarithm function is defined as

$\displaystyle{\rm
Li}_{n}(z)\equiv\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}}=\frac{z}{1^{n}}+\frac{z^{2}}{2^{n}}+\frac{z^{3}}{3^{n}}+\cdots,$
(23)

the total coherence of a GHZ state when one of the qubits is in a non-inertial
frame can be expressed as,

$\displaystyle C(\hat{\rho})=2\sin\theta\cos\theta\frac{{\rm
Li}_{-1/2}(\tanh^{2}r)}{\sinh^{2}r\,\cosh r}.$ (24)

When Charlie’s qubit is not accelerated (i.e the $r\rightarrow 0$ limit) the
corresponding quantum coherence is,

$\displaystyle C(\hat{\rho})=2\cos\theta\sin\theta.$ (25)

Figure 3: The variation of coherence in a non-inertial frame of reference is
studied for a GHZ state under the variation of (a) the acceleration parameter
$r$ and (b) the generalisation parameter $\theta$.

The variation of quantum coherence as a function of the acceleration parameter
$r$ is shown in Fig. 3(a) for different $\theta$ values. From the plot for any
given value of $\theta$ we notice that quantum coherence has a maximum value
when $r=0$, corresponding to the situation where Charlie’s qubit has not yet
been accelerated. The coherence then decreases with increase in $r$ and
saturates to a finite value. The decrease in coherence due to uniform
acceleration is because some part of it becomes inaccessible as it lies in a
causally disconnected region. The saturation value depends on the value of
coherence in the inertial frame ($r=0$). Hence, higher the value of the
coherence in the inertial frame, the higher the saturation value of coherence
in the non-inertial frame.

In Fig. 3(b), we study the change of the quantum coherence as a function of
the generalisation parameter $\theta$ for different values of the acceleration
parameter $r$. The coherence is maximum at $\theta=(2n+1)\pi/4$ where
$n\in\mathbb{Z}$ and is zero at $\theta=n\pi/2$, but the maximum value depends
on the value of $r$.

To study the situation, when more than one party is under acceleration, we
consider the setting where both Bob’s and Charlie’s qubits are being
accelerated. The quantum state when two qubits are accelerated is,

$\displaystyle|\mathrm{GHZ}\rangle$ $\displaystyle=\frac{1}{\cosh
r_{1}}\frac{1}{\cosh
r_{2}}\sum_{n,m=0}^{\infty}\tanh^{n}r_{1}\tanh^{m}r_{2}\bigg{[}\cos\theta|0\rangle|m\rangle_{I}|m\rangle_{II}|n\rangle_{I}|n\rangle_{II}$
$\displaystyle+\frac{\sqrt{m+1}}{\cosh r_{2}}\frac{\sqrt{n+1}}{\cosh
r_{1}}\sin\theta\times|1\rangle|m+1\rangle_{I}|m\rangle_{II}|n+1\rangle_{I}|n\rangle_{II}\bigg{]},$
(26)

where the pairs ($r_{2}$,$m$) and ($r_{1}$,$n$) are the acceleration parameter
and mode corresponding to Bob’s and Charlie’s qubits respectively. To
construct the density matrix we first trace out the contributions from the
causally disconnected Rindler II mode,

$\displaystyle\hat{\rho}$
$\displaystyle=\frac{1}{\cosh^{2}r_{1}\cosh^{2}r_{2}}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\tanh^{2n}r_{1}\tanh^{2m}r_{2}\bigg{[}\cos^{2}\theta|0\rangle|m\rangle|n\rangle\langle
0|\langle m|\langle n|+\frac{\sqrt{m+1}\sqrt{n+1}}{\cosh r_{1}\cosh
r_{2}}\cos\theta\sin\theta$
$\displaystyle\bigg{(}|0\rangle|m\rangle|n\rangle\langle 1|\langle m+1|\langle
n+1|+|1\rangle|m+1\rangle|n+1\rangle\langle 0|\langle m|\langle
n|\bigg{)}+\frac{(m+1)(n+1)}{\cosh^{2}r_{1}\cosh^{2}r_{2}}\sin^{2}\theta$
$\displaystyle|1\rangle|m+1\rangle|n+1\rangle\langle 1|\langle m+1|\langle
n+1|\bigg{]},$ (27)

From the density matrix in Eq. (27) and using the $\ell_{1}$-norm measure we
can determine the total quantum coherence of the GHZ state when both Bob’s and
Charlie’s qubits are accelerated,

$\displaystyle
C(\hat{\rho})=2\sin\theta\cos\theta\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\sinh^{2}r_{1}\cosh
r_{1}}\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\sinh^{2}r_{2}\cosh r_{2}},$ (28)

where we find to be a product of two polylogarithm functions one each
corresponding to Bob’s and Charlie’s qubits. If Bob and Charlie have the same
acceleration, which would translate to $r_{1}=r_{2}=r$, the coherence would
then be,

$\displaystyle
C(\hat{\rho})=2\sin\theta\cos\theta\left[\frac{Li_{-1/2}(\tanh^{2}r)}{\sinh^{2}r\cosh
r}\right]^{2}.$ (29)

The variation of quantum coherence with the parameters $r_{1}$ and $r_{2}$
corresponding to Bob’s and Charlie’s acceleration is shown through the contour
plots in Fig. 4. The contour plots Fig. 4(a), 4(b) and 4(c) correspond to
$\theta$ of $\pi/4$, $\pi/5$ and $\pi/6$ respectively. We find that with the
increase in $r_{1}$ and $r_{2}$, the coherence decreases and saturates to a
finite value. The maximal value and saturation value of coherence are
dependent on the value of $\theta$ the parameter generalizing the GHZ state.

Figure 4: The variation of quantum coherence of GHZ state with both $r_{1}$
and $r_{2}$ is shown for (a) $\theta=\pi/4$, (b) $\theta=\pi/5$ and (c)
$\theta=\pi/6$ respectively.

In literature, the most commonly discussed GHZ state is
$(|000\rangle+|111\rangle)/\sqrt{2}$, corresponding to the generalised GHZ
state with $\theta=\pi/4$, which has quantum coherence,

$\displaystyle C(\hat{\rho})$
$\displaystyle=\frac{Li_{-1/2}(\tanh^{2}r)}{\sinh^{2}r\,\cosh
r}\leavevmode\nobreak\ ,$ $\displaystyle C(\hat{\rho})$
$\displaystyle=\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\sinh^{2}r_{1}\cosh
r_{1}}\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\sinh^{2}r_{2}\cosh r_{2}}.$ (30)

Finally we note that the two qubit reduced density matrices corresponding to
the GHZ state are diagonal in nature and hence are incoherent. Thus all the
coherences in the GHZ state are genuinely multipartite in nature and the loss
of even one qubit removes all the coherence in the system.

In Ref. [31], the authors considered a two qubit system of which one qubit is
moving with a constant acceleration. The entanglement of the bipartite system
vanished in the infinite acceleration limit. It is well known that
entanglement is just one of the different types of quantum correlations. Out
of the different quantifiers of quantum correlations, quantum discord measures
the total quantum correlations in the system. A study of quantum discord in
relativistic system [55] show that total quantum correlations does not vanish
in the infinite acceleration limit. Hence a bipartite system in the infinite
acceleration limit has quantum correlations beyond the entanglement type. An
investigation on accelerating tripartite systems was carried out in Ref. [76],
where the GHZ and W-states were characterised using the $\pi$-tangle measure
of entanglement. The entanglement of the tripartite system did not go to zero
in the infinite acceleration limit, rather it showed a decrease initially and
then attained a saturation value. This is in stark contrast to the behaviour
of the bipartite entanglement. In Ref. [76] it was suggested that the
incomplete definition of $\pi$-tangle could be the reason as to why the
tripartite entanglement shows a different qualitative behaviour when compared
to the bipartite entanglement as measured in Ref. [31]. Through the present
work we observe that the quantum coherence of the GHZ state is qualitatively
similar to the change in its tripartite entanglement. But here the saturation
is due to the non-vanishing behaviour of local quantum correlations. However
for quantum coherence, the $\ell_{1}$-norm measures the entire quantum
coherence and the saturation value is a physical feature of the system. In the
case of entanglement, the saturation value may not be an actual physical
feature since $\pi$-tangle is inadequate to measure entanglement.

Next we can consider the three qubit GHZ state given in Eq. (2) in which two
qubits are under uniform acceleration. The GHZ state can be rewritten as
$(|0\mathbf{0}\rangle+|1\mathbf{1}\rangle)/\sqrt{2}$, where
$|\mathbf{0}\rangle=|00\rangle$ and $|\mathbf{1}\rangle=|11\rangle$ are
redundantly-encoded logical qubits. Since the GHZ state can be written in
terms of the Bell state, we would expect the coherences of both these states
to be equal. But the coherence of the GHZ state when the logical qubit is
accelerated is,

$\displaystyle C_{\ell_{1}}=[Li_{-1/2}(\tanh^{2}r)/(\sinh^{2}r\cosh r)]^{2},$
(31)

which implies that $C(|\psi_{Bell}\rangle)\neq C(|\psi_{GHZ}\rangle)$ an
relativistic system. This is because in the GHZ state, the logical non-
inertial qubit is bigger than the regular qubit considered in the Bell state.
So, we find that the quantum coherence coupled with uniform acceleration helps
in distinguishing between Bell states and GHZ states.

Figure 5: The variation of quantum coherence as a function of $r$ for fixed
values of $\theta$ and $\phi$ is given for (a) the tripartite generalized
W-state and (b) the two qubit reduced systems $\rho_{AB}$, $\rho_{BC}$ and
$\rho_{AC}$. Here the notation $W(\rho_{AB})$ means the reduced state
$\rho_{AB}$ corresponding to the symmetric $W$-state.

###  III.2 W class

The W-states exhibit non-maximal multipartite entanglement which is not
locally equivalent to GHZ-type entanglement, and may therefore be considered a
distinct entanglement class. Unlike GHZ states, W-states with large numbers of
qubits are highly robust against qubit loss. The generalised form of the
tripartite W-state is,

$\displaystyle|W\rangle_{g}=\sin\theta\cos\phi|100\rangle+\sin\theta\sin\phi|010\rangle+\cos\theta|001\rangle,$
(32)

where $\theta=[0,\pi]$ and $\phi=[0,2\pi)$ are the parameters generalising the
W-state.

First let us consider the situation when only one qubit is in a non-inertial
frame of reference. Towards this end, we consider the situation when Charlie’s
qubit is in a non-inertial frame while Alice and Bob’s qubits are in an
inertial frame. Naturally the Minkowski modes of Charlie’s qubit are replaced
by their corresponding Rindler modes. It is well known that the Minkowski
region can be divided into two causally disconnected Rindler modes. So, we
trace out the second Rindler region from the quantum state, since it is
inaccessible for measuerment. The total quantum coherence is then measured
using the $\ell_{1}$-norm of coherence,

$\displaystyle C(\hat{\rho})$
$\displaystyle=2\sin\theta\cos\theta(\sin\phi+\cos\phi)\frac{Li_{-1/2}(\tanh^{2}r)}{\sinh^{2}r\,\cosh
r}$ $\displaystyle+2\sin^{2}\theta\sin\phi\cos\phi\leavevmode\nobreak\ .$ (33)

Figure 6: Contour plot of the Quantum coherence $C(\rho)$ as a function of
$\theta$ and $\phi$ of the generalised W-state when one qubit is in a non-
inertial frame is shown for (a) $r=0.01$ and (b) $r=4.0$.

To understand the relativistic effects on a quantum state with bipartite
distribution, we compute the reduced density matrices corresponding to the
$W$-state. Here, we can trace out either Alice or Bob’s qubit and the quantum
coherence corresponding to the reduced state is,

$\displaystyle
C(\hat{\rho}_{BC})=C(\hat{\rho}_{AC})=2\sin\theta\cos\theta\sin\phi\frac{Li_{-1/2}(\tanh^{2}r)}{\sinh^{2}r\cosh
r}.$ (34)

The reduced density matrix obtained after tracing out Charlie’s qubit does not
have any effects of non-inertial nature and the coherence of the joint state
of Alice and Bob is,

$\displaystyle C(\hat{\rho}_{AB})=2\sin^{2}\theta\sin\phi\cos\phi,$ (35)

which is the expected standard result. The most common form the three qubit
W-state is the tripartite state of the form
$|W\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$, and of
all the generalised W-states it has the maximum amount of coherence and
entanglement. Also the total coherence and entanglement are distributed in a
symmetric manner. The quantum coherence of a tripartite system when Charlie’s
qubit is accelerated is ,

$\displaystyle C(\hat{\rho})=\frac{4\,Li_{-1/2}(\tanh^{2}r)}{3\cosh
r\,\sinh^{2}r}+\frac{2}{3}.$ (36)

In the $r\rightarrow 0$ limit (inertial limit), it reduces to the standard
value of $C(\hat{\rho})=2$. The change in coherence of the tripartite
generalised W-state is given as a function of the acceleration parameter $r$
for different values of $\theta$ and $\phi$ in Fig. 5(a). From the plots we
notice that the quantum coherence is initially maximum at $r=0$ and then it
decreases and reaches a saturation value at large values of $r$. The decrease
in coherence is due to the bifurcation of the Minkowski mode into two Rindler
modes, of which one is disconnected from the rest of the system. Any coherence
due to this mode is inaccessible to measurements. The saturation value is the
amount of accessible coherence which can never be lost to relativistic motion.
The maximal value and saturation value are dependent on the generalisation
parameter. The
$|W\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$ state has
the maximal coherence at $r=0$ and also has the maximal saturation value. The
quantum coherence of the reduced density matrices of the generalised W-state
is shown in Fig. 5(b). The quantum coherence of the reduced state
$\hat{\rho}_{AB}$ is a constant since the accelerated qubit $C$ has already
been traced out. In the case of $\hat{\rho}_{BC}$ we find that the coherence
is maximal at $r=0$ and saturates to a finite value at large accelerations.
The curve represents the coherence of $\hat{\rho}_{BC}$ and the dashed lines
denote the coherence in the state $\hat{\rho}_{AB}$. In Fig. 6 the contour
plot shows the variation of quantum coherence as a function of the
generalisation parameter $\theta$ and $\phi$ for the acceleration parameters
$r=0.01$ and $r=4.0$.

Next we look at the case where two qubits, the ones corresponding to Bob and
Charlie are being accelerated. Adopting a similar procedure of replacing the
Minkowski states by Rindler modes and tracing out the Rindler second mode, we
measure the total quantum coherence of the system. The quantum coherence of
the tripartite system is,

$\displaystyle C(\hat{\rho})$ $\displaystyle=2\sin\theta\cos\theta\sin\phi$
$\displaystyle\times\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\sinh^{2}r_{1}\cosh
r_{1}}\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\sinh^{2}r_{2}\cosh r_{2}}$
$\displaystyle+2\sin\theta\cos\theta\cos\phi\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\sinh^{2}r_{1}\cosh
r_{1}}$
$\displaystyle+2\sin^{2}\theta\sin\phi\cos\phi\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\sinh^{2}r_{2}\cosh
r_{2}}.$ (37)

Figure 7: In the generalised W-state when both Bob’s and Charlie’s qubits are
accelerated, the variation of quantum coherence as a function of the
acceleration parameters $r_{1}$ and $r_{2}$ is given for (a) $W$-state, (b)
$\theta=\pi/5$ , $\phi=\pi/5$, (c) $\theta=\pi/6$ , $\phi=\pi/6$ and (d)
$\theta=\pi/3$ , $\phi=\pi/6$.

From Eq. (37) we notice that the quantum coherence is a sum of three terms.
The first contains the non-inertial decoherence from both Bob’s and Charlie’s
qubit. The non-inertial contributions from Bob and Charlie appear in the
second and third terms respectively. At the bipartite level, there are two
possibilities namely: (i) when Alice’s qubit is traced out; (ii) when either
Bob’s qubit or Charlie’s qubit is traced out. The quantum coherence
corresponding to these different situations are,

$\displaystyle C(\hat{\rho}_{BC})$
$\displaystyle=2\sin\theta\cos\theta\sin\phi\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\sinh^{2}r_{1}\cosh
r_{1}}$
$\displaystyle\times\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\sinh^{2}r_{2}\cosh
r_{2}},$ (38) $\displaystyle C(\hat{\rho}_{AC})$
$\displaystyle=2\sin\theta\cos\theta\sin\phi\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\sinh^{2}r_{1}\cosh
r_{1}},$ (39) $\displaystyle C(\hat{\rho}_{AB})$
$\displaystyle=2\sin^{2}\theta\sin\phi\cos\phi\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\sinh^{2}r_{2}\cosh
r_{2}}.$ (40)

Here Eq. (38) refers to the situation where Alice’s qubit is traced out and in
the resulting bipartite system both the qubits are in non-inertial frames.
When Bob’s qubit is traced out the total quantum coherence is given through
Eq. (39), and similarly when Charlie’s qubit is traced out Eq. (40) gives the
quantum coherence of the resulting bipartite state. For the $W$-state of the
form$\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$ when both Bob’s
and Charlie’s qubits are accelerated the total quantum coherence of the system
is,

$\displaystyle C(\hat{\rho})$
$\displaystyle=\frac{2}{3}\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\sinh^{2}r_{1}\cosh
r_{1}}\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\sinh^{2}r_{2}\cosh r_{2}}$
$\displaystyle+\frac{2}{3}\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\sinh^{2}r_{1}\cosh
r_{1}}+\frac{2}{3}\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\sinh^{2}r_{2}\cosh
r_{2}}.$ (41)

Eq. (41) reduces to the standard value of $C(\hat{\rho})=2$ for
$r_{1},r_{2}\rightarrow 0$ and the single qubit acceleration limit when either
$r_{1}$ or $r_{2}$ tends to zero. In the contour plot shown in Fig. 7 we
analyse the variation of the quantum coherence as a function of $r_{1}$ and
$r_{2}$ namely Charlie’s and Bob’s acceleration parameter for different values
of the generalisation parameters $\theta$ and $\phi$. We find that coherence
is maximum when $r_{1}$ and $r_{2}$ are zero and decreases with increase in
their value.

A very interesting limiting case of the generalised W-state occurs when
$\theta=\pi/2$ and $\phi=\pi/4$. The generalised W-state corresponding to this
value is $(|100\rangle+|010\rangle)/\sqrt{2}$. We can observe that this
quantum state is of the biseparable form AB-C, where the AB pair is entangled
and C is separable. In this state Charlie’s qubit is accelerated, the total
quantum coherence is $C=1$. This is because the only coherence contribution in
the system comes because of the correlation between Alice and Bob’s qubit. On
the contrary, when Bob is in a non-inertial frame of reference, the total
quantum coherence in the system is,

$\displaystyle C(\hat{\rho})=\frac{Li_{-1/2}(\tanh^{2}r)}{\sinh^{2}r\cosh r}.$
(42)

When both Bob and Charlie are in non-inertial frames, the resulting coherence
is same as when Bob is in a non-inertial frame. This is because Charlie’s
qubit does not contribute to the coherence in the system.

For the W-state, the tripartite entanglement measured by the $\pi$-tangle
saturates at a finite value [76]. This is in stark contrast with the results
obtained for the bipartite entanglement in Ref. [31]. This is again due to the
insufficiency in using $\pi$-tangle as a measure of entanglement. For a
$W$-state, the variation of the quantum coherence due to uniform acceleration
is similar to the change in the entanglement of the system. But the
$\ell_{1}$-norm measure of coherence is a complete measure unlike the
$\pi$-tangle measure of entanglement used in Ref. [76]. The saturation nature
of quantum coherence is due to the presence of non-classical correlations at
relativistic velocities. This result holds for single qubit as well as two
qubit accelerated systems.

##  IV Separable states

The SLOCC class of classification applies to entangled states. But in general
quantum states may not be entangled. An example of a separable state is
$|000\rangle$ and for this state $\hat{\rho}=\hat{\rho}_{d}$ in the
computational basis and so it is an incoherent state. Here we would like to
mention that one of the crucial properties of quantum coherence is that it is
a basis-dependent quantity. Hence studying a separable quantum state
$|+++\rangle$ where $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ in the
$\sigma^{z}$-basis we find quantum coherence in the system. To find the
relativistic effects, we accelerate either one or two of these qubits. For the
accelerated qubits we replace the Minkowski states by their corresponding
Rindler modes. We already know that there are two causally disconnected
Rindler modes, so we trace out one of the modes and compute the quantum
coherence of the rest of the system. Here when only Charlie’s qubit is
accelerated the quantum coherence computed using the $\ell_{1}$-norm measure
is,

$\displaystyle C(\hat{\rho})=3+\frac{4\;Li_{-1/2}(\tanh^{2}r)}{\cosh
r\sinh^{2}r}.$ (43)

In the $r\rightarrow 0$ limit, the inertial value of the $\ell_{1}$-norm
coherence of the system is recovered. When both Charlie’s and Bob’s qubit are
accelerated the quantum coherence is,

$\displaystyle C(\hat{\rho})$
$\displaystyle=1+\frac{2\;Li_{-1/2}(\tanh^{2}r_{1})}{\cosh
r_{1}\sinh^{2}r_{1}}+\frac{2\;Li_{-1/2}(\tanh^{2}r_{2})}{\cosh
r_{2}\sinh^{2}r_{2}}$ $\displaystyle+\frac{2\;Li_{-1/2}(\tanh^{2}r_{1})}{\cosh
r_{1}\sinh^{2}r_{1}}\;\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\cosh
r_{2}\sinh^{2}r_{2}}.$ (44)

From this result, in the appropriate limiting conditions we can recover the
single qubit result as well as the inertial value.

The coherence measured in the GHZ state, W-state and the separable
$|+++\rangle$ state is the total amount of quantum coherence present in the
states. But the type of coherence in the GHZ and W-states is fundamentally
different from the type of coherence present in the $|+++\rangle$ state. In
the GHZ and W-states, the coherence arises due to the correlation between the
qubits. Meanwhile the coherence in the $|+++\rangle$ state is because of the
superposition between the levels within each qubit. These two fundamentally
different forms of coherences were identified in Ref. [14]. The coherence
arising due to the correlation between the qubits is the global coherence of
the system and the coherence resulting from the superposition of the levels
within a qubit are called local coherence. These two forms of coherences are
complementary to each other in a sense that the increase in the global
coherence causes a decrease in the local coherence and vice versa. With
increase in the acceleration the quantum coherence decreases and it saturates
at a finite value. To understand this for the $|+++\rangle$ state, we can
consider the single qubit system $|+\rangle$ system which we can write as,

$\displaystyle|+\rangle=\sum_{n=0}^{\infty}\frac{\tanh^{n}r}{\sqrt{2}\cosh
r}\left(|n\rangle_{I}|n\rangle_{II}+\frac{\sqrt{n+1}}{\cosh
r}|n+1\rangle_{I}|n\rangle_{II}\right).$ (45)

Here we can observe that the initial superposition between two levels is
spread out between the four modes. Of these two modes corresponding to Rindler
region II are traced out and hence there is a loss of superposition and
consequently a loss of coherence. This loss is the inaccessible coherence when
the system undergoes uniform acceleration.

The GHZ and W-states represents one extreme where the global coherence is
maximum with zero local coherence and the $|+++\rangle$ denotes the other
extreme where the local coherence is maximum with no global coherence. But
there are some pure quantum states in which both these types of coherence
coexist. For such states we can quantify the global and local coherence using
the formula,

$\displaystyle C_{G}$ $\displaystyle=\|\hat{\rho}-\pi(\hat{\rho})\|_{l_{1}},$
(46) $\displaystyle C_{L}$
$\displaystyle=\|\pi(\hat{\rho})-[\pi(\hat{\rho})]_{d}\|_{l_{1}}.$ (47)

Here $\pi(\hat{\rho})=\hat{\rho}_{1}\otimes\dots\otimes\hat{\rho}_{N}$ is the
product state of the density matrix $\hat{\rho}$ where the reduced density
matrix $\hat{\rho}_{1}={\rm Tr}_{(2,...,N)}\ \hat{\rho}$. The quantum state
$[\pi(\hat{\rho})]_{d}$ is the decohered density matrix corresponding to
$\pi(\hat{\rho})$ the product state. The total quantum coherence of the system
$C_{T}$ measures coherence contributions from correlations as well as local
superpositions. In Ref. [14] a trade-off was observed between the global
coherence and local coherence.

Figure 8: A plot of the total coherence $(C_{T})$, global coherence $(C_{G})$
and local coherence $(C_{L})$ for (a) the $W\bar{W}$ state, (b) Star state
when the central qubit is accelerating and (c) Star state when the peripheral
qubit is accelerating.

##  V Tripartite pure states with local and global coherence

In this section we study the non-inertial effects when both local and global
coherences are present in the system. Towards this end we analyze the quantum
coherence of tripartite $W\bar{W}$ and star states. The $W\bar{W}$ state is a
symmetric tripartite state with both local and global coherence. On the
contrary the star state is an asymmetric state with both local and global
coherences.

###  V.1 $W\bar{W}$ state

The three qubit $W\bar{W}$ state [77] is a linear superposition of the
tripartite $W$ state and the $\bar{W}$ state as shown below,

$\displaystyle|W\bar{W}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|W\rangle+|\bar{W}\rangle),$
$\displaystyle|W\rangle$
$\displaystyle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle),$
$\displaystyle|\bar{W}\rangle$
$\displaystyle=\frac{1}{\sqrt{3}}(|011\rangle+|101\rangle+|110\rangle).$ (48)

First we consider the situation, where a single qubit is in a noninertial
frame that is to say only Charlie’s qubit is being accelerated. After tracing
out one of the Rindler region, the total quantum coherence of the resulting
density matrix is,

$\displaystyle C_{T}(\hat{\rho})=2+\frac{3\;Li_{-1/2}(\tanh^{2}r)}{\cosh
r\sinh^{2}r}.$ (49)

In the $r\rightarrow 0$ limit the quantum coherence recovers the inertial
limit.

The single qubit reduced density matrices of the $W\bar{W}$ state in Eq. (V.1)
are,

$\displaystyle\hat{\rho}_{A}=\hat{\rho}_{B}=\hat{\rho}_{C}=\begin{pmatrix}\frac{1}{2}&\frac{1}{3}\\\
\frac{1}{3}&\frac{1}{2}\end{pmatrix}.$ (50)

From the reduced density matrix we notice that the single qubit state has a
finite amount of quantum coherence. Consequently the product state
$\hat{\rho}_{A}\otimes\hat{\rho}_{B}\otimes\hat{\rho}_{C}$ will also have some
coherence which is the local coherence of the system. Using Equations (46) and
(47) we find the global and local coherence when one of the qubit is in non-
inertial frame,

$\displaystyle C_{G}(\hat{\rho})$
$\displaystyle=\frac{2}{9}+\frac{31\;Li_{-1/2}(\tanh^{2}r)}{27\cosh
r\sinh^{2}r},$ (51) $\displaystyle C_{L}(\hat{\rho})$
$\displaystyle=\frac{16}{9}+\frac{50\;Li_{-1/2}(\tanh^{2}r)}{27\cosh
r\sinh^{2}r}.$ (52)

The behaviour of the total coherence, global coherence and local coherence are
shown in Fig. 8(a). From the plots we find that all the different forms of
coherences show identical behaviour and they decrease with increase in
acceleration and attain a saturation value at higher values of acceleration.
The results show that both global and local coherence have similar decoherence
properties. Hence Unruh decoherence cannot distinguish between the two forms
of coherence viz the one arising due to the correlations between the qubits
and the quantumness due to the superposition between the levels within a
qubit.

To study the relativistic effects when two qubits are in noninertial frames,
we uniformly accelerate both Bob’s and Charlie’s qubits. The total quantum
coherence of the $W\bar{W}$ state in this scenario is,

$\displaystyle C_{T}(\hat{\rho})$
$\displaystyle=\frac{4}{6}+\frac{8\;Li_{-1/2}(\tanh^{2}r_{1})}{6\cosh
r_{1}\sinh^{2}r_{1}}+\frac{8\;Li_{-1/2}(\tanh^{2}r_{2})}{6\cosh
r_{2}\sinh^{2}r_{2}}$
$\displaystyle+\frac{10Li_{-1/2}(\tanh^{2}r_{1})\;Li_{-1/2}(\tanh^{2}r_{2})}{6\cosh
r_{1}\sinh^{2}r_{1}\;\cosh r_{2}\sinh^{2}r_{2}}.$ (53)

The total coherence is present as both global coherence arising from inter-
qubit correlations and also as local coherence coming from intraqubit
superpositions. The measured values of the global and local coherence of the
system are,

$\displaystyle C_{G}(\hat{\rho})$
$\displaystyle=\frac{2\;Li_{-1/2}(\tanh^{2}r_{1})}{9\cosh
r_{1}\sinh^{2}r_{1}}+\frac{2\;Li_{-1/2}(\tanh^{2}r_{2})}{9\cosh
r_{2}\sinh^{2}r_{2}}$
$\displaystyle+\frac{25\;Li_{-1/2}(\tanh^{2}r_{1})\;Li_{-1/2}(\tanh^{2}r_{2})}{27\cosh
r_{1}\sinh^{2}r_{1}\;\cosh r_{2}\sinh^{2}r_{2}},$ (54) $\displaystyle
C_{L}(\hat{\rho})$
$\displaystyle=\frac{2}{3}+\frac{10\;Li_{-1/2}(\tanh^{2}r_{1})}{9\cosh
r_{1}\sinh^{2}r_{1}}+\frac{10\;Li_{-1/2}(\tanh^{2}r_{2})}{9\cosh
r_{2}\sinh^{2}r_{2}}$
$\displaystyle+\frac{20\;Li_{-1/2}(\tanh^{2}r_{1})\;Li_{-1/2}(\tanh^{2}r_{2})}{27\cosh
r_{1}\sinh^{2}r_{1}\;\cosh r_{2}\sinh^{2}r_{2}}.$ (55)

We find that the total coherence, global coherence and local coherence reduce
to their respective inertial values when $r_{1},r_{2}\rightarrow 0$. Also we
find that the total coherence is a sum of the global and local coherence for
the $\ell_{1}$-norm of coherence.

###  V.2 Star state

A star state is an asymmetric quantum state [78, 79], so called because a
central qubit is entangled with the peripheral qubits collectively, but upon
tracing out the central qubit the peripheral ones have a separable form,

$\displaystyle\hat{\rho}_{B,C}=\mathrm{tr}_{A}(\ket{S}_{A,B,C}\bra{S}_{A,B,C})=\hat{\rho}_{B}\otimes\hat{\rho}_{C}.$
(56)

In the tripartite star state we have a central qubit $A$ which is entangled to
other qubits $B$ and $C$ individually. The qubits $B$ and $C$ are not
entangled with each other and are referred to as peripheral qubits. The form
of the tripartite star state is,

$\displaystyle|S\rangle=\frac{1}{2}(|000\rangle+|100\rangle+|101\rangle+|111\rangle).$
(57)

When the central qubit $A$ is accelerated, the total quantum coherence in the
system is,

$\displaystyle C_{T}(\hat{\rho})=1+\frac{2\;Li_{-1/2}(\tanh^{2}r)}{\cosh
r\sinh^{2}r}.$ (58)

The star state has both global coherence and local coherence and they are

$\displaystyle C_{G}(\hat{\rho})$
$\displaystyle=\frac{-1}{4}+\frac{7\;Li_{-1/2}(\tanh^{2}r)}{8\cosh
r\sinh^{2}r},$ (59) $\displaystyle C_{L}(\hat{\rho})$
$\displaystyle=\frac{5}{4}+\frac{9\;Li_{-1/2}(\tanh^{2}r)}{8\cosh
r\sinh^{2}r}.$ (60)

When the peripheral qubit $B$ is being accelerated, the total coherence of the
system calculated using the $\ell_{1}$-norm of coherence is,

$\displaystyle
C_{T}(\hat{\rho})=\frac{3}{2}+\frac{3\;Li_{-1/2}(\tanh^{2}r)}{2\cosh
r\sinh^{2}r}.$ (61)

The corresponding global of the system is,

$\displaystyle
C_{G}(\hat{\rho})=\frac{1}{4}+\frac{3\;Li_{-1/2}(\tanh^{2}r)}{8\cosh
r\sinh^{2}r}.$ (62)

The local coherence of the system on accelerating the peripheral qubit is same
as the local coherence when the central qubit is being under acceleration, the
expression for which is given in Eq. (60). In both the cases namely when the
central qubit and the peripheral qubit in the $r\rightarrow 0$ limit the
results corresponding to the inertial frame are recovered. For the star
states, the variation of quantum coherence is qualitatively similar when both
the central and peripheral qubits are accelerated. The global local and total
coherence have a maximum value in the inertial frame and decreases with
acceleration and attain a saturation value for large values of the
acceleration as shown in Fig. 8. From the analysis of the two different
situations where either the central qubit or the peripheral qubit is being
accelerated, we find that both global and local coherence have similar
decoherence properties. This shows that Unruh decoherence affects all kinds of
quantumness equally.

Next we look at the situation where two qubits are in non-inertial frames of
reference. Due to the asymmetry of the star states, there are two
possibilities for this situation viz: (i) when a peripheral qubit and a
central qubit are accelerated ; (ii) when both the peripheral qubits are
accelerated. The first situation can be analysed when both Bob’s and Charlie’s
qubits are simultaneously accelerated. The total coherence of the system in
this case is,

$\displaystyle C_{T}(\hat{\rho})$
$\displaystyle=\frac{1}{2}+\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\cosh
r_{1}\sinh^{2}r_{1}}+\frac{1}{2}\;\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\cosh
r_{2}\sinh^{2}r_{2}}$ $\displaystyle+\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\cosh
r_{1}\sinh^{2}r_{1}}\;\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\cosh
r_{2}\sinh^{2}r_{2}}.$ (63)

Under the same conditions the global and local coherence of the star state
are,

$\displaystyle C_{G}(\hat{\rho})$
$\displaystyle=\frac{1}{2}+\frac{1\;Li_{-1/2}(\tanh^{2}r_{1})}{4\cosh
r_{1}\sinh^{2}r_{1}}-\frac{1\;Li_{-1/2}(\tanh^{2}r_{1})}{4\cosh
r_{2}\sinh^{2}r_{2}}$
$\displaystyle+\frac{5\;Li_{-1/2}(\tanh^{2}r_{1})\;Li_{-1/2}(\tanh^{2}r_{2})}{8\cosh
r_{1}\sinh^{2}r_{1}\;\cosh r_{2}\sinh^{2}r_{2}},$ (64) $\displaystyle
C_{L}(\hat{\rho})$
$\displaystyle=\frac{1}{2}+\frac{3\;Li_{-1/2}(\tanh^{2}r_{1})}{4\cosh
r_{1}\sinh^{2}r_{1}}+\frac{3\;Li_{-1/2}(\tanh^{2}r_{2})}{4\cosh
r_{2}\sinh^{2}r_{2}}$
$\displaystyle+\frac{3\;Li_{-1/2}(\tanh^{2}r_{1})\;Li_{-1/2}(\tanh^{2}r_{2})}{8\cosh
r_{1}\sinh^{2}r_{1}\;\cosh r_{2}\sinh^{2}r_{2}}.$ (65)

In the second case when Alice and Bob’s qubits are being accelerated, the
total coherence of the system is,

$\displaystyle C_{T}(\hat{\rho})$
$\displaystyle=\frac{1}{2}+\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\cosh
r_{1}\sinh^{2}r_{1}}+\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\cosh
r_{2}\sinh^{2}r_{2}}$
$\displaystyle+\frac{1}{2}\;\frac{Li_{-1/2}(\tanh^{2}r_{1})}{\cosh
r_{1}\sinh^{2}r_{1}}\;\frac{Li_{-1/2}(\tanh^{2}r_{2})}{\cosh
r_{2}\sinh^{2}r_{2}}.$ (66)

The corresponding global and local coherences are,

$\displaystyle C_{G}(\hat{\rho})$
$\displaystyle=\frac{\;Li_{-1/2}(\tanh^{2}r_{1})}{4\cosh
r_{1}\sinh^{2}r_{1}}+\frac{Li_{-1/2}(\tanh^{2}r_{2})}{4\cosh
r_{2}\sinh^{2}r_{2}}$
$\displaystyle+\frac{Li_{-1/2}(\tanh^{2}r_{1})\;Li_{-1/2}(\tanh^{2}r_{2})}{8\cosh
r_{1}\sinh^{2}r_{1}\;\cosh r_{2}\sinh^{2}r_{2}},$ (67) $\displaystyle
C_{L}(\hat{\rho})$
$\displaystyle=\frac{1}{2}+\frac{3\;Li_{-1/2}(\tanh^{2}r_{1})}{4\cosh
r_{1}\sinh^{2}r_{1}}+\frac{3\;Li_{-1/2}(\tanh^{2}r_{2})}{4\cosh
r_{2}\sinh^{2}r_{2}}$
$\displaystyle+\frac{3\;Li_{-1/2}(\tanh^{2}r_{1})\;Li_{-1/2}(\tanh^{2}r_{2})}{8\cosh
r_{1}\sinh^{2}r_{1}\;\cosh r_{2}\sinh^{2}r_{2}}.$ (68)

All the different quantum coherences attain their inertial values in the limit
$r\rightarrow 0$. From the expressions of the total, global and local
coherence we find that the $C_{T}=C_{G}+C_{L}$ for all the different cases of
star states.

In the tripartite systems like $W\bar{W}$-states and star states, the total
coherence in the system is distributed as global and local coherence. Here by
the word global coherence we mean coherence arising due to all the types of
non-classical correlations (both local and entanglement type) between the
qubits. Hence the global coherence does not fall to zero in the infinite
acceleration limit and rather saturates to a finite value due to the presence
of local correlations. The effect of non-inertial motion on local coherence
has not been investigated before. From our results we can see that the intra-
qubit superpositions do not completely vanish in the infinite acceleration
limit which is a very interesting result. In the tripartite systems when
either one or two of the parties are in non-inertial frames of reference, the
accelerated qubits get split into two modes corresponding to the two Rindler
regions. The degradation of quantum coherence is because part of the coherence
becomes inaccessible to experimental measurement. Both the global and local
coherence decay at the same rate. Hence we find that the acceleration affects
the interqubit and intraqubit properties in the same manner.

Figure 9: A plot of the variation of the normalized decoherence (a) with the
acceleration parameter $r$ and (b) with the number of accelerated qubits $n$
for a fixed values of $r=1.5$ and $r=2.0$. The total number of qubits in the
system is $N=11$.

##  VI Applications

###  VI.1 Relativistic effects on multipartite state distributed over a
network

Let us consider a network of satellites sharing a multipartite quantum state.
An important study, could be about the effect of acceleration on the quantum
coherence of the multipartite system. To understand this, we measure the
quantum coherence of the $N$-partite GHZ and W-state. In the $N$ partite
state, we have $n$ number of uniformly accelerated qubits. First let us
consider a $N$-partite GHZ state,

$\displaystyle|\mathrm{GHZ}\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes
N}+|1\rangle^{\otimes N}).$ (69)

The total quantum coherence is $C_{\ell_{1}}(\hat{\rho})=1$ for this state.
Out of the $N$-qubits if $n$ qubits undergo acceleration, the quantum
coherence of the state is,

$\displaystyle
C(\hat{\rho})=\prod_{k=1}^{n}\frac{Li_{-1/2}(\tanh^{2}r_{k})}{\sinh^{2}r_{k}\cosh
r_{k}}.$ (70)

Let $C_{k}(\hat{\rho})$ be the quantum coherence of the $N$-partite system
when only the $k^{th}$ qubit is being accelerated. Using this the coherence of
the $N$-partite system with $n$ accelerating qubits can be written as,

$\displaystyle C(\hat{\rho})=\prod_{k=1}^{n}C_{k}(\hat{\rho}).$ (71)

When all the satellites have the same acceleration, the quantum coherence
simplifies to $C(\hat{\rho})=[C_{k}(\hat{\rho})]^{n}$. From this result we can
see that the quantum coherence of the $N$-partite GHZ state falls
exponentially with the increase in the number of accelerating satellites. This
exponential fall is a manifestation of genuinely multipartite form of quantum
coherence where the loss of even a single qubit makes the state completely
incoherent. Hence a small acceleration of each of the $m$ qubits leads to a
huge loss of coherence in the whole system. To put this into a numerical
perspective let us consider a $11$ qubit GHZ state without any of the qubits
under acceleration. The quantum coherence of such a system is
$C_{\ell_{1}}(\hat{\rho})=1$ and let us consider $r=2.0$ for each of the
qubit. The coherence when only one qubit is accelerating is
$C_{\ell_{1}}(\hat{\rho})=0.8988$. Now when $10$ qubits start accelerating, we
can see that the total coherence reduced to
$C_{\ell_{1}}(\hat{\rho})=(0.8988)^{10}=0.3439$. For very large values of $n$,
the quantum coherence $C_{\ell_{1}}\rightarrow 0$. Since the reduction of
quantum coherence depends on its distribution in multipartite system, the
results obtained here might hold for entanglement as well. This is because
coherence and entanglement are distributed in a similar manner. So we can
expect entanglement to fall exponentially similar to quantum coherence. Hence
when quantum information is shared between a network of satellites, the
quantum coherence is a function of the number of accelerating satellites and
the amount of acceleration.

Next we consider the $N$-partite W-state,

$\displaystyle|W\rangle=\frac{1}{\sqrt{N}}(|00\cdots 01\rangle+|00\cdots
10\rangle+\cdots+|10\cdots 00\rangle).$ (72)

In the non-relativistic scenario the total quantum coherence of this state in
the non-inertial frame of reference is $C_{\ell_{1}}(\hat{\rho})=N-1$. Here if
$n$ qubits start accelerating, the quantum coherence of the $N$-partite state
changes to,

$\displaystyle C_{\ell_{1}}(\hat{\rho})$
$\displaystyle=\frac{2}{N}\sum\limits_{\begin{subarray}{c}1\leq i,j\leq n\\\
i<j\end{subarray}}\frac{Li_{-1/2}(\tanh^{2}r_{i})}{\sinh^{2}r_{i}\cosh
r_{i}}\frac{Li_{-1/2}(\tanh^{2}r_{j})}{\sinh^{2}r_{j}\cosh r_{j}}$
$\displaystyle+\frac{2}{N}(N-n)\sum_{i=1}^{N}\frac{Li_{-1/2}(\tanh^{2}r_{i})}{\sinh^{2}r_{i}\cosh
r_{i}}$ $\displaystyle+\frac{(N-n)(N-(n+1))}{N}.$ (73)

In the large $N$ limit, when $n\approx N$ the first term which is a product of
two polylog functions dominates over the second and the third term. Hence we
would see the quantum coherence falling as polynomial function of second
order. When $n\ll N$ in the large $N$ limit, the third term dominates over the
first and second term. Consequently the decrease of quantum coherence due to
the relativistic effects will be minimal under such situations. If
acceleration is the same for all the satellites then we have,

$\displaystyle C_{\ell_{1}}(\hat{\rho})$
$\displaystyle=\frac{2}{N}\frac{n(n-1)}{2}\left[\frac{Li_{-1/2}(\tanh^{2}r_{i})}{\sinh^{2}r_{i}\cosh
r_{i}}\right]^{2}$
$\displaystyle+\frac{2}{N}(N-n)n\left[\frac{Li_{-1/2}(\tanh^{2}r_{i})}{\sinh^{2}r_{i}\cosh
r_{i}}\right]$ $\displaystyle+\frac{(N-n)(N-(n+1))}{N}.$ (74)

Since the quantum coherence of the $N$-partite W-state is $(N-1)$, it is not
possible to compare it with the coherence loss of the GHZ state. To make a
comparison between the coherence loss of the GHZ and W-states we define,

$\displaystyle
C_{\ell_{1}}^{N}(\hat{\rho})=\frac{C_{\ell_{1}}^{R}(\hat{\rho})}{C_{\ell_{1}}^{NR}(\hat{\rho})},$
(75)

where $C_{\ell_{1}}^{N}$ is the normalized amount of quantum coherence and
$C_{\ell_{1}}^{R}$ is the amount of relativistic coherence and
$C_{\ell_{1}}^{NR}$ is the quantum coherence of the non-relativistic state.
For the GHZ state $C_{\ell_{1}}^{NR}=1$ and so
$C_{\ell_{1}}^{N}(\hat{\rho})=C_{\ell_{1}}^{R}(\hat{\rho})$. In the case of
the W-state, for $r=2.0$, $C_{\ell_{1}}^{R}(\hat{\rho})=8.2430$ and
$C_{\ell_{1}}^{NR}=N-1$. The normalized amount of quantum coherence in the
system when $10$ out of $11$ qubits are being accelerated is
$C_{\ell_{1}}^{N}=0.8243$. Hence, for the same number of quantum states and
acceleration, we find that the quantum coherence of W-state is more robust to
Unruh decoherence when compared with GHZ state. This implies that in a
multipartite system sharing quantum coherence in a bipartite manner protects
it from decoherence due to relativistic effects.

Figs. 9(a) and 9(b) depicts the relativistic quantum coherence of the system.
In Fig. 9(a) we plot the change in quantum coherence with $r$ for both GHZ and
W-states. Here we consider two situations where one qubit is being accelerated
or $10$ qubits are being accelerated. In the first case, where one qubit is
under acceleration the loss of coherence is minimal. Compared to the first
case, we find that the coherence loss is much higher when $10$ qubits are
accelerated. Also we find that among these states, the W-state with increasing
accelerating qubits is more robust to decoherence. In Fig. 9 (b) we plot the
coherence as a function of the number of accelerating qubits. We find that the
loss of coherence is higher with increase in the number of accelerating
qubits. Also for a given number of accelerating qubits, the GHZ states
experience a higher loss of coherence.

###  VI.2 Coherence degradation at infinite acceleration

The entanglement degradation in the highly relativistic situation has been
explained in Ref. [31] in the context of a bipartite systems. In Ref. [31],
the authors consider the two qubit case where Alice is an inertial observer
and Bob being non-inertial with uniform acceleration. In the infinite
acceleration limit, non-inertial Bob is close to the Rindler horizon Ref.
[31], which can be considered as the event horizon from the perspective of a
black hole. It is well known that a spherical non-rotating static black hole
can be described using a Schwarzchild space-time. This can be approximated by
Rindler co-ordinate in Minkowski space-time in the infinite acceleration limit
[80]. Hence in the existing scenario, we consider Alice and Bob to be close to
the event horizon, such that inertial Alice falls into a static black hole and
accelerating Bob close to event horizon escapes from falling into it.

Similarly our work investigates the quantum coherence degradation in
tripartite systems due to relativistic effects. Here we consider three qubits,
one each in the possession of Alice, Bob and Charlie. Degradation of quantum
coherence has been observed when either a single qubit or two qubits are being
accelerated. From our results in Sec. III, for both GHZ and W-states we notice
a decrease in quantum coherence, but the coherence freezes at a finite amount
in the infinite acceleration limit. The entire coherence in the GHZ and
W-state is due to the correlations between the qubits. In a similar context to
understand the relativistic effects on local coherence we study the
$|+++\rangle$ separable state. The qualitative behavior of the local coherence
in the $|+++\rangle$ state is similar to that of the global coherence of the
GHZ and W-states. Hence we conclude that when an inertial observer falls into
a black hole, the quantum correlations and quantum superposition shared with
the non-inertial observer does not vanish completely.

##  VII Results & discussion

The relativistic effects on the quantum coherence of a multipartite system is
investigated using the $\ell_{1}$-norm measure of coherence. Initially all the
qubits are in an inertial frame and can be described using Minkowski space-
time coordinates. When some of the qubits are under acceleration, the
Minkowski space corresponding to them is divided into two causally
disconnected regions. From the Minkowski modes, the Rindler modes can be
obtained via the Unruh modes. When we consider a Gaussian Minkowski smearing
function it can be approximated by a single Unruh mode. Under the
monochromatic approximation we can describe the states using Rindler co-
ordinates in the Minkowski space time. Hence the quantum coherence initially
present in the Minkowski co-ordinates is distributed between the two Rindler
co-ordinates of which only one is experimentally accessible. The coherence
which can be measured in the accessible Rindler region is called the
accessible coherence. The coherence corresponding to the other Rindler region
is called the inaccessible coherence since it cannot be experimentally
measured.

First we investigate the tripartite quantum systems. Based on the SLOCC
classification, the tripartite states are classified into the GHZ class and
the W classes. For both the GHZ and W-states we find that the quantum
coherence decreases with increase in acceleration and attains a saturation
value for very high accelerations. Here we note that coherence can exist due
to correlations between qubits which is known as global coherence. Also
coherence may arise due to superposition between the levels within a qubit,
referred to as local coherence. The global coherence is the only coherence
present in the GHZ and the W-states.

Next we consider a separable state of the form $|+++\rangle$, to understand
the relativistic effects on local coherence. The local coherence also
decreases with increase in acceleration and saturates at higher acceleration.
The local coherence and the global coherence are complementary to each other
and the total coherence is a combination of these two types of coherences.
Some tripartite quantum states have both local and global coherence. As an
example we consider two such quantum states namely $W\bar{W}$ and star states.
In the $W\bar{W}$ state, the global coherence is distributed equally between
the three qubits. There is an asymmetric distribution of quantum coherence in
the star states. For both these states, all the different forms of coherence
namely the global, local and total coherence decreases with an increase in
acceleration.

Figure 10: A schematic illustration of the bifurcation of the Bell state which
displays linear loss of coherence into two extreme classes of states viz. GHZ
class with exponential loss of coherence and W class with polynomial loss of
coherence.

The entanglement of the bipartite system decreases due to relativistic effects
and reaches zero in the infinite acceleration limit [31]. But the tripartite
entanglement measured in Ref. [76] using the $\pi$-tangle does not go to zero
for higher values of acceleration. Due to relativistic effects the quantum
discord of a system also does not go to zero in the infinite acceleration
limit [55]. For the tripartite entanglement, the non-vanishing nature is
because the $\pi$-tangle is not a complete measure of entanglement. But for
the quantum discord this is a fundamental feature of the underlying quantum
correlations. From our work we find that there is a qualitative relationship
between the total quantum correlations and the quantum coherence of a
relativistic quantum system.

In quantum information theory, it is not possible to distinguish between Bell
states and the GHZ states through an estimation of entropy. But we can
calculate the quantum coherence of these two states when they are in an
inertial frame of reference. Here we again find that
$C({|\psi_{Bell}\rangle})=C({|\psi_{GHZ}\rangle})$ and so we will not be able
to distinguish between these two states. This is because the GHZ state
$\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$ can be rewritten as an
equivalent Bell state
$\frac{1}{\sqrt{2}}(|0\mathbf{0}\rangle+|1\mathbf{1}\rangle)$ where $|{\bf
0}\rangle=|00\rangle$ and $|{\bf 1}\rangle=|11\rangle$ are redundantly encoded
logical qubits encoding the same quantum information. Next we consider the
situation where part of the system is moving with constant acceleration. When
the single qubit in GHZ and Bell states move with constant acceleration it is
not possible to distinguish between them. On the contrary when the logical
qubit of the GHZ state is in an accelerated motion, then we have
$C({|\psi_{Bell}\rangle})\neq C({|\psi_{GHZ}\rangle})$ and the reason for this
is that the size of the logical qubit under uniform acceleration is bigger
than that of the regular qubit of the Bell state. This result is very
interesting from a quantum information theory perspective where in general we
cannot distinguish between Bell and GHZ states or between multipartite GHZ
states with different number of qubits. But a measurement of quantum coherence
combined with relativistic motion can differentiate them. Thus coherence
measurement of accelerated systems can be used in quantum state
discrimination.

Next we consider a $N$-partite GHZ state and W-state and estimate the quantum
coherence when $n<N$ qubits are accelerated. The GHZ and W-state represent the
two extremes of coherence sharing. In the GHZ state, the coherence is present
in a maximally entangled multipartite manner such that the loss of a single
qubit destroys the entire coherence in the system. We find that in a GHZ state
when the number of accelerating qubits $n$ increases, the coherence falls
exponentially. The coherence in a W-state is shared in a local bipartite
fashion. Here in the case of a W-states we observe that the coherence falls
polynomially with increase in the number of accelerating qubits. In Fig. 10,
through a flow chart style we illustrate the quantum coherence decrease as we
move from $n=2$ to $n>2$ quantum system.

The Bell states describe the maximally entangled states in a two qubit system.
When one of the qubits becomes non-inertial, then the coherence of the Bell
state is given by $C_{\ell_{1}}=Li_{-1/2}(\tanh^{2}r)/(\sinh^{2}r\cosh r)$.
There is only one way to entangle or correlate in a bipartite state. When we
move to multipartite systems, there is more than one way to correlate the
states. The GHZ and W-states denote the two extreme forms of correlation
sharing where it is shared in a multipartite fashion in the former case and in
a bipartite manner in the later one. When more than one qubit is accelerated,
the coherence falls exponentially in the GHZ state and polynomially in the
W-state. As we move from the bipartite to the multipartite case, the linear
fall of coherence observed in Bell type states changes to exponential fall for
the GHZ states and polynomial decrease for the W-state. Hence the Bell type
maximally entangled state clearly bifurcates into two extreme classes of
entangled states with exponential fall (GHZ class) and polynomial fall
(W-state) of coherence. This also proves that the W-state is more robust to
Unruh decoherence compared to the GHZ state. This is because the coherence is
shared in a genuinely multipartite manner in a GHZ state, but in a W-state it
is shared only in a bipartite way. For large $N$ W-state, the loss of a few
qubits results in a W-like state. This result might be useful for satellite
based quantum communication, where some of the satellites are moving with very
high velocities. In Ref. [81] the quantum coherence of multipartite W-states
for Dirac field has been calculated in terms of the Kruskal modes in certain
limiting cases. But in our work, we consider multipartite W-states of bosonic
modes and compute quantum coherence in a very general setting where arbitrary
number of qubits are being accelerated. In our method the Minkowski modes are
converted into Rindler modes in the noninertial frame.

A spherical non-rotating blackhole is described by a Schwarzchild space-time.
In the infinite acceleration limit, the Schwarzchild space-time can be
approximately described by Rindler co-ordinates in Minkowski space time. Hence
our investigation in the infinite acceleration limit can be used to analyze
quantum coherence in the context of black holes. Our study shows that when
Alice falls into a black hole, she might still share quantum coherence with
Bob and Charlie who are escaping from the black-hole. Hence we conclude that
quantum correlations and quantum superposition present in a system do not
completely vanish in the relativistic limit.

In our work we have used the single mode approximation where we use single
frequency global modes. This can be experimentally realized using localized
sources and detectors. Some proposals towards this end might include the use
of localized projective measurements [82], homodyne detection [51] and
accelerated cavities [83, 84]. Here we note that for a pure state, the $l_{1}$
norm measure of coherence is equal to the robustness of coherence [73], which
can be directly estimated using the interference of fringes. A possible method
is the use of Berry phase atomic interferometry experiment [85, 86], in which
one of arm of the interferometer is under non-inertial motion. The phase
difference between the inertial and non-inertial arms can give us the change
in coherence due to the non-inertial motion. Hence an experimental
verification of relativistic effects on quantum coherence might be an
interesting future work. On the theoretical side an investigation on the
quantum coherence effects in curved space time is also an interesting topic to
explore.

## Acknowledgements

Chandrashekar Radhakrishnan was supported in part by a seed grant from IIT
Madras to the Centre for Quantum Information, Communication and Computing.
Peter Rohde is funded by an ARC Future Fellowship (project FT160100397). We
are thankful to the anonymous reviewer for their comments on the single mode
approximation, which was very useful.
_Note: After submission of this manuscript we became aware of the work[87]
which independently derived similar results._

## References

  * Shor [1998] P. W. Shor, Quantum computing, Documenta Mathematica 1, 467 (1998).
  * Deutsch and Jozsa [1992] D. Deutsch and R. Jozsa, Rapid solution of problems by quantum computation, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 439, 553 (1992).
  * Grover [1996] L. K. Grover, A fast quantum mechanical algorithm for database search, in  _Proceedings of the twenty-eighth annual ACM symposium on Theory of computing_ (1996) pp. 212–219, quant-ph/9605043 .
  * Shor [1999] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Review 41, 303 (1999), quant-ph/9508027 .
  * Giovannetti  _et al._ [2006] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Physical Review Letters 96, 010401 (2006), quant-ph/0509179 .
  * Bennett  _et al._ [1993] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels, Physical Review Letters 70, 1895 (1993).
  * Bouwmeester  _et al._ [1997] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Experimental quantum teleportation, Nature 390, 575 (1997), 1901.11004 .
  * Bennett and Brassard [1984] C. H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and con tos5, in  _Proceedings of the International Conference on Computers, Systems and Signal Processing_ (1984) pp. 175–179.
  * Bennett  _et al._ [1992] C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, Experimental quantum cryptography, Journal of Cryptology 5, 3 (1992).
  * Bancal  _et al._ [2009] J.-D. Bancal, C. Branciard, N. Gisin, and S. Pironio, Quantifying multipartite nonlocality, Physical Review Letters 103, 090503 (2009), 0903.2715 .
  * Skrzypczyk  _et al._ [2014] P. Skrzypczyk, M. Navascués, and D. Cavalcanti, Quantifying einstein-podolsky-rosen steering, Physical Review Letters 112, 180404 (2014).
  * Ollivier and Zurek [2001] H. Ollivier and W. H. Zurek, Quantum discord: a measure of the quantumness of correlations, Physical Review Letters 88, 017901 (2001).
  * Baumgratz  _et al._ [2014] T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Physical Review Letters 113, 140401 (2014).
  * Radhakrishnan  _et al._ [2016] C. Radhakrishnan, M. Parthasarathy, S. Jambulingam, and T. Byrnes, Distribution of quantum coherence in multipartite systems, Physical Review Letters 116, 150504 (2016).
  * Winter and Yang [2016] A. Winter and D. Yang, Operational resource theory of coherence, Physical Review Letters 116, 120404 (2016), 1506.07975 .
  * Brandão and Gour [2015] F. G. Brandão and G. Gour, Reversible framework for quantum resource theories, Physical Review Letters 115, 070503 (2015).
  * Chitambar and Gour [2016] E. Chitambar and G. Gour, Critical examination of incoherent operations and a physically consistent resource theory of quantum coherence, Physical Review Letters 117, 030401 (2016).
  * Chitambar and Hsieh [2016] E. Chitambar and M.-H. Hsieh, Relating the resource theories of entanglement and quantum coherence, Physical Review Letters 117, 020402 (2016).
  * Chanda and Bhattacharya [2016] T. Chanda and S. Bhattacharya, Delineating incoherent non-markovian dynamics using quantum coherence, Annals of Physics 366, 1 (2016), 1505.05810 .
  * Radhakrishnan  _et al._ [2019] C. Radhakrishnan, Z. Lü, J. Jing, and T. Byrnes, Dynamics of quantum coherence in a spin-star system: Bipartite initial state and coherence distribution, Physical Review A 100, 042333 (2019).
  * Hillery [2016] M. Hillery, Coherence as a resource in decision problems: The deutsch-jozsa algorithm and a variation, Physical Review A 93, 012111 (2016), 1512.01874 .
  * Zhang  _et al._ [2019] C. Zhang, T. R. Bromley, Y.-F. Huang, H. Cao, W.-M. Lv, B.-H. Liu, C.-F. Li, G.-C. Guo, M. Cianciaruso, and G. Adesso, Demonstrating quantum coherence and metrology that is resilient to transversal noise, Physical Review Letters 123, 180504 (2019), 1907.10540 .
  * Ma  _et al._ [2019] J. Ma, Y. Zhou, X. Yuan, and X. Ma, Operational interpretation of coherence in quantum key distribution, Physical Review A 99, 062325 (2019), 1810.03267 .
  * Peres and Terno [2004] A. Peres and D. R. Terno, Quantum information and relativity theory, Reviews of Modern Physics 76, 93 (2004).
  * Dunningham  _et al._ [2009] J. Dunningham, V. Palge, and V. Vedral, Entanglement and nonlocality of a single relativistic particle, Physical Review A 80, 044302 (2009).
  * Mann and Ralph [2012] R. B. Mann and T. C. Ralph, Relativistic quantum information, Classical and Quantum Gravity 29, 220301 (2012).
  * Alsing and Milburn [2002] P. M. Alsing and G. J. Milburn, Lorentz invariance of entanglement, arXiv preprint quant-ph/0203051 10.26421/qic2.6-4 (2002), quant-ph/0203051 .
  * Alsing and Fuentes [2012] P. M. Alsing and I. Fuentes, Observer-dependent entanglement, Classical and Quantum Gravity 29, 224001 (2012), 1210.2223 .
  * Li and Du [2003] H. Li and J. Du, Relativistic invariant quantum entanglement between the spins of moving bodies, Physical Review A 68, 022108 (2003), quant-ph/0211159 .
  * Shi [2004] Y. Shi, Entanglement in relativistic quantum field theory, Physical Review D 70, 105001 (2004), hep-th/0408062 .
  * Fuentes-Schuller and Mann [2005] I. Fuentes-Schuller and R. B. Mann, Alice falls into a black hole: entanglement in noninertial frames, Physical Review Letters 95, 120404 (2005), quant-ph/0410172 .
  * Bruschi  _et al._ [2010] D. E. Bruschi, J. Louko, E. Martín-Martínez, A. Dragan, and I. Fuentes, Unruh effect in quantum information beyond the single-mode approximation, Phys. Rev. A 82, 042332 (2010).
  * Dragan  _et al._ [2013a] A. Dragan, J. Doukas, E. Martín-Martínez, and D. E. Bruschi, Localized projective measurement of a quantum field in non-inertial frames, Classical and Quantum Gravity 30, 10.1088/0264-9381/30/23/235006 (2013a).
  * Qian Dong [2020] G.-H. S. . S.-H. D. Qian Dong, Ariadna J. Torres-Arenas, Tetrapartite entanglement features of w-class state in uniform acceleration, Front. Phys. 15, 11602 (2020).
  * Unruh [1976] W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D 14, 870 (1976).
  * Downes  _et al._ [2013a] T. G. Downes, T. C. Ralph, and N. Walk, Quantum communication with an accelerated partner, Physical Review A 87, 10.1103/physreva.87.012327 (2013a).
  * Kabat [1995] D. Kabat, Black hole entropy and entropy of entanglement, Nuclear Physics B 453, 281 (1995), hep-th/9503016 .
  * Henderson  _et al._ [2018] L. J. Henderson, R. A. Hennigar, R. B. Mann, A. R. Smith, and J. Zhang, Harvesting entanglement from the black hole vacuum, Classical and Quantum Gravity 35, 21LT02 (2018), 1712.10018 .
  * Martín-Martínez  _et al._ [2010a] E. Martín-Martínez, L. J. Garay, and J. León, Unveiling quantum entanglement degradation near a schwarzschild black hole, Physical Review D 82, 064006 (2010a).
  * Martín-Martínez  _et al._ [2010b] E. Martín-Martínez, L. J. Garay, and J. León, Quantum entanglement produced in the formation of a black hole, Physical Review D 82, 064028 (2010b).
  * Terashima [2000] H. Terashima, Entanglement entropy of the black hole horizon, Physical Review D 61, 104016 (2000), gr-qc/9911091 .
  * Verlinde and Verlinde [2013] E. Verlinde and H. Verlinde, Black hole entanglement and quantum error correction, Journal of High Energy Physics 2013, 107 (2013), 1211.6913 .
  * Ahmadi  _et al._ [2014a] M. Ahmadi, D. E. Bruschi, C. Sabín, G. Adesso, and I. Fuentes, Relativistic quantum metrology: Exploiting relativity to improve quantum measurement technologies, Scientific Reports 4, 1 (2014a).
  * Ahmadi  _et al._ [2014b] M. Ahmadi, D. E. Bruschi, and I. Fuentes, Quantum metrology for relativistic quantum fields, Physical Review D 89, 065028 (2014b), 1312.5707 .
  * Tian  _et al._ [2015] Z. Tian, J. Wang, H. Fan, and J. Jing, Relativistic quantum metrology in open system dynamics, Scientific Reports 5, 1 (2015), 1501.06676 .
  * Du and Mann [2021] H. Du and R. B. Mann, Fisher information as a probe of spacetime structure: relativistic quantum metrology in (a) ds, Journal of High Energy Physics 2021, 1 (2021), 2012.08557 .
  * Friis  _et al._ [2013] N. Friis, A. R. Lee, K. Truong, C. Sabín, E. Solano, G. Johansson, and I. Fuentes, Relativistic quantum teleportation with superconducting circuits, Physical Review Letters 110, 113602 (2013).
  * Lin  _et al._ [2015] S.-Y. Lin, C.-H. Chou, and B. Hu, Quantum teleportation between moving detectors, Physical Review D 91, 084063 (2015).
  * Grochowski  _et al._ [2017] P. T. Grochowski, G. Rajchel, F. Kiałka, and A. Dragan, Effect of relativistic acceleration on continuous variable quantum teleportation and dense coding, Physical Review D 95, 105005 (2017).
  * Koga  _et al._ [2018] J.-i. Koga, G. Kimura, and K. Maeda, Quantum teleportation in vacuum using only unruh-dewitt detectors, Physical Review A 97, 062338 (2018), 1804.01183 .
  * Downes  _et al._ [2013b] T. Downes, T. Ralph, and N. Walk, Quantum communication with an accelerated partner, Physical Review A 87, 012327 (2013b), 1203.2716 .
  * Landulfo [2016] A. G. Landulfo, Nonperturbative approach to relativistic quantum communication channels, Physical Review D 93, 104019 (2016).
  * Kravtsov  _et al._ [2018] K. Kravtsov, I. Radchenko, S. Kulik, and S. Molotkov, Relativistic quantum key distribution system with one-way quantum communication, Scientific Reports 8, 1 (2018), 1801.02896 .
  * Radchenko  _et al._ [2014] I. Radchenko, K. Kravtsov, S. Kulik, and S. Molotkov, Relativistic quantum cryptography, Laser Physics Letters 11, 065203 (2014), 1403.3122 .
  * Datta [2009] A. Datta, Quantum discord between relatively accelerated observers, Physical Review A 80, 052304 (2009), 0905.3301 .
  * Jung  _et al._ [2013] E. Jung, M.-R. Hwang, and D. Park, Quantum discord and quantum entanglement in the presence of an asymptotically flat static black hole, International Journal of Quantum Information 11, 1350061 (2013).
  * Mattle  _et al._ [1996] K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, Dense coding in experimental quantum communication, Physical Review Letters 76, 4656 (1996).
  * Ursin  _et al._ [2007] R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, _et al._ , Entanglement-based quantum communication over 144 km, Nature Physics 3, 481 (2007).
  * Jin  _et al._ [2010] X.-M. Jin, J.-G. Ren, B. Yang, Z.-H. Yi, F. Zhou, X.-F. Xu, S.-K. Wang, D. Yang, Y.-F. Hu, S. Jiang, _et al._ , Experimental free-space quantum teleportation, Nature Photonics 4, 376 (2010).
  * Brito  _et al._ [2021] S. Brito, A. Canabarro, D. Cavalcanti, and R. Chaves, Satellite-based photonic quantum networks are small-world, PRX Quantum 2, 010304 (2021).
  * Liao  _et al._ [2017] S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, _et al._ , Satellite-to-ground quantum key distribution, Nature 549, 43 (2017).
  * Yin  _et al._ [2012] J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, _et al._ , Quantum teleportation and entanglement distribution over 100-kilometre free-space channels, Nature 488, 185 (2012), 1205.2024 .
  * Villar  _et al._ [2020] A. Villar, A. Lohrmann, X. Bai, T. Vergoossen, R. Bedington, C. Perumangatt, H. Y. Lim, T. Islam, A. Reezwana, Z. Tang, _et al._ , Entanglement demonstration on board a nano-satellite, Optica 7, 734 (2020), 2006.14430 .
  * Yin  _et al._ [2017a] J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, _et al._ , Satellite-to-ground entanglement-based quantum key distribution, Physical Review Letters 119, 200501 (2017a).
  * Yin  _et al._ [2020] J. Yin, Y.-H. Li, S.-K. Liao, M. Yang, Y. Cao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, S.-L. Li, _et al._ , Entanglement-based secure quantum cryptography over 1,120 kilometres, Nature 582, 501 (2020).
  * Yin  _et al._ [2017b] J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, _et al._ , Satellite-based entanglement distribution over 1200 kilometers, Science 356, 1140 (2017b), 1707.01339 .
  * Wang  _et al._ [2016] J. Wang, Z. Tian, J. Jing, and H. Fan, Irreversible degradation of quantum coherence under relativistic motion, Physical Review A 93, 062105 (2016), 1601.03238 .
  * Huang and Situ [2018] Z. Huang and H. Situ, Quantum coherence behaviors of fermionic system in non-inertial frame, Quantum Information Processing 17, 1 (2018), 1609.02295 .
  * He  _et al._ [2018] J. He, Z.-Y. Ding, J.-D. Shi, and T. Wu, Multipartite quantum coherence and distribution under the unruh effect, Annalen der Physik 530, 1800167 (2018).
  * Zeng and Cao [2021] H.-S. Zeng and H.-M. Cao, Distribution and evolution of quantum coherence for open multi-qubit systems in non-inertial frames, Annalen der Physik 533, 2000606 (2021).
  * Rana  _et al._ [2016] S. Rana, P. Parashar, and M. Lewenstein, Trace-distance measure of coherence, Physical Review A 93, 012110 (2016), 1511.01854 .
  * Shao  _et al._ [2015] L.-H. Shao, Z. Xi, H. Fan, and Y. Li, Fidelity and trace-norm distances for quantifying coherence, Physical Review A 91, 042120 (2015).
  * Wang  _et al._ [2017] Y.-T. Wang, J.-S. Tang, Z.-Y. Wei, S. Yu, Z.-J. Ke, X.-Y. Xu, C.-F. Li, and G.-C. Guo, Directly measuring the degree of quantum coherence using interference fringes, Physical Review Letters 118, 020403 (2017).
  * Napoli  _et al._ [2016] C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. Johnston, and G. Adesso, Robustness of coherence: an operational and observable measure of quantum coherence, Physical Review Letters 116, 150502 (2016), 1601.03781 .
  * Piani  _et al._ [2016] M. Piani, M. Cianciaruso, T. R. Bromley, C. Napoli, N. Johnston, and G. Adesso, Robustness of asymmetry and coherence of quantum states, Physical Review A 93, 042107 (2016), 1601.03782 .
  * Hwang  _et al._ [2011] M.-R. Hwang, D. Park, and E. Jung, Tripartite entanglement in a noninertial frame, Physical Review A 83, 012111 (2011), 1010.6154 .
  * Sudha  _et al._ [2012] Sudha, A. R. U. Devi, and A. K. Rajagopal, Monogamy of quantum correlations in three-qubit pure states, Phys. Rev. A 85, 012103 (2012).
  * Plesch and Bužek [2003a] M. Plesch and V. Bužek, Entangled graphs: Bipartite entanglement in multiqubit systems, Phys. Rev. A 67, 012322 (2003a).
  * Plesch and Bužek [2003b] M. Plesch and V. Bužek, Entangled graphs. ii. classical correlations in multiqubit entangled systems, Phys. Rev. A 68, 012313 (2003b).
  * Wald [1984] R. M. Wald,  _General Relativity_ (The University of Chicago Press, 1984).
  * Wu  _et al._ [2020] S.-M. Wu, Z.-C. Li, and H.-S. Zeng, Quantum coherence of multipartite w-state in a schwarzschild spacetime, EPL (Europhysics Letters) 129, 40002 (2020).
  * Dragan  _et al._ [2013b] A. Dragan, J. Doukas, E. Martin-Martinez, and D. E. Bruschi, Localized projective measurement of a quantum field in non-inertial frames, Classical and Quantum Gravity 30, 235006 (2013b).
  * Downes  _et al._ [2011] T. G. Downes, I. Fuentes, and T. Ralph, Entangling moving cavities in noninertial frames, Physical review letters 106, 210502 (2011).
  * Friis  _et al._ [2012] N. Friis, A. R. Lee, D. E. Bruschi, and J. Louko, Kinematic entanglement degradation of fermionic cavity modes, Physical Review D 85, 025012 (2012).
  * Rideout  _et al._ [2012] D. Rideout, T. Jennewein, G. Amelino-Camelia, T. F. Demarie, B. L. Higgins, A. Kempf, A. Kent, R. Laflamme, X. Ma, R. B. Mann, _et al._ , Fundamental quantum optics experiments conceivable with satellites—reaching relativistic distances and velocities, Classical and Quantum Gravity 29, 224011 (2012).
  * Martín-Martínez  _et al._ [2011] E. Martín-Martínez, I. Fuentes, and R. B. Mann, Using berry’s phase to detect the unruh effect at lower accelerations, Physical Review Letters 107, 131301 (2011).
  * Wu  _et al._ [2021] S.-M. Wu, H.-S. Zeng, and H.-M. Cao, Quantum coherence and distribution of n-partite bosonic fields in noninertial frame, Classical and Quantum Gravity 38, 185007 (2021).