# Observer dependent entanglement

Paul M. Alsing  Air Force Research Laboratory, Information Directorate, Rome,
N.Y., USA  Ivette Fuentes  School of Mathematical Sciences, University of
Nottingham, Nottingham NG7 2RD United Kingdom

###### Abstract

Understanding the observer-dependent nature of quantum entanglement has been a
central question in relativistic quantum information. In this paper we will
review key results on relativistic entanglement in flat and curved spacetime
and discuss recent work which shows that motion and gravity have observable
effects on entanglement between localized systems.

##  1 Introduction

In quantum information non-classical properties such as entanglement are
exploited to improve information tasks. A prototypical example of this is
quantum teleportation where two observers Alice and Bob use two quantum
systems in an entangled state to transmit information about the state of a
third system. Impressively, cutting-edge experiments involving entanglement
based communications are reaching regimes where relativistic effects can no
longer be neglected. Such is the case of protocols which involve distributing
entanglement over hundreds of kilometers [1, 2]. Understanding entanglement in
relativistic settings has been a key question in relativistic quantum
information. Early results show that entanglement is observer-dependent [3, 4,
5, 6]. The entanglement between two field modes is degraded by the Unruh
effect when observers are in uniform acceleration. We also learned that the
spatial degrees of freedom of global fields are entangled, including the
vacuum state [7, 8, 9, 10]. This entanglement can be extracted by point-like
systems and in principal be used for quantum information processing (see for
example [11, 12, 13, 14]). Most of the early studies on relativistic
entanglement in non-inertial frames involved global modes. However, more
recently, researchers in the field have focused their attention on
understanding entanglement between fields or systems which are localized in
space and time. The motivation for this is that entangled localized systems
can be in principle measured, transformed and exploited for quantum
information tasks. Among the most popular systems considered for this purpose
are moving cavities [15, 16, 17, 18, 19, 20, 21], point-like detectors [11,
22, 23, 24, 25] and localized wave-packets [26, 27, 28]. In this paper we will
review global mode entanglement in flat and curved spacetime which constitutes
the first step in the study of entanglement in quantum field theory. We will
then discuss more recent ideas on entanglement which show that motion and
gravity have observable effects on quantum correlations between localized
systems [16, 17, 18, 19, 20, 21]. Interestingly, in these settings it is
possible to generate quantum gates through motion in spacetime [19, 21, 29].

The observer-dependent nature of entanglement is a consequence of the particle
content being different for different observers in quantum field theory [30,
31]. In flat spacetime, all inertial observers agree on particle number and
therefore, on entanglement. Entanglement is well defined in that case since
inertial observers play a special role. However, in the case of curved
spacetime, the entanglement in a given state varies even for inertial
observers (see discussion in [31]).

In special relativity one also finds that quantum correlations are observer-
dependent. The entanglement between two spin particles is invariant only when
the spin and momentum of the particles are considered to be a single
subsystem. If only spin degrees of freedom are considered, different inertial
observers would disagree on the amount of the entanglement between the
particles. Some works show that spin entanglement in transformed into momentum
entanglement under Lorentz transformations while some recent papers argue that
considering spin degrees of freedom alone (by tracing over momentum) lead to
inconsistencies (this will be discussed further in section 6).

The paper is organized as follows: in the section (2) we will introduce
technical tools in quantum field theory and quantum information. We will
review the basics of field quantization focusing on the free bosonic massless
case. We will describe the interaction of the field with point-like systems
better known as Unruh-DeWitt detectors. By imposing boundary conditions we
will describe fields contained within moving mirrors (cavities) and show how
to construct wave-packets that are localized in space and time. A brief
discussion on fields in curved spacetimes will be presented. We will end the
section by reviewing measures of entanglement in the pure and mixed case as
well as introduce the covariant matrix formalisms which allows for relatively
simple entanglement computations in quantum field theory. In section (3) of
this paper we will review the results on free mode entanglement in non-
inertial frames, in an expanding universe and in a black hole spacetime. We
will present ideas on how to extract field entanglement using Unruh-DeWitt
detectors in section (4). In section (5) we will present a more modern view on
the study of entanglement in quantum field theory where the entanglement
between the modes of moving cavities is analyzed and review recent work on how
localized wave-packets can be used to implement quantum information protocols.
For completeness, in section (6) we review the concept of observer dependent
entanglement for the case of zero acceleration. Here we discuss the Wigner
rotation, the change in state under Lorentz transformations and their effect
on entanglement for spin $\frac{1}{2}$ particles and photons. Finally, in
section (7) we will point out open questions, discuss work in progress and
future directions in the understanding of entanglement in quantum field
theory.

##  2 Technical tools

###  2.1 Quantum field theory

The theoretical framework in which questions of relativistic entanglement are
analyzed is quantum field theory in flat and curved spacetime. In the absence
of a consistent quantum theory of gravity, quantum field theory allows the
exploration of some aspects of the overlap of relativity and quantum theory by
considering quantum fields on a classical spacetime. The most important lesson
we have learned from quantum field theory is that fields are fundamental,
while particles are derived notions (if at all possible) [30]. Field
quantization is inequivalent for different observers and therefore, the
particle content of the field may vary for different observers. For example,
the Minkowski vacuum seen by inertial observers in flat spacetime corresponds
to a state populated with a thermal distribution of particles for observers in
uniform acceleration [32]. The temperature, known as the Unruh temperature, is
a function of the observer’s acceleration. As we will see, a consequence of
this is that the entanglement of free field modes in flat spacetime is
observer-dependent [3, 4], and effects quantum information processing tasks
such a teleporation [5, 33, 11]. Another interesting example is that of an
expanding universe [30]. The vacuum state for observers in the asymptotic past
is populated by particles as seen by observers in the future infinity [30].
The expansion of the universe creates particles and these particles are
entangled [31, 34]. It might at first sight seam surprising that the dynamical
Casimir effect is closely related to the Unruh effect [35, 36]. Both effects
are predictions of quantum field theory. The vacuum state of an inertial
cavity defined by inertial observers is inequivalent to the vacuum state of
the cavity undergoing uniform acceleration as seen by observers moving along
with the cavity (Rindler observers)[15, 16, 17, 18, 19, 20, 21]. Therefore, if
a cavity is at rest and the field is in the vacuum state, entangled particles
will be created when the cavity subsequently undergoes non-uniform accelerated
motion [16, 18]. Related to this effect is the dynamical Casimir effect where
the mirrors of the cavity oscillate [36, 37]. Before we discuss in more detail
the entanglement between the modes of a quantum field in these and other
scenarios we will revisit basic concepts in quantum field theory, considering
the simplest case: the massless uncharged bosonic field (which we denote
$\phi$ ) in a flat (1+1)-dimensional spacetime. Throughout our paper we will
work in natural units $c=\hbar=1$ and the signature of the metric $(+,-)$.

####  2.1.1 Global fields

The massless real bosonic quantum field obeys the Klein-Gordon equation
$\square\phi=0$, where the d’Alambertian operator $\square$ is defined as

$\square\phi:=\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi),$
(1)

where $g=det(g_{ab})$ and $\partial_{\mu}=\frac{\partial}{\partial x^{\mu}}$.
In flat $(1+1)$-dimensional spacetime the metric is
$g_{\mu\nu}=\eta_{\mu\nu}=\\{+-\\}$ and thus,
$\square\phi=\partial^{2}_{t}-\partial^{2}_{x}$. Minkowski coordinates $(t,x)$
are a convenient choice for inertial observers. The solutions to the equation
are plane waves

$u_{\omega,M}(t,x)=\frac{1}{\sqrt{4\pi\omega}}e^{-i\omega(t-\epsilon x)},$ (2)

where the label $M$ stands for Minkowski and $\epsilon$ takes the value $+1$
for modes with positive momentum (right movers) and $-1$ for modes with
negative momentum (left movers). The modes of frequency $\omega>0$ are
orthonormal with respect to the Lorentz invariant inner product

$(\phi,\psi)=-i\int_{\Sigma}(\psi^{*}\partial_{\mu}\phi-(\partial_{\mu}\psi^{*})\phi)d\Sigma^{\mu},$
(3)

where $\Sigma$ is a spacelike hypersurface. These solutions are known as
global field modes.

To quantize the field the notion of a time-like Killing vector field is
required. A Killing vector field $K^{\mu}$ is the tangent field to a flow
induced by a transformation which leaves the metric invariant. This means that
the Lie derivative of the metric tensor defined by

$\mathcal{L}_{K}g_{\mu\nu}=K^{\lambda}\partial_{\lambda}g_{\mu\nu}+g_{\mu\lambda}\partial_{\nu}K^{\lambda}+g_{\nu\lambda}\partial_{\mu}K^{\lambda},$

must vanish. When a spacetime admits such a structure it is possible to find a
special basis for the solutions of $\square\phi=0$ such that

$\displaystyle\mathcal{L}_{K}u_{k,M}$ $\displaystyle=$ $\displaystyle
K^{\mu}\partial_{\mu}u_{k,M}=-i\omega u_{k,M},$

where we have considered the action of a Lie derivative on a function. Vectors
lying within the light cone at each point are called time-like. Therefore, if
$K^{\mu}$ is a timelike Minkowski vector field, the Lie derivative corresponds
to $\partial_{t}$. By the action of the Lie derivative on the solutions of the
Klein-Gordon equation we can identify the parameter $\omega>0$ with a
frequency, and classify the plane waves such that $u_{k,M}$ are positive
frequency solutions and $u_{k,M}^{*}$ are negative frequency solutions. A few
words about the physical significance of the existence of a Killing vector
field are in order. If a spacetime has as Killing vector $K^{\mu}$, one can
always find a coordinate system in which the metric is independent of one of
the coordinates and the quantity $E=p_{\mu}K^{\mu}$ is constant along a
geodesic with tangent vector $p^{\mu}$ [38]. The quantity $E$ can be
considered as the conserved energy of a photon with 4-momentum $p^{\mu}$. For
static observers, i.e. those whose 4-velocity $U^{\mu}=dx^{\mu}/d\tau$ is
proportional to the timelike Killing vector $K^{\mu}$ as
$K^{\mu}=V(x)U^{\mu}$, one defines the “redshift” factor
$V=(K^{\mu}K_{\mu})^{1/2}$ as the norm of the Killing vector (since
$U^{\mu}U_{\mu}=1$). The frequency $\omega$ of the photon measured by a static
observer with 4-velocity $U^{\mu}$ is given by $\omega=p_{\mu}U^{\mu}$, and
hence $\omega=E/V$. A photon emitted by a static observer $1$ will be observed
by a static observer $2$ to have frequency $\omega_{2}=\omega_{1}V_{1}/V_{2}$.
Note that along the orbit of the Killing vector $K^{\mu}$ (not necessarily a
geodesic), $V$ is constant. For a general $1+1$ spacetime with coordinates
$x=(x^{0},x^{1})$, a photon $p^{\mu}=(\omega_{0},\pm k(x))$ of frequency
$\omega_{0}>0$ and wavevector of magnitude
$k(x)=\omega_{0}\,\sqrt{-g_{00}(x)/g_{1}(x)}$ (such that
$g_{\mu\nu}p^{\mu}p^{\nu}=0$) will be measured to have frequency
$\omega_{K}(x)=\omega_{0}\,\sqrt{g_{00}(x)}\,\sqrt{(1\pm\alpha)/(1\mp\alpha)}$
with $x$-dependent Doppler factor
$\alpha=\sqrt{-g_{00}(x)/g_{1}(x)}\,\big{(}K^{1}(x)/K^{0}(x)\big{)}$ by a
static observer along the orbit of the Killing vector
$K=K^{0}(x)\partial_{x^{0}}+K^{1}(x)\partial_{x^{1}}$. In particular, in flat
Minkowski spacetime with metric $g_{\mu\nu}=(+,-)$ in $(t,x)$ coordinates a
photon of frequency $\omega_{0}$ emitted by an inertial Minkowski observer
will be measured to have the frequency
$\omega_{K}(x)=\omega\,\,\sqrt{(1\pm\alpha)/(1\mp\alpha)}$ with Doppler factor
$\alpha=K^{1}(x)/K^{0}(x)$.

If the metric is static ($\partial_{0}g_{\mu\nu}=0$ and $g_{0\nu}=0$) then the
metric components are independent of the time coordinates $t$ and the Klein-
Gordon equation can be separated into space and time components as
$f_{\omega}(t,\vec{x})=e^{-i\omega t}\bar{f}_{\omega}(\vec{x})$ (here $(t,x)$
are general $1+1$ coordinates). The modes $(f_{\omega},f^{*}_{\omega})$ form a
basis of the wave equation from which to define the notion of particles. By
definition, a detector measures the proper time $\tau$ along its trajectory.
If the detectors’s trajectory follows the orbit of the Killing field (i.e. the
static observers defined above) the proper time will be proportional to the
Killing time $t$. Modes that are positive frequency with respect to this
Killing vector serve as a natural basis for describing the Fock space of
particles [38]. Most importantly, under Lorentz transformations, timelike
vectors are transformed into timelike vectors, so that the separation of modes
into positive and negative frequencies remains invariant under boosts. In a
general curved spacetime, the non-existence of a Killing field implies that
the separation of modes into positive and negative frequencies is different
along each point of the detectors’s trajectory, and hence the concept of
“particle” is lost (for further details, see [30] and Chap. 9 of [38]). Note
that the photons of measured frequency $\omega_{K}(x)$ in the previous
paragraph are not pure plane waves along the Killing orbit, and therefore must
be decomposed into the natural positive and negative frequency modes
$(f_{\omega},f^{*}_{\omega})$.

Having identified positive and negative modes, the quantized field satisfying
$\square\hat{\phi}=0$ is then given by the following operator value function

$\hat{\phi}=\int(u_{k,M}a_{k,M}+u_{k,M}^{*}a_{k,M}^{\dagger})dk,$

where the creation and annihilation Minkowski operators $a_{k,M}^{\dagger}$
and $a_{k,M}$ satisfy the commutation relations
$[a^{\dagger}_{k,M},a_{k^{\prime},M}]=\delta_{k,k^{\prime}}$. Note that the
solutions have been treated differently by associating creation and
annihilation operators with negative and positive frequency modes,
respectively. The vacuum state is defined by the equation
$a_{k,M}{|0\rangle}^{\mathcal{M}}=0$ and can be written as
${|0\rangle}^{\mathcal{M}}=\prod_{k}{|0_{k}\rangle}^{\mathcal{M}}$ where
${|0_{k}\rangle}^{\mathcal{M}}$ is the ground state of mode $k$. Particle
states are constructed by the action of creation operators on the vacuum state

$|n_{1},...,n_{k}\rangle^{\mathcal{M}}=(n_{1}!,...,n_{k}!)^{-1/2}\,a_{1,M}^{\dagger
n_{1}}...a_{k,M}^{\dagger n_{k}}|0\rangle^{\mathcal{M}}.$

Only when there exists a time-like Killing vector field it is meaningful to
define particles. Observers flowing along timelike Killing vector fields are
those who can properly describe particle states. This has important
consequences to relativistic quantum information since the notion of particles
(and therefore, subsystems) are indispensable to store information and thus,
to define entanglement. However, in the most general case, curved spacetimes
do not admit time-like Killing vector fields.

Interestingly, in the case where the spacetime admits a global timelike
Killing vector field, the vector field is not necessarily unique. Consider two
time-like Killing vector fields $\partial_{T}$ and $\partial_{\hat{T}}$. It is
then possible to find in each case a basis for the solutions to the Klein-
Gordon equation $\\{{u}_{k},{u}_{k}^{*}\\}$ and
$\\{\bar{u}_{k},\bar{u}_{k}^{*}\\}$ such that classification into positive and
frequency solutions is possible with respect to $\partial_{T}$ and
$\partial_{\hat{T}}$ respectively. The field is equivalently quantized in both
bases, therefore

$\hat{\phi}=\int(u_{k}a_{k}+u_{k}^{*}a_{k}^{\dagger})dk=\int(\bar{u}_{k^{\prime}}\bar{a}_{k^{\prime}}+\bar{u}_{k^{\prime}}^{*}\bar{a}_{k^{\prime}}^{\dagger})dk^{\prime}.$

Using the inner product, one obtains a transformation between the mode
solutions and correspondingly, between the creation and annihilation
operators,

$a_{k}=\sum_{k^{\prime}}(\alpha^{\ast}_{kk^{\prime}}\bar{a}_{k^{\prime}}-\beta^{\ast}_{kk^{\prime}}\bar{a}_{k^{\prime}}^{\dagger}),$

where $\alpha_{kk^{\prime}}=(u_{k},\bar{u}_{k^{\prime}})$ and
$\beta_{kk^{\prime}}=-(u_{k},\bar{u}^{\ast}_{k^{\prime}})$ are called
Bogoliubov coefficients. Since the vacua are given by
${a}_{k}{|0\rangle}={\bar{a}}_{k}{\bar{|0\rangle}}=0$ it is possible to find a
transformation between the states in the two bases. We note that as long as
one of the Bogoliubov coefficients $\beta_{kk^{\prime}}$ is non-zero, and the
un-barred state is the vacuum state, the state in the bared basis is populated
with particles. Therefore, different Killing observers observe a different
particle content in the field, i.e. particles are observer-dependent notions.

In flat spacetime there are two kinds of observers who can meaningfully
describe particles for all times: inertial observers who follow straight lines
and observers in uniform acceleration who’s trajectories are given by
hyperbolas parameterized for example by

$x=\chi\cosh\left(a\eta\right),\qquad t=\chi\sinh\left(a\eta\right),$ (4)

where $a$ is the proper acceleration at the reference worldline $\chi=1/a$
with proper time $\eta$. (The notion of defining particles in a general curved
spacetime is addressed in e.g. [30, 38]. For the other special cases when the
acceleration (i) is asymptotically uniform in the past/future see e.g. [30,
38, 39], or (ii) asymptotically zero in the past but asymptotically uniform in
the future and see e.g. [40]). The transformation suggests that a suitable
choice of coordinates for uniformly accelerated observers are $(\eta,\chi)$
which are known as Rindler coordinates.

Figure 1:  Rindler space-time diagram: lines of constant position $\chi$ are
hyperbolae and all curves of constant $\eta$ are straight lines that come from
the origin. An uniformly accelerated observer Rob travels along a hyperbola
constrained to either region $I$ or region $II$.

The transformation in Eq. (4) is defined in the region $|x|\geq t$ known as
the (right) Rindler wedge I. When $\eta\rightarrow\infty$ then
$t/x=\tanh(a\eta)\rightarrow 1\Rightarrow x=t$. Uniformly accelerated
observers asymptotically approach the speed of light and are constrained to
move in wedge I. Since the transformation does not cover all of Minkowski
spacetime, one must define a second region called (left) Rindler wedge II by
considering a coordinate transformation which differs from Eq. (4) by an
overall sign in both coordinates. Rindler regions I and II are causally
disconnected, and the lines $x\pm t=0$ at 45 degrees define the Rindler
horizon, Fig.(1).

The metric in Rindler coordinates takes the form
$ds^{2}=(a^{2}\chi^{2}\,d\eta^{2}-d\chi^{2})$ where $a^{2}\chi^{2}$ acts as an
effective gravitational potential $g_{\eta\eta}(\chi)$ for the Rindler
observer’s local redshift factor. The Klein-Gordon equation in Rindler
coordinates is $(a^{-2}\partial_{\eta}^{2}-\partial_{ln(a\chi)}^{2})\phi=0$
and the solutions [41, 42, 43] are again plane waves, though now with
logarithmic spatial dependence $\ln\chi$, (compare with (2))

$\displaystyle u_{\omega,I}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega}}\,e^{i(\epsilon(\omega/a)\ln\chi-\omega\eta)}=\frac{1}{\sqrt{4\pi\Omega}}\,\left(\frac{x-\epsilon
t}{l_{\Omega}}\right)^{i\epsilon\Omega}\equiv u_{\Omega,I},$ (5a)
$\displaystyle u^{*}_{\omega,I}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega}}\,e^{-i(\epsilon(\omega/a)\ln\chi-\omega\eta)}=\frac{1}{\sqrt{4\pi\Omega}}\,\left(\frac{x-\epsilon
t}{l_{\Omega}}\right)^{-i\epsilon\Omega}\equiv u^{*}_{\Omega,I}.$ (5b)

In the above $\omega>0$, $\epsilon=1$ corresponds to modes propagating to the
right along lines of constant $x-t$, and $\epsilon=-1$ to modes propagating to
the left along lines of constant $x+t$. In the second equality we have
introduced a positive constant $l_{\Omega}$ of dimension length, and defined
the dimensionless positive constant $\Omega=\omega/a$. Some authors [42]
choose to label the Rindler mode by the (positive) frequency $\omega$, while
other authors [41, 43] label the modes by the (positive) dimensionless
quantity $\Omega$. (Note that $-\infty<\epsilon\omega/a=\epsilon\Omega<\infty$
acts as the effective wavevector for the Unruh modes, if one where to push the
analogy with the inertial Minkowski modes (2)). Here we follow derivations
from [43] and throughout this work, it will be understood that a wavevector
subscript $k$ on Minkowski modes ($u_{k,M}$, etc…) takes values in the range
$-\infty$ to $\infty$, while for Unruh modes ($u_{k,I}$, $u_{k,II}$ etc…) it
takes values from $0$ to $\infty$.

The solutions $u_{k,I}$ and $u^{*}_{k,I}$ are identified as positive and
negative frequency solutions, respectively, with respect to the timelike
Killing vector field $\partial_{\eta}$. These solutions have support only in
the right Rindler wedge and therefore are labeled by the subscript $I$. Note
that they do not constitute a complete set of solutions. The transformation
which defines Rindler region $II$ also gives rise to the same spacetime.
However, the future-directed timelike Killing vector field which in this case
is given by $\partial_{(-\eta)}=-\partial_{\eta}$, and the solutions are

$\displaystyle u_{\omega,II}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega}}\,e^{i(-\epsilon(\omega/a)\ln(-\chi)+\omega\eta)}=\frac{1}{\sqrt{4\pi\Omega}}\,\left(\frac{\epsilon
t-x}{l_{\Omega}}\right)^{-i\epsilon\Omega}\equiv u_{\Omega,II},$ (5fa)
$\displaystyle u^{*}_{\omega,II}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega}}\,e^{-i(-\epsilon(\omega/a)\ln(-\chi)+\omega\eta)}=\frac{1}{\sqrt{4\pi\Omega}}\,\left(\frac{\epsilon
t-x}{l_{\Omega}}\right)^{i\epsilon\Omega}\equiv u^{*}_{\Omega,II},$ (5fb)

with support in region $II$. The solutions of region $I$ together with the
solutions in region II form a complete set of orthonormal solutions.
Therefore, we can quantize the field in this basis as well,

$\hat{\phi}=\int(u_{\Omega,{I}}a_{\Omega,I}+u_{\Omega,{II}}a_{\Omega,{II}}+h.c.)\,d\Omega.$

Since region $I$ is causally disconnected from region $II$, the mode operators
in the separated wedges commute
$[a_{\Omega,I},a^{\dagger}_{\Omega^{\prime},II}]=0$, etc. The vacuum state in
the Rindler basis is
$\left|0\right>_{R}=\left|0\right>^{I}\otimes\left|0\right>^{II}$ where
$a_{k,{I}}\left|0\right>^{I}=0$ and $a_{k,{II}}\left|0\right>^{II}=0$. Making
use of the inner product we find the Bogoliubov transformations,

$\displaystyle a_{k,M}$ $\displaystyle=$
$\displaystyle\int(u_{k,M},u_{\Omega,{I}})a_{\Omega,{I}}+(u_{k,M},u_{\Omega,I}^{\ast})a_{\Omega,I}^{\dagger}$
$\displaystyle+$
$\displaystyle(u_{k,M},u_{\Omega,II})a_{\Omega,II}+(u_{\Omega,M},u_{\Omega,II}^{*})a_{\Omega,II}^{\dagger}d\Omega,$

where, for example,
$(u_{k,M},u_{\Omega,I})=-i\,\int(u^{*}_{k,M}\,\partial_{t}u_{\Omega,I}-(\partial_{t}u_{\Omega,I})\,u_{k,M}^{*})dx$.
Upon computing the inner products [41, 43] the above formula can be written as

$\displaystyle a_{\omega,M}$ $\displaystyle=$
$\displaystyle\int^{\infty}_{0}d\Omega\,[(\alpha_{\omega\Omega}^{R})^{*}(\cosh(r_{\Omega})a_{\Omega,I}-\sinh(r_{\Omega})a^{\dagger}_{\Omega,II})$
(5fg) $\displaystyle+$
$\displaystyle(\alpha_{\omega\Omega}^{L})^{*}(-\sinh(r_{\Omega})a^{\dagger}_{\Omega,I}+\cosh(r_{\Omega})a_{\Omega,II})],$

where

$\displaystyle\alpha_{\omega\Omega}^{R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi\omega}}\,(\omega
l)^{i\epsilon\Omega},\qquad\alpha_{\omega\Omega}^{L}=\frac{1}{\sqrt{2\pi\omega}}\,(\omega
l)^{-i\epsilon\Omega},$

are the Bogoliubov coefficients for the massless case, and $l$ is an overall
constant of dimension length, independent of $\Omega$ and $\epsilon$.

The Minkowski creation and annihilation operators result in an infinite sum of
Rindler operators. An alternative basis for the inertial observers known as
the Unruh basis can significantly simplify the transformations between
inertial and uniformly accelerated observers. The Unruh modes $a_{\Omega,R}$,
$a_{\Omega,L}$ are appropriately chosen linear combinations of right-moving
and left-moving Rindler modes respectively such that they are analytic across
both regions $I$ and $II$. That is, $u_{\Omega,I}$ and $u^{*}_{\Omega,II}$ are
both proportional to $(x-\epsilon t)^{i\epsilon\Omega}$ when
$(-1)^{i\epsilon\Omega}=(e^{i\pi})^{i\epsilon\Omega}=e^{-\epsilon\pi\Omega}$
is factored out of the latter region $II$ mode. The Unruh modes are given by
the direct Bogoliubov transformation with region $I$ and $II$ Rindler modes
for each value of $\Omega$ as

$\displaystyle a_{\Omega,R}$ $\displaystyle=$
$\displaystyle\cosh(r_{\Omega})a_{\Omega,I}-\sinh(r_{\Omega})a^{\dagger}_{\Omega,II},$
$\displaystyle a^{\dagger}_{\Omega,L}$ $\displaystyle=$
$\displaystyle-\sinh(r_{\Omega})a_{\Omega,I}+\cosh(r_{\Omega})a^{\dagger}_{\Omega,II},$
(5fh)

where $\tanh(r_{\Omega})=e^{-\pi\Omega}$. Here $a_{\Omega,R}$ annihilates a
right ($R$) moving Unruh mode traveling along lines of constant $x-t$ in both
wedges $I$ and $II$, while $a_{\Omega,L}$ annihilates a left ($L$) moving
Unruh mode traveling along lines of constant $x+t$, with
$[a_{\Omega,R},a^{\dagger}_{\Omega,R}]=1$,
$[a_{\Omega,L},a^{\dagger}_{\Omega,L}]=1$ and all cross commutators vanishing.
In terms of mode functions, the Bogoliubov transformation from the Rindler to
the Unruh modes is given by

$\displaystyle u_{\Omega,R}$ $\displaystyle=$
$\displaystyle\cosh({r})u_{\Omega,I}+\sinh({r})u^{\ast}_{\Omega,II},$
$\displaystyle u^{*}_{\Omega,L}$ $\displaystyle=$
$\displaystyle\sinh({r})u_{\Omega,I}+\cosh({r})u^{\ast}_{\Omega,II},$ (5fi)

which are analytic in $x-t$ across both Rindler wedges $I$ and $II$. Note that
the sign of the momentum $k$ in region $II$ is opposite of that in region $I$,
but coupled with utilizing the complex conjugate of the region $II$ Rindler
mode, renders the resulting Unruh modes $u_{\Omega,R}$ and $u^{*}_{\Omega,L}$
right-movers (see [38] Chap. 9.5 for further details).

The most general Unruh annihilation operator of purely positive Minkowski
frequency is a linear combination of the two $R,L$ Unruh creation operators,

$a_{\Omega,U}=q_{L}a_{\Omega,L}+q_{R}a_{\Omega,R},$ (5fj)

where $q_{L}$ and $q_{R}$ are complex numbers with
$|q_{L}|^{2}+|q_{R}|^{2}=1$. The introduction of the Unruh modes allows us to
write the Minkowski annihilation operator (5fg) as a linear combination of
only Unruh annihilation operators

$a_{\omega,M}=\int^{\infty}_{0}d\Omega\,[(\alpha_{\omega\Omega}^{R})^{*}a_{\Omega,R}+(\alpha_{\omega\Omega}^{L})^{*}a_{\Omega,L}].$
(5fk)

Hence, both Minkowski and Unruh annihilation operators annihilate the
Minkowski vacuum, i.e. $a_{\omega,M}|0\rangle_{\mathcal{M}}=0$,
$a_{\Omega,R}|0\rangle_{\mathcal{M}}=0$, and
$a_{\Omega,L}|0\rangle_{\mathcal{M}}=0$, and therefore, the Unruh vacuum and
the Minkowski vacuum coincide.

Using the direct Bogoliubov transformation between the Unruh and Rindler
annihilation operators it is straight forward to show that (see e.g. [32, 38,
41])

$\left|0_{k}\right>^{\mathcal{M}}=\frac{1}{\cosh(r)}\sum_{n}\tanh^{n}(r)\left|n_{k}\right>^{I}\left|n_{k}\right>^{II},$
(5fl)

where $\tanh r\equiv e^{-\pi\omega/a}$. The vacuum state in the Rindler basis
corresponds to a two mode squeezed state. Since the accelerated observer is
constrained to move in region $I$ one must trace over the states in (the
causally disconnected) region $II$. The density matrix of the Minkowski vacuum
is given by $\rho_{0}=\left|0_{k}\right>\left<0_{k}\right|^{\mathcal{M}}$ and
therefore, the state in region $I$ corresponds to the following reduced
density matrix

$\displaystyle\rho_{I}$ $\displaystyle=$
$\displaystyle\frac{1}{\cosh^{2}(r)}\sum_{n}\tanh^{2n}(r)\left|n\right>^{I}\left<n\right|^{I},$
$\displaystyle=$
$\displaystyle(1-e^{-2\pi\omega/a})\sum_{n}(e^{-2\pi\omega/a})^{n}\left|n\right>^{I}\left<n\right|^{I},$

which corresponds to a thermal state with temperature $T_{U}=\frac{a}{2\pi
k_{B}}$ (where $k_{B}$ is the Boltzmann constant) proportional to the
observer’s acceleration. The temperature is known as the Unruh temperature.
This is the well known Unruh effect [32]: the vacuum state as seen by inertial
observers is a thermal state for observers in uniform acceleration.

####  2.1.2 Unruh-DeWitt detectors

In order to give a more physical interpretation to the Unruh effect, Unruh-
DeWitt detectors were introduced [32, 41, 42, 44]. The detectors consist of a
point-like system endowed with an internal structure which can be either a two
level system or a harmonic oscillator [45, 46]. The detector moves in
spacetime following a classical trajectory given by $x(\tau)$ where $\tau$ is
the detector’s proper time. The detector couples to the field locally via it’s
monopole moment. Therefore, the interaction Hamiltonian for a harmonic
oscillator detector is given by

$\hat{H}_{I}(\tau)=\int
dk\lambda(\tau)(d\,e^{i\Omega\tau}+d^{\dagger}\,e^{-i\Omega\tau})\left(\hat{a}_{k}e^{i(kx(\tau)-\omega
t(\tau))}+\hat{a}_{k}^{\dagger}e^{-i(kx(\tau)-t(\tau))}\right),$ (5fm)

where $d$ and $d^{\dagger}$ are annihilation and creation operators for the
internal degrees of freedom of the detector, $\Omega$ the frequency of the
detector and $\omega=|k|$ the frequency of the field modes. The coupling
function $\lambda(\tau)$ between the field and the detector can be chosen such
that the interaction is switched on and off adiabatically. This removes
transient effects.

Standard calculations [30] consider a detector that follows either an inertial
trajectory (say $x(t)=vt$ for a detector with zero acceleration) or a
uniformly accelerated one ($x=(t^{2}+a^{-2})^{1/2}$). At time $t=-\tau_{0}$
the field is in the vacuum state $|0\rangle_{f}$ and the detector in it’s
ground state $|0\rangle_{d}$. The detector is then turned on and the
transition probability of the detector to an exited state at time $\tau_{0}$
is calculated using perturbation theory. The transition rate per unit of
proper time of the detector to a state with $n=1$ excitations at first order
is given by

$\displaystyle\mathcal{P}=\sum_{n,\,\psi}\frac{1}{2\tau_{0}}\int_{-\infty}^{\infty}d\tau|A_{n}|^{2},\quad
A_{n}={}_{d}\\!\langle
1_{n}|{}_{f}\\!\langle\psi|H_{I}(\tau)|0\rangle_{d}|0\rangle_{f},$ (5fn)

where $|\psi\rangle_{f}$ is the final state of the field which is not
observed, and hence, averaged over all possible outcomes. This leads to the
detector response function [30, 41]

$F(\omega)=\lim_{s\downarrow
0}\lim_{\tau_{0}\uparrow\infty}\frac{1}{2\tau_{0}}\int_{-\tau_{0}}^{\tau_{0}}\int_{-\tau^{\prime}_{0}}^{\tau^{\prime}_{0}}\,e^{-i\omega(\tau-\tau^{\prime})-s|\tau|-s|\tau^{\prime}}\,g(\tau,\tau^{\prime}),$

taking $\lambda(\tau)=\exp(-s|\tau|)$, and defining
$g(\tau,\tau^{\prime})=G(x(\tau),x(\tau^{\prime}))$ where
$G(x,x^{\prime})={}_{f}\\!\langle
0|\,\hat{\phi}(x)\,\hat{\phi}(x^{\prime})\,|0\rangle_{f}$ is the positive
frequency Wightman function (WF), and the field $\hat{\phi}(x)$ is given by
the integral over all momentum of the rightmost term in large parentheses in
(5fm). In the above, the response function $F(\omega)$ is a consequence of
Fermi’s golden rule and depends only on the field and not on the structure of
the detector, since the latter has been completely factored out [30]. In
general the WF is inversely proportional to the squared geodesic distance
$\sigma(\Delta x)=g_{\mu\nu}(x-x^{\prime})^{\mu}(x-x^{\prime})^{\nu}$ and
hence is singular as $x\rightarrow x^{\prime}$. This singular behavior is
typically regularized by the “$i\epsilon$” prescription which treats
$F(\omega)$ as a contour integral in the complex
$\Delta\tau=\tau-\tau^{\prime}$ plane. For an inertial trajectory $x=vt$ one
has [30] $g(\Delta\tau)=-[2\pi(\Delta\tau-i\epsilon)]^{-2}$. For $\omega>0$
the contour is closed in the lower half-plane for the integral to be
convergent. However, since the pole in $g(\Delta\tau)$ is in the upper half-
plane, no contributions are obtained resulting in $F(\omega)=0$ as expected.
For the uniformly accelerated trajectory $x(\tau)=(t^{2}(\tau)+a^{-2})^{1/2}$
with $t(\tau)=a^{-1}\sinh(a\tau)$ one obtains
$g(\Delta\tau)=-[(4\pi/a)\sinh(a\Delta\tau/2-i\epsilon a)]^{-2}$. A contour
integration [30] now picks up contributions from the poles along the negative
imaginary axis, leading to a response function proportional to
$\omega/(\exp(2\pi\omega/a)-1)^{-1}$. The appearance of the Planck factor
$(\exp(2\pi\omega/a)-1)^{-1}$ indicates that the equilibrium reached between
the accelerated detector and the field $\hat{\phi}$ in the vacuum state
$|0\rangle_{f}$ is identical to the case when the detector remained
unaccelerated, but was immersed in a bath at (Unruh) temperature
$T_{U}=a/(2\pi k_{B})$. Transition rates for Unruh-DeWitt detectors in curved
spacetime have been considered in [47]. We will see in the next section that
such accelerated detectors can be used to extract entanglement from the field.

Note that since the detector is point-like it couples with the same strength
to every frequency field mode. However this is not a very realistic situation.
A more physical situation corresponds to a detector which has a spatial
profile [41, 48, 49]. Such detectors couple to a distribution of field modes
which depends on the specifics of the profile. The use of detectors with
infinitesimal, though non-zero, spatial extent can be utilized to give a more
physical interpretation of the mathematically formal “$i\epsilon$”
prescription used above in the computation of the detector response function.
In brief, the use of infinitesimal sized detectors smears out the field
$\hat{\phi}(x)$ along the detector trajectory, thus regularizing the singular
behavior of the WF as $x\rightarrow x^{\prime}$. Following Takagi (see §3.2 of
[41]), one can consider the instantaneous non-rotating rest frame of the
detector (the Fermi-Walker (FW) frame, see §13.6 of [50]). The coordinates
transformation $t(\tau,\zeta)=(a^{-1}+\zeta)\sinh(a\tau)$ and
$x(\tau,\zeta)=(a^{-1}+\zeta)\cosh(a\tau)$ with metric
$ds^{2}=(1+a\zeta)^{2}d\tau^{2}-d\zeta^{2}$, describes the observer’s “local
laboratory” where the detector sits at the origin $\zeta=0$ of the local
spatial coordinates. For a “rigid” detector, its monopole field
${\cal{M}}(x(\tau,\zeta))$ (the term in leftmost parentheses in (5fm) can be
written in a factorized form
${\cal{M}}(x(\tau,\zeta))=\lambda(\tau)\,M(\tau)\,f(\zeta)$. The detector-
field interaction is now given by $H_{int}=\int d\tau\int
d\zeta{\cal{M}}(x(\tau,\zeta))\,\hat{\phi}(x(\tau,\zeta))$ which is now of the
form $\lambda(\tau)\,M(\tau)\,\hat{\phi}^{\prime}(\tau)$, where
$\hat{\phi}^{\prime}(\tau)=\int d\zeta f(\zeta)\,\hat{\phi}(x(\tau,\zeta)$ is
the smeared out quantum field. Using the mode expansion
$\hat{\phi}(x(\tau))=\int dk\,a_{k}U_{k}(x(\tau))$ with plane waves
$U_{k}(x(\tau))=e^{ik_{\mu}x^{\mu}(\tau)}$ allows the smeared field to be
written as $\hat{\phi}^{\prime}(\tau)=\int
dk\,a_{k}U_{k}(x(\tau))\tilde{f}_{k}(\tau)+h.c.$ Here
$\tilde{f}_{k}(\tau)=\int d\zeta
e^{ik_{\mu}[x^{\mu}(\tau,\zeta)-x^{\mu}(\tau)]}$ $=\int d\zeta
e^{ik_{\mu}[\partial x^{\mu}(\tau,\zeta)/\partial\zeta^{j}]\,\zeta^{j}}$ are
wavepackets that can be effectively replaced with $\tilde{f}_{k}\approx
e^{-\epsilon\omega_{k}/2}$ (upon substitution of the coordinate
transformations to the FW frame) where $\epsilon$ is a small positive quantity
on the order $1/a$. This latter form of $\tilde{f}_{k}$ is just the
“$i\epsilon$” prescription, and shows the advantage of utilizing smeared
fields/wavepackets to regularize divergent quantities, as well as lending
physical interpretation to mathematical procedures. Work in progress shows
that a uniformly accelerated detector with a Gaussian spatial profile
naturally couples to a peak distribution of Rindler modes [51]. Such a
detector also naturally couples to distributions of Unruh modes which in this
case yields a frequency distribution with two peaks corresponding to left and
right moving Unruh modes. This detector model will help in the understanding,
from physical perspective, the nature of Unruh modes which have been used to
analyze the degradation of entanglement in non-inertial frames.

####  2.1.3 Moving cavities

As mentioned previously, in order to preform quantum information tasks
observers need to have access to the state and therefore, localizing field
modes in space and time becomes relevant. This can be achieved by confining
quantum fields in cavities which can move in spacetime [15, 16, 17, 18, 19,
20, 21]. To confine a massless field within a cavity of length $L$ appropriate
Dirichlet boundary conditions must be imposed on the solutions of the Klein-
Gordon equation given by (2) at the cavity mirrors placed at $x_{l}$ and
$x_{r}=x_{l}+L$. The normalized field solutions in this case correspond to
sine functions

$\displaystyle U_{n,M}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n\pi}}\sin\left(\omega_{n}[x-x_{l}]\right)e^{-i\omega_{n}t},$
(5fo)

where $n\in N$ labels the energy states which now have a discrete spectrum
given by $\omega_{n}=n\pi/L$. The Killing vector field is $\partial_{t}$ which
classifies $U_{n,M}$ and $U^{\ast}_{n,M}$ as positive and negative frequency
solutions. The normalization of the solutions gives
$[U_{n,M},U_{m,M}]=[U^{\ast}_{n,M},U^{\ast}_{m,M}]=\delta_{nm}$ and the mix
products vanish. The field is therefore,

$\hat{\phi}=\sum_{n}(U_{n,M}a_{n,M}+U_{n,M}^{*}a_{n,M}^{\dagger}),$

where creation and annihilation operators satisfy standard bosonic commutation
relations. We have used capitalized letters for the cavity modes to
distinguish them from global modes.

We can also consider the field confined to a cavity in uniform acceleration by
imposing uniformly accelerating boundary conditions to the modes in Eq. (5a)
at $\chi=\chi_{r}$ and $\chi=\chi_{l}$. Here we considered Rindler coordinates
$(\eta,\chi)$ which are a suitable choice for this case once more. We consider
the cavity to move in region $I$ without loss of generality. The cavity is
constructed such that it has a constant proper length as measured by a co-
moving observer. Therefore, the mirrors move with different proper
acceleration $A_{r}=1/x_{r}$ and $A_{l}=1/x_{l}$. Taking advantage of the
invariance under the boost Killing vector $\partial_{\eta}$ during
acceleration we write

$\displaystyle U_{n,R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n\pi}}\sin\left(\Omega_{n}[\chi-\chi_{l}]\right)e^{-i\omega_{n}\eta},$

where $\Omega_{n}=n\pi/L^{\prime}$ with
$L^{\prime}=\chi_{r}-\chi_{l}=a^{-1}\ln(1+aL)$. For the case $aL/c^{2}\ll 1$
we have $L^{\prime}\approx L$, and the small difference of the acceleration at
the ends of the mirror can be ignored with respect to the acceleration at the
center of the cavity. Since we are only interested in the field within the
mirrors, we dropped the index $I$ and instead included $R$ to denote the modes
of the cavity, which is undergoing uniform acceleration. The quantum field
inside the cavity as seen by a co-moving observer is given by
$\hat{\phi}_{R}(\eta,\xi)=\sum_{n}(U_{n,R}(\eta,\xi)a_{n,R}+U_{n,R}^{*}(\eta,\xi)a_{n,R}^{\dagger})$
where $a_{n,R}^{\dagger}$ and $a_{n,R}$ are once more creation and
annihilation operators with
$[a_{n,R},a^{\dagger}_{n^{\prime},R}]=\delta_{n,n^{\prime}}$. The ground
state, in this case, is defined by $a_{n,R}|0\rangle^{R}=0$, $\forall n$. Note
that the Rindler coordinates completely cover the region inside the cavity and
that the horizon always lies outside the cavity for all values of $a$. We
assume the cavity’s mirrors to be perfectly reflecting therefore, the states
inside the cavity it will remain the same for all times [15]. This is in
agreement with Schutzhold and Unruh’s [52] comment concerning how the non-
transmissive cavity protects the state from Unruh radiation. If one prepares a
state, for example a pure state, as long as the cavity remains either inertial
or in uniform acceleration, the state remains unchanged, i.e. pure.

More interesting is to consider the case in which the cavity is placed in a
spaceship initially at rest which uniformly accelerates for a finite amount of
time and finally moves at constant velocity, as illustrated in figure 2.

Figure 2: Non-uniformly accelerated trajectory from initial constant velocity
to uniformly accelerated motion.

This trajectory is interesting since it is more realistic (the uniformly
accelerated trajectory assumes constant acceleration from past infinity to
future infinity). Further, general travel scenarios can be crafted by
considering sequences of it, and therefore it is called the basic building
block trajectory[16]. The modes inside an cavity initially at rest will be
affected by the non-uniform accelerated motion giving rise to particle
creation and entanglement [16, 17, 18, 19, 20, 21],. It is possible to find
Bogoliubov transformations between the modes of the inertial cavity and the
modes of the cavity after any given travel scenario. The new modes will be
given by

$a_{m,F}=\sum_{n}{\alpha^{\ast}_{mn}a_{n,M}-\beta^{\ast}_{mn}a^{\dagger}_{n,M}},$
(5fp)

where $F$ denotes the modes in the final region, and
$\alpha_{mn}=(U_{m,F},U_{n,M})$ and $\beta_{mn}=-(U_{m,F},U^{\ast}_{n,M})$ are
Bogoliubov coefficients. To study the above case in more detail it is
convenient to work in the covariant matrix formalism which will be introduced
in the section on quantum information. In the simple case of the building
block trajectory the coefficients are given by

$\displaystyle\alpha^{B}_{mn}=\frac{1}{L}\sqrt{\frac{n}{m}}F_{mn}+\frac{1}{\ln(\frac{x_{r}}{x_{l}})}\sqrt{\frac{m}{n}}G_{mn},$
(5fqa)
$\displaystyle\beta^{B}_{mn}=\frac{1}{L}\sqrt{\frac{n}{m}}F_{mn}-\frac{1}{\ln(\frac{x_{r}}{x_{l}})}\sqrt{\frac{m}{n}}G_{mn},$
(5fqb)

where

$\displaystyle F_{mn}:$ $\displaystyle=$
$\displaystyle\int_{L_{0}}^{R_{0}}dx\sin\left(\omega_{n}(x-L_{0})\right)\sin\left(\Omega_{m}\ln(\frac{x}{L_{0}})\right)$
$\displaystyle G_{mn}:$ $\displaystyle=$
$\displaystyle\int_{L_{0}}^{R_{0}}\frac{dx}{x}\sin\left(\omega_{n}(x-L_{0})\right)\sin\left(\Omega_{m}\ln(\frac{x}{L_{0}})\right).$

Here $B$ stands for basic building block. These coefficients are difficult to
handle both numerically and analytically. However, in the case where $h=aL$ is
small, the terms in the integrand can be expanded in a Maclaurin series such
that, to second order, the coefficients take a simple form [16, 17, 18].

####  2.1.4 Curved spacetime

For a general curved space metric $g_{\mu\nu}(x)$ the d’Alambertian (see
section 2.1.1) gives rises to a more complicated, and very likely non-
separable, Klein-Gordon equation. However, in some special cases solutions can
be found [30]. An example is a $(1+1)$-dim expanding Robertson-Walker
universe. This spacetime does not admit a global Killing vector field and
therefore, it is not possible to define particles globally. This is also true
for most spacetimes. However, the Robertson-Walker universe has a special
property in that it is asymptotically flat in the past and future infinity
regions where timelike Killing vector fields can be defined and employed to
distinguish positive and negative solutions to the Klein-Gordon equation. We
will consider that a scalar field living in this spacetime is in the vacuum
state as seen by observers in the past infinity region, and show that the
state will be populated with particles in the future infinity.

The spacetime of a Robertson-Walker universe in $(1+1)$-dim is given by
$ds^{2}=dt^{2}-a^{2}(t)d\chi^{2}$ where the spatial sections of the spacetime
are expanding (or contracting) uniformly according to the function $a^{2}(t)$.
Considering the infinitesimal coordinate transformation $d\eta=dt/a(t)$ and
defining $a^{2}(t)=c^{2}(\eta)$ we obtain the metric
$ds^{2}=c^{2}(\eta)(d\eta^{2}-d\chi^{2})$. Consider
$c(\eta)=1+\epsilon(1+\tanh(\sigma\eta))$ where $\epsilon$ and $\sigma$ are
constants. This describes a toy model of a universe undergoing a period of
smooth expansion. The parameter $\epsilon$ is known as the expansion volume
and $\sigma$ is the expansion rate. In the limit $\eta\rightarrow-\infty$ the
metric is $ds^{2}=(d\eta^{2}-d\chi^{2})$ and in the limit
$\eta\rightarrow\infty$ then $ds^{2}=(1+2\epsilon)(d\eta^{2}-d\chi^{2})$. The
metric is flat in these regions where the vector field $\partial_{\eta}$ has
Killing properties.

We now consider the massive Klein-Gordon equation $(\square+m^{2})\phi=0$ in
the Robertson-Walker spacetime described above where $m$ is the mass of the
field. The metric tensor $g_{\mu\nu}$ has components

$g_{\mu\nu}=c(\eta)\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),$

therefore, $g=det(g_{\mu\nu})=-c^{2}(\eta)$ and the Klein-Gordon equation
takes the form

$((\partial_{\eta}^{2}-\partial_{\chi}^{2})+c(\eta)m^{2})\phi=0.$ (5fqr)

Exploiting spatial translational invariance we separate the solutions into

$u_{k}=\frac{1}{\sqrt{2\pi\omega}}e^{ik\chi}\xi_{k}(\eta),$

so that the equation becomes

$\partial^{2}_{\eta}\xi_{k}(\eta)+(k^{2}+c(\eta)m^{2})\xi_{k}(\eta)=0.$ (5fqs)

This equation can be solved in terms of two hypergeometric functions. We find
two types of solutions [30]

$\displaystyle u^{(1)}_{k}(\eta,\chi)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega_{in}}}e^{ik\chi-i\omega_{+}\eta-\left(\frac{i\omega_{-}}{\sigma}\right)\ln
2\cosh(\sigma\eta)}{{}_{2}}F_{1}(\alpha,\beta,\gamma_{1},\delta_{+}),$ (5fqta)
$\displaystyle u^{(2)}_{k}(\eta,\chi)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{4\pi\omega_{out}}}e^{ik\chi-i\omega_{+}\eta-\left(\frac{i\omega_{-}}{\sigma}\right)\ln
2\cosh(\sigma\eta)}{{}_{2}}F_{1}(\alpha,\beta,\gamma_{2},\delta_{-}),$ (5fqtb)

where the constants above are defined as

$\alpha=1+\frac{i\omega_{-}}{\sigma},\quad\beta=\frac{i\omega_{-}}{\sigma},\quad\gamma_{1}=1-\frac{i\omega_{in}}{\sigma},\quad\gamma_{2}=1+\frac{i\omega_{out}}{\sigma}\quad\delta_{\pm}=\frac{1}{2}(1\pm\tanh(\sigma\eta)),$
(5fqtu)

and the frequencies $\omega_{in}=[k^{2}+m^{2}]^{\frac{1}{2}}$,
$\omega_{out}=[k^{2}+m^{2}(1+2\epsilon)]^{\frac{1}{2}}$ and
$\omega_{\pm}=\frac{1}{2}(\omega_{out}\pm\omega_{in})$. We note that in the
limit $\eta\rightarrow-\infty$ the first solution becomes

$u_{k}^{(1)}\rightarrow\frac{1}{\sqrt{4\pi\omega_{in}}}e^{ik\chi-i\omega_{in}\eta},$

while in the case $\eta\rightarrow+\infty$ the second solution is

$u_{k}^{(2)}\rightarrow\frac{1}{\sqrt{4\pi\omega_{out}}}e^{ik\chi-i\omega_{out}\eta}.$

One can see that in these limits the asymptotic solutions $u_{k}^{(1)}$ and
$u_{k}^{(2)}$ to the Klein-Gordon equation are plane waves which can then be
associated with positive mode solutions. The negative mode solutions
correspond to $u_{k}^{1*}$ and $u_{k}^{2*}$. Since $u^{1}_{k}$ is associated
with a plane wave at $\eta\rightarrow-\infty$ (past infinity) we call these
solutions in-waves $u_{k}^{(1)}\equiv u_{k}^{(in)}$. The solutions
$u_{k}^{(2)}\equiv u_{k}^{(out)}$ which are associated with plane waves at
$\eta\rightarrow+\infty$ (future infinity) will be called out-waves.

Using the linear transformation properties of hypergeometric functions one can
write $u_{k}^{(in)}$ in terms of $u_{k}^{(out)}$. This is easier than
calculating the Bogoliubov coefficients via brute force direct integration
using the inner product. One obtains

$u_{k}^{(in)}(\eta,\chi)=\alpha_{k}u_{k}^{(out)}(\eta,\chi)+\beta_{k}u_{k}^{(out*)}(\eta,\chi),$
(5fqtv)

where

$\displaystyle\alpha_{k}$ $\displaystyle=$
$\displaystyle\left(\frac{\omega_{out}}{\omega_{in}}\right)^{\frac{1}{2}}\frac{\Gamma\left(1-\frac{i\omega_{in}}{\sigma}\right)\Gamma\left(-\frac{i\omega_{out}}{\sigma}\right)}{\Gamma\left(-\frac{i\omega_{+}}{\sigma}\right)\Gamma\left(1-\frac{i\omega_{+}}{\sigma}\right)},$
(5fqtwa) $\displaystyle\beta_{k}$ $\displaystyle=$
$\displaystyle\left(\frac{\omega_{out}}{\omega_{in}}\right)^{\frac{1}{2}}\frac{\Gamma\left(1-\frac{i\omega_{in}}{\sigma}\right)\Gamma\left(\frac{i\omega_{out}}{\sigma}\right)}{\Gamma\left(\frac{i\omega_{-}}{\sigma}\right)\Gamma\left(1+\frac{i\omega_{-}}{\sigma}\right)}.$
(5fqtwb)

Here $\Gamma$ is the Gamma functions with the properties
$\Gamma(1+z)=z\,\Gamma(z)$, $\Gamma(1-ix)=\pi/[\Gamma(ix)\sinh(\pi x)]$ and
$|\Gamma(ix)|^{2}=\pi/[x\sinh(\pi x)]$ for $z$ complex and $x$ real. From the
above expressions we can read off the Bogoliubov coefficients
$\alpha_{kk^{\prime}}=\alpha_{k}\delta_{kk^{\prime}}$ and
$\beta_{kk^{\prime}}=\beta_{k}\delta_{-kk^{\prime}}$. Therefore the
transformation between annihilation operators yields
$a^{in}_{k}=\alpha_{k}^{*}a^{out}_{k}-\beta^{*}_{k}a_{-k}^{out\dagger}.$ We
now consider that the state of the field in the past infinity is the vacuum
state,

${|0\rangle}^{in}=\bigotimes_{k=-\infty}^{\infty}{|0\rangle_{k}}^{in,}$
(5fqtwx)

and use the expression for the in-mode annihilation operator to calculate the
state in the future infinity. Since the transformation between in and out
annihilation operators only mixes modes of frequency $k$ and $-k$, one finds
that the state seen by observers in the future infinity is,

$|0\rangle_{k}^{in}=\sqrt{1-\gamma}\sum_{n}\gamma^{n}|n\rangle_{k}^{out}|n\rangle_{-k}^{out}.$
(5fqtwy)

where,

$\gamma=\frac{\sinh^{2}(\pi\omega_{-}/\sigma)}{\sinh^{2}(\pi\omega_{+}/\sigma)}.$

The vacuum state from the perspective of observers in the remote past has
particles in the remote future. Due to the expansion of the universe there has
been particle creation. The state $|0\rangle^{in}$ is (once again) a two-mode
squeezed state, and the particle creation process just described has a strong
analogy with the quantum optical process of spontaneous parametric down
conversion if one describes only the signal photons emerging from the end of a
nonlinear crystal, while ignoring the idler photons [53, 33]. (Note: here the
laser pump acts as as the “source” driving the expansion of the universe).
Before reviewing work on entanglement in all of the above scenarios, we will
introduce the basic tools to quantify it.

###  2.2 Quantum information

The main aim of quantum information is to learn how to store, process and read
information using quantum systems. In this section we will briefly revise
basic concepts of quantum entanglement for pure and mixed states which are
considered to be key resources in quantum information.

####  2.2.1 Entanglement

Entanglement is a quantum property which is a consequence of the superposition
principle and the tensor product structure of the Hilbert space. The pure bi-
partite case is well understood. The state of two particles A and B is a
vector in a $(d_{a}\times d_{b})$-dimensional Hilbert space
$\mathcal{H}_{ab}=\mathcal{H}_{a}\otimes\mathcal{H}_{b}$. The space
$\mathcal{H}_{ab}$ is the tensor product of the subspaces $\mathcal{H}_{a}$
and $\mathcal{H}_{b}$ of each particle. An pure state element of the space
$\mathcal{H}_{ab}$ is written as
$|\psi_{ab}\rangle=\sum_{i,j}A_{ij}|i\rangle_{a}\otimes|j\rangle_{b}$.

A state $|\psi_{ab}\rangle\in\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ is
separable if $|\psi_{ab}\rangle=|\phi\rangle_{a}\otimes|\varphi\rangle_{b}$.
Separable states can be prepared by local operations and classical
communication. This means that observers manipulate each particle
independently by making measurements, or applying unitary transformations of
the form $|\psi_{ab}\rangle=U_{a}\otimes
U_{b}\,|\phi\rangle_{a}\otimes|\varphi\rangle_{b}$ where $U_{a}$ and $U_{b}$
are unitaries acting on particles A and B, respectively. The observers are
also allowed to exchange classical information. If the state is not separable
then it is entangled. An entangled state cannot be prepared by local
operations and classical communication, observers must make global operations
on the systems. To determine whether or not a general pure state
$|\psi_{ab}\rangle=\sum A_{ij}|i\rangle_{a}|j\rangle_{b}$ is entangled we
consider the following theorem:

> Schmidt decomposition: Let $\mathcal{H}_{a}$ and $\mathcal{H}_{a}$ be
> Hilbert spaces of dimension $d_{a}$ and $d_{b}$, respectively. For any
> vector $|\psi_{ab}\rangle\in\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ there
> exists a sets of orthonormal vectors
> $\\{|j\rangle_{a}\\}_{a=1,\ldots,d_{a}}\\!\subset\\!\mathcal{H}_{a}\mbox{
> and }\\{|l\rangle_{b}\\}_{b=1,\ldots,d_{b}}\\!\subset\\!\mathcal{H}_{b}$
> such that we can write
> $|\psi_{ab}\rangle=\sum_{i=1,\ldots,d}\lambda_{i}|i\rangle_{a}|i\rangle_{b}$.
> The Schmidt coefficients $\lambda_{i}$ are non-negative scalars such that
> $\sum_{i}\lambda^{2}_{i}=1$, and the sum runs over the minimum dimension
> $d=\textrm{min}(d_{a},d_{b})$ of the two Hilbert spaces.

This special basis is called the Schmidt basis [54, 55]. Note that the
correlations between systems A and B are now made explicit. For example, if
$\lambda_{i}=1$ and $\lambda_{j\neq i}=0$ then the state is separable. In the
case that all the $\lambda_{i}$’s are equal and $\lambda_{i}=1/\sqrt{d}$ then
the state is maximally entangled. It is clear that the distribution of the
Schmidt coefficients determine how entangled the state is. Therefore, to
quantify entanglement in the pure bi-partite case we need a monotonous and
continuous function of the $\lambda_{i}$’s such that

  1. 1.

$S(\lambda_{i})=0$ for separable states,

  2. 2.

$S(\lambda_{i})=\log(d)$ for maximally entangled states.

Considering the density matrix $\rho_{ab}=|\psi_{ab}\rangle\langle\psi_{ab}|$
and it’s reduced density matrix $\rho_{b}=Tr_{a}(\rho_{ab})$, we find that the
von Neumann entropy

$\displaystyle S(\rho_{b})$ $\displaystyle=$ $\displaystyle-
Tr\left(\rho_{b}\log_{2}\rho_{b}\right)=-\sum_{i}|\lambda_{i}|^{2}\log_{2}|\lambda_{i}|^{2},$

quantifies the entanglement between system A and B. We observe from the
Schmidt decomposition that it is equivalent to trace over either system A or B
and therefore, $S(\rho_{a})=S(\rho_{b})$. The von Neumann entropy of a pure
state is $S(\rho_{ab})=0$.

Quantifying entanglement in the mixed case is more involved since now there is
no analog to the Schmidt decomposition. However, it is still possible to
define a separable mixed state. A bipartite mixed state is separable if we can
write its density matrix as
$\rho_{ab}=\sum_{i}p_{i}\rho_{a}^{i}\otimes\rho_{b}^{i}$ where
$\sum_{i}p_{i}=1$.

To determine whether or not a general mixed state is entangled it is
convenient to define the partial transpose of a density matrix. Consider the
general bipartite mixed state

$\rho_{ab}=\sum_{ijkl}C_{ijkl}\,|i\rangle_{a}|j\rangle_{b}\,{}_{b}\langle
k|{}_{a}\langle l|\equiv\sum_{ijkl}C_{ijkl}\,|i\rangle_{a}{}_{a}\langle
l|\otimes|j\rangle_{b}{}_{b}\langle k|.$

The partial transpose $\rho^{PT}_{ab}$ of $\rho_{ab}$ is (transposing on
system $a$)

$\rho_{ab}^{PT}=\sum_{ijkl}C_{ljki}\,|i\rangle_{a}|j\rangle_{b}{}_{b}\langle
k|_{a}\langle l|,$

or equivalently (transposing on system $b$)

$\rho_{ab}^{PT}=\sum_{ijkl}C_{ikjl}|i\rangle_{a}|j\rangle_{b}{}_{b}\langle
k|_{a}\langle l|.$

Since the partial transpose of a separable state has positive eigenvalues it
is possible to construct a bipartite separability criterion: If the
eigenvalues of $\rho_{ab}^{PT}\geqslant 0$ then $\rho_{ab}$ is separable.
However, this criterion is only sufficient for $2\times 2$ and $2\times 3$
systems [56]. For systems of higher dimension the criterion is only necessary,
meaning that there are (bound) entangled states with positive partial
transpose. Such states are known as bound entangled states.

Adding up the negative eigenvalues gives an estimate of how entangled a state
is. Therefore, we will now define the negativity and logarithmic negativity
[57, 58] which are two entanglement monotones. The negativity of a density
matrix $\rho_{ab}$ is defined as the sum of the absolute values of the
negative eigenvalues of the partial transpose $\rho_{ab}^{PT}$,

$N(\rho_{ab}):=\frac{\|{\rho^{PT}_{ab}}\|-1}{2},$

where $\|\cdot\|$ denotes the trace norm
$\|X\|:=Tr\left[\sqrt{X^{\dagger}X}\right]$. The logarithmic negativity of a
density matrix $\rho_{ab}$ is defined as
$E_{N}(\rho_{ab}):=\log_{2}\|\rho_{ab}^{PT}\|.$

####  2.2.2 Covariant Matrix formalism

The covariant matrix formalism is usually employed in quantum optics [59]. It
is a framework that involves simple mathematical tools which are applicable to
systems consisting of a finite number of harmonic oscillators. While in
general computing entanglement can be very involved, it has been shown that in
the case of Gaussian states (such as coherent, squeezed and thermal states)
the covariant matrix formalism can be employed to produce computable measures
of quantum information [60]. Gaussian states are described by quasi-
probability distributions of Gaussian shape in phase space [61]. This
technique is very useful in relativistic quantum information since Bogoliubov
transformations map Gaussian states back onto Gaussian states [20, 62, 63]. As
long as one considers initially Gaussian states and a finite number of field
modes it is possible to calculate entanglement and other interesting
quantities using the measures developed in this framework. In the standard
Fock space description of quantum field theory quantum states correspond to
density matrices which contain all the information pertaining to the state. In
the covariant matrix formalism the notion of density matrix is replaced by a
real symplectic covariant matrix $\sigma$ defined by

$\sigma_{ij}\,=\,\left\langle\,\mathrm{X}_{i}\mathrm{X}_{j}\,+\,\mathrm{X}_{j}\mathrm{X}_{i}\,\right\rangle\,-\,2\,\left\langle\,\mathrm{X}_{i}\,\right\rangle\left\langle\,\mathrm{X}_{j}\,\right\rangle,$
(5fqtwz)

where the operators $\mathrm{X}_{i}$ are the generalized positions and
momenta, i.e.

$\displaystyle\mathrm{X}_{(2n-1)}$
$\displaystyle=\frac{1}{\sqrt{2}}(a_{n}\,+\,a_{n}^{\dagger})\,,$ (5fqtwaaa)
$\displaystyle\mathrm{X}_{(2n)}$
$\displaystyle=\frac{-i}{\sqrt{2}}(a_{n}\,-\,a_{n}^{\dagger})\,,$ (5fqtwaab)

and the index $n=1,2,3,\ldots$ labels the modes, and
$\langle\,\mathcal{O}\,\rangle$ denotes the expectation value of the operator
$\mathcal{O}$ with respect to the initial Gaussian state. The matrix $\sigma$
together with the vector of first moments $\langle\,\mathrm{X}_{i}\,\rangle$
completely characterizes all Gaussian states. However, the covariance matrix
is sufficient to compute entanglement. Unitary transformations in the Fock
space are replaced in this formalism by symplectic transformations in phase
space. A transformation $S$ is called symplectic, if it leaves the symplectic
form $\Omega$ (a numerical matrix) invariant, i.e., $S\,\Omega\,S^{T}=\Omega$,
where $[X_{i},X_{j}]=2i\Omega_{ij}$ and $(S_{ij})^{T}=(S_{ji})$. The
expression for the symplectic representation of a Bogoliubov transformation in
terms of its general coefficients $\alpha_{mn}$ and $\beta_{mn}$ takes a
simple form [20]. The matrix $S$ is decomposed into $2\times 2$ blocks
${\mathcal{M}}_{mn}$ as

$S=\left(\begin{array}[]{cccc}{\mathcal{M}}_{11}&{\mathcal{M}}_{12}&{\mathcal{M}}_{13}&\ldots\\\
{\mathcal{M}}_{21}&{\mathcal{M}}_{22}&{\mathcal{M}}_{23}&\ldots\\\
{\mathcal{M}}_{31}&{\mathcal{M}}_{32}&{\mathcal{M}}_{33}&\ldots\\\
\vdots&\vdots&\vdots&\ddots\end{array}\right),$ (5fqtwaaab)

where the sub-blocks are given by

$\mathcal{M}_{mn}\,=\,\left(\begin{array}[]{cc}\Re(\alpha_{mn}\,-\,\beta_{mn})&\Im(\alpha_{mn}\,+\,\beta_{mn})\\\\[4.2679pt]
-\Im(\alpha_{mn}\,-\,\beta_{mn})&\Re(\alpha_{mn}\,+\,\beta_{mn})\\\
\end{array}\right)\,.$ (5fqtwaaac)

Here $\Re(z)$ and $\Im(z)$ denote the real part and imaginary part of the
complex number $z$ respectively. The transformed covariance matrix
$\tilde{\sigma}$ is then simply obtained as
$\tilde{\sigma}\,=\,S\,\sigma\,S^{T}\,.$ This transformation ensures that if
$\sigma$ is Gaussian, $\tilde{\sigma}$ remains Gaussian. Fortunately, for
Gaussian states computable measures of entanglement have been developed [60,
64]. For example, the negativity between two modes is given by

$\mathcal{N}=\max\\{0,(1-\widehat{\nu}_{-})/2\widehat{\nu}_{-}\\},$
(5fqtwaaad)

where $\widehat{\nu}_{-}$ is the smallest symplectic eigenvalue of the partial
transpose matrix
$\,\widehat{\sigma}_{kk^{\prime}}=T_{k^{\prime}}{\sigma}_{kk^{\prime}}\,T_{k^{\prime}}\,$,
where $T_{k^{\prime}}=\textrm{diag}\\{1,1,1,-1\\}$ partially transposes mode
$k^{\prime}$, see Ref. [61].

Computing the negativity between two modes $k$ and $k^{\prime}$ involves first
tracing over all other modes. Impressively, the partial trace over any subset
of modes is computed by eliminating all rows and columns corresponding to all
modes other than $k$ and $k^{\prime}$. The smallest symplectic eigenvalue
$\widehat{\nu}_{-}$ is then obtained by diagonalizing the matrix
$i\Omega\,\widehat{\sigma}_{kk^{\prime}}$ by a symplectic operation $D$,
yielding the two eigenvalues $0\leq\widehat{\nu}_{-}\leq\widehat{\nu}_{+}$.
The state ${\sigma}_{kk^{\prime}}$ is entangled if $0\leq\widehat{\nu}_{-}<1$.

##  3 Entanglement of global modes

###  3.1 Flat spacetime entanglement

In the Unruh effect the Minkowski vacuum is a thermal state for uniformly
accelerated observers. Since the Rindler regions $I$ and $II$ are causally
disconnected, uniformly accelerated observers in one Rindler region (wedge)
have no access to information from the other Rindler region. Therefore, the
state of the Rindler observer, adapted to his/her particular wedge by tracing
out over the unaccessible wedge, is mixed. The state appears more mixed for
observers with increasing acceleration. By this we mean that the temperature
associated to the state by a particular observer is higher for observers with
larger proper accelerations. This has an effect on entanglement [3, 4].
Consider a pure entangled Minkowski state of two field modes each of them
labeled by a different Minkowski frequency. We assume that two inertial
observers, Alice and Bob, are able to distinguish these modes and agree that
the state is maximally entangled. As typically considered in the literature,
Alice and a Rindler observer in region $I$, called Rob, also analyze the
state. Since Rindler observers in Region $I$ must trace over the region $II$
part of the state, the correlations of the state will be partially lost. Since
the vacuum itself exhibits greater noise for observers with higher
accelerations one also expects other states to appear more mixed and less
correlated for higher accelerations. Therefore, there is a degradation of
entanglement due to the Unruh effect. Let us point out that general region $I$
states are not necessarily thermal after tracing out over region $II$ states,
however, they are mixed. A thermal state is a particular type of mixed state
in which the fraction of the ensemble in each pure state is given by a
Boltzmann distribution. In the case of Gaussian states such as the vacuum,
thermal states are obtained after tracing over modes. Calculations involving
Minkowski states are very involved because a single Minkowski mode corresponds
to an infinite superposition (continuous) of Rindler modes (5fg) making the
trace operation highly non-trivial. It is therefore, more convenient from a
mathematical point of view to analyze inertial states involving Unruh modes
such as the Bell state,

$|\Psi\rangle=\frac{1}{\sqrt{2}}\left(|0_{k}\rangle^{\mathcal{M}}|0_{\Omega}\rangle^{\mathcal{U}}+|1_{k}\rangle^{\mathcal{M}}|1_{\Omega}\rangle^{\mathcal{U}}\right),$
(5fqtwaaae)

where ${\mathcal{M}}$ and $\mathcal{U}$ label Minkowski and Unruh states
respectively with frequencies $k$ and $\Omega=\omega/a$. This state was
introduced in Bruschi, et.al [43]. For more details on the use of Unruh modes
to analyze entanglement and the single mode approximation see the appendix,
section 8. For inertial observers the modes $k$ and $\Omega>0$ (which must be
two distinguishable modes) are maximally correlated since the Bell state is
maximally entangled. It is then interesting to investigate to what degree the
state is entangled when described by observers in uniform acceleration. In the
simplest scenario shown in Fig. (1), Alice is considered to be inertial and a
uniformly accelerated observer Rob is introduced, who analyzes mode $\Omega$.
To study this situation, the states corresponding to Rob must be transformed
into the appropriate basis, in this case, the Rindler basis. We have already
calculated the transformation for the vacuum state in (2.1.1), and with that
in hand, we find the single particle Unruh state

$a_{\Omega,U}^{\dagger}|0_{\Omega}\rangle^{\mathcal{U}}=\left|1_{\Omega}\right>^{\mathcal{U}}=\frac{1}{\cosh(r)}\sum_{n}\tanh^{n}(r)\sqrt{n+1}\left|(n+1)_{\Omega}\right>_{I}\left|n_{\Omega}\right>_{II},$
(5fqtwaaaf)

where we have chosen $|q_{R}|=1$ in Eq. (5fj). Thus, the maximally entangled
state seen by inertial Alice and accelerated Rob is

$\displaystyle|\Psi\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left|0_{k}\right>^{\mathcal{M}}\otimes\frac{1}{\cosh(r)}\sum_{n}\tanh^{n}(r)\left|n_{\Omega}\right>_{I}\left|n_{\Omega}\right>_{II}$
(5fqtwaaag) $\displaystyle+$
$\displaystyle\frac{1}{\sqrt{2}}\left|1_{k}\right>^{\mathcal{M}}\otimes\frac{1}{\cosh(r)}\sum_{n}\tanh^{n}(r)\sqrt{n+1}\left|(n+1)_{\Omega}\right>_{I}\left|n_{\Omega}\right>_{II}.$

Since Rob, with trajectory in region $I$, is causally disconnected from region
$II$ we must take the trace over region $II$. The density matrix for the
Alice-Rob subsystem is

$\rho_{AR}=\frac{1}{\cosh^{2}(r)}\sum_{n=0}^{\infty}\tanh^{2n}(r)\rho_{n},$
(5fqtwaaah)

where

$\displaystyle\rho_{n}$ $\displaystyle=$
$\displaystyle\left|0_{k},n_{\Omega}\right>\left<0_{k},n_{\Omega}\right|+\frac{\sqrt{n+1}}{\cosh(r)}\left(\left|0_{k},n_{\Omega}\right>\left<1_{k},(n+1)_{\Omega}\right|+\left|1_{k},(n+1)_{\Omega}\right>\left<0_{k},n_{\Omega}\right|\right)$
$\displaystyle+$
$\displaystyle\frac{n+1}{\cosh^{2}(r)}\left|1_{k},(n+1)_{\Omega}\right>\left<1_{k},(n+1)_{\Omega}\right|,$

involves only ${\mathcal{M}}$ and region $I$ states. The entanglement between
the modes can be quantified by the logarithmic negativity yielding
$\mathcal{N}(\rho_{AR})=\log_{2}((1/2\cosh^{2}(r))+\Sigma)$ where

$\Sigma=\sum_{n}\frac{\tanh^{2n}(r)}{2\cosh^{2}(r)}\sqrt{\left(\frac{n}{\sinh^{2}(r)}+\tanh^{2}(r)\right)^{2}+\frac{4}{\cosh^{2}(r)}}.$
(5fqtwaaai)

One observes that entanglement is degraded for observers in uniform
acceleration and therefore, it is observer dependent. In the flat spacetime
case, one can argue that this is an effect of Rob being non-inertial. Rob must
be in a spaceship to be accelerated and energy must be supplied to accelerate
the system. In the flat spacetime case, inertial observers play a special role
and therefore, a well defined notion of entanglement corresponds to the
entanglement described from the inertial perspective. However, in curved
spacetime different observers describe a different particle content in the
field which results in different degrees of entanglement in the field. In that
case, there is no well-defined notion of entanglement.

Using Unruh rather than Minkowski states considerably simplifies the
mathematical techniques involved in the analysis of entanglement. However, it
introduces interpretational issues [65]. The physical interpretation of Unruh
states requires deeper understanding. Work in progress shows that finite size
accelerated detectors naturally couple to peaked distributions of Unruh modes
[51]. Such detectors will provide further insights on the states considered in
Eq. (5fqtwaaae). Currently, it is only possible to say that the work of
Bruschi, et.al. [43] studies the entanglement of a family of maximally
entangled states parameterized by the dimensionless $\Omega=\omega/a$ in non-
inertial frames. By fixing the physical frequency $\omega$ and changing $a$
one analyzes the entanglement in a family of states, all of which have the
same frequency $\omega$ as seen by observers with different proper
acceleration $a$. An alternative view is that the entanglement corresponds to
a family of states with different physical frequency $\omega$ as seen by the
same observer moving with fixed proper acceleration $a$ [66].

The analysis of entanglement in non-inertial frames has been generalized in
several directions and different quantum fields have been considered. For
example, the work by Adesso, et. al. [62] considers two-mode squeezed states
in the inertial frame (instead of the Bell states analyzed above) allowing
them to employ the covariance matrix formalism which is only applicable for
Gaussian states. The authors studied the case when both observers accelerate
finding that entanglement vanishes at finite acceleration. This effect is also
known in the literature as sudden death of entanglement [67]. The effect is
similar to that which occurs in parametric down conversion when a two-mode
entangled state is feed into an amplification channel (see also [68]). It was
shown that the rate of entanglement degradation depends on the entanglement in
the inertial frame. States with higher degrees of inertial entanglement
degrade more in non-inertial frames. The degradation of entanglement has also
been analyzed for inertial thermal states and non-maximally entangled states
[69].

Other types of correlations relevant to quantum information theory have also
been studied in non-inertial frames. For example, it was found that classical
correlations are conserved in case one observer is non-inertial [3, 4] and
degraded when both observers accelerate [62]. More recently, the quantum
discord [70, 71], a type of purely quantum correlations which are present even
in separable states, was analyzed in non-inertial frames. Datta [72] found
that in a regime where there is no distillable entanglement there is a finite
amount of quantum discord. Interestingly, in the limit of infinite
acceleration the discord does not vanish. Multi-partite entanglement has also
been subject of study for observers in uniform acceleration [4, 62]. The bi-
partite entanglement between modes is degraded in non-inertial frames,
however, three and four-partite correlations are created between modes in
regions $I$ and $II$ [62, 73]. Multipartite entanglement has also been
analyzed when the state contains tri-partite correlations in the inertial
frame [74].

The analysis of entanglement in non-inertial frames has been considered for
different types of fields. For example, Dirac fields [4, 69, 74, 75, 76, 77,
78], the electromagnetic field [79] and charged bosonic fields have been
studied [66]. The results for fermionic fields show that entanglement is
degraded in non-inertial frames however, the entanglement remains finite in
the infinite acceleration limit [4, 69, 77, 80]. Interestingly, for fermions
the entanglement in the infinite acceleration limit does not violate Bell’s
inequalities [78]. Multipartite entanglement also presents interesting
differences. In the bosonic case genuine tripartite entanglement is generated
amongst the modes of Alice, Rob and Anti-Rob (an observer with uniform
acceleration in region $II$) [62]. These correlations increase with
acceleration. However, in the fermionic case tripartite entanglement is always
zero[4]. Recent work [74] considers an inertial tripartite state which
contains genuine tripartite correlations, as measured by the Svetlichny
inequality [81]. These correlations persist for any finite acceleration for
both GGHZ and MS three-qubit states provided the respective control parameters
are appropriately chosen, vanishing only in the infinite acceleration limit.
This indicates that tripartite entanglement and its associated nonlocal
correlations are more robust to relativistic effects than their bipartite
counterparts. It is important to mention that an ambiguity on the definition
of entanglement between fermionic fields has been pointed out in [82, 83].
However, these results are subject of a present debate [84, 85]. (Note that in
[4], signs arising from the transposition of anti-commuting fermionic
operators could always be absorbed into an explicit phase that was introduced,
and hence into a redefinition of the fermionic creation operators).

The main differences in the entanglement degradation for scalar and fermionic
fields arise due to particle statistics [66, 77, 86, 87] and not to
dimensionality of the Hilbert space. In the fermionic case, a redistribution
of entanglement between particle and antiparticle states seams to prevent the
degradation of entanglement in the infinite acceleration limit [77]. Such
redistribution does not occur in the bosonic case [66]. Initially it was
conjectured that the Hilbert space dimensionality played a role [4, 69]. In
the bosonic case the Hilbert space is infinite dimensional while in the case
of Grassmann scalars, the lowest dimensional fermions, the fermionic Hilbert
space is of dimension two. One might suspect that entanglement is degraded
stronger in the bosonic case since for higher accelerations the bosonic
harmonic oscillators become populated with increasing temperature while
Grassmann scalars are always in mixed states which involve only two states.
Considering higher dimensional fermions [88] and bosonic fields with truncated
Hilbert spaces [86] showed that it is statistics rather than dimensionality
which produces the observed differences in the degradation of entanglement in
non-inertial frames. In the case of entanglement degradation for
electromagnetic fields it was found that photon helicity entangled states do
not degrade with acceleration [79].

###  3.2 Entanglement in curved spacetime.

General curved spacetimes do not have global timelike Killing vector fields
and therefore it is not possible to define subsystems. Without the notion of
subsystem it is not possible to study entanglement. In spite of this, special
cases have been analyzed. For instance, black hole spacetimes where the
spacetime is approximately flat at the horizon, and Robertson-Walker universes
which have two asymptotically flat spacetime regions.

####  3.2.1 Entanglement across black hole horizons

The Schwarzchild spacetime of an eternal black hole describes the geometry of
a spherical non-rotating mass $m$ [30]. Considering only the radial component,
the metric is

$\displaystyle ds^{2}=(1-(2m/R))dT^{2}-(1-(2m/R))^{-1}dR^{2}.$

where the horizon of the black hole is at $R=2m$. Considering the following
coordinate change $R-2m=x^{2}/8m$, such that

$1-(2m/R)=x^{2}/8mR=\frac{(x^{2}/8m)}{(x^{2}/8m+2m)}=\frac{(Ax)^{2}}{(1+(Ax)^{2})}\approx(Ax)^{2}$

when $x\approx 0$ with $A=1/4m$. This means that $dR^{2}=(Ax^{2})dx^{2}$.
Therefore, very close to the horizon $R\approx 2m$, the Schwarzschild
spacetime can be approximated by Rindler spacetime

$\displaystyle ds^{2}=-(Ax)^{2}dT^{2}+dx^{2},$

where the acceleration parameter $a=A^{-1}=4m$.

This means that, very close to the horizon of the black hole, we can consider
Alice as being an inertial observer and falling into the black hole, while Rob
escapes the fall by being a stationary, accelerated observer. If Alice claims
that the state of the field is the vacuum state, then Rob detects a thermal
state of the form

$\left|0_{k}\right>^{\mathcal{\mathcal{M}}}=\frac{1}{\cosh(q)}\sum_{n}\tanh^{n}(q)\left|n_{k}\right>^{in}\left|n_{k}\right>^{out},$
(5fqtwaaaj)

where $\cosh(q)=(1-e^{2m\pi\omega})^{-1/2}$ is a function of the mass of the
black hole $m$. Here $in$ and $out$ denote the modes inside and outside the
black hole. $\left|0_{k}\right>^{\mathcal{\mathcal{M}}}$ is the state detected
by Alice. If we then consider the case where two modes of the field in Alice’s
frame are maximally entangled, Rob will detect less entanglement between the
modes due to the Hawking effect [3, 73, 89, 90]. Using the covariant matrix
formalism an inertial two-mode squeezed state was analyzed [73]. The
entanglement between the modes is degraded for observers outside the black
hole, however genuine four-partite entanglement between the modes inside and
outside the black hole are created. For low black hole masses entanglement
vanishes and classical correlations also degrade. The analysis of entanglement
degradation has also been considered taking into account the distance of the
observer to the event horizon [91] by writing the metric in terms of the
proper time $\tau_{0}$ of a stationary observer placed at $r_{0}$,

$\displaystyle ds^{2}=-(\tilde{A}x)^{2}d\tau_{0}^{2}+dx^{2}$

which yields a modified acceleration
$\tilde{A}=A/\sqrt{1-2m/r_{0}}=(4m\sqrt{1-2m/r_{0}})^{-1}$ giving rise to the
same state as in Eq. (5fqtwaaaj) with a modified squeezing parameter
$\tilde{q}$ given by
$\cosh(\tilde{q})=(1-e^{2m\pi\omega/\sqrt{1-2m/r_{0}}})^{-1/2}$. This study
has been also carried out in the fermionic case [87, 88], and in higher
dimensional black holes [92].

####  3.2.2 Entanglement in an expanding universe

Our second curved space example is the Robertson-Walker universe [30]. In this
spacetime particles can be defined in two asymptotically flat regions.
Considering the field in the remote past to be in the vacuum state, it is
found that in the distant future the state contains particles. In the interim
region, where the universe is undergoing expansion, no sensible notion of
particles exist. In [31] it was shown that in the future infinity region
entanglement has been created between field modes. Interestingly, it is
possible to learn about the expansion parameters of the Universe from the
entanglement generated. The vacuum state in the past infinity corresponds to
the following two mode states in the future infinity

$|0\rangle^{in}=\sqrt{1-\gamma}\sum_{n}\gamma^{n}|n\rangle_{k}^{out}|n\rangle_{-k}^{out}.$

Since the state is pure, we can employ the von Neumann entropy to quantify the
entanglement generated the field modes $k$ and $-k$. In order to do this we
need to compute the reduced density matrix for one of the modes. The density
matrix for the state is

$\displaystyle\rho_{0}$ $\displaystyle=$ $\displaystyle|0\rangle^{in}\langle
0|^{in}=(1-\gamma)\sum_{n,m}\gamma^{(n+m)}|n\rangle_{k}^{out}|n\rangle_{-k}^{out}\langle
m|_{k}^{out}\langle m|_{-k}^{out}.$ (5fqtwaaak)

The reduced density matrix for mode $k$ is obtained by tracing over mode $-k$

$\displaystyle\rho_{k}$ $\displaystyle=$ $\displaystyle
tr_{-k}[\rho_{0}]=(1-\gamma)\sum_{n,m}\gamma^{(n+m)}\delta_{nm}|n\rangle_{k}^{out}\langle
m|_{k}^{out}$ (5fqtwaaal) $\displaystyle=$
$\displaystyle(1-\gamma)\sum_{n,m}\gamma^{2n}|n\rangle_{k}^{out}\langle
n|_{k}^{out}.$

Since the reduced denstiy matrix is already in diagonal form with eigenvalues
$\lambda_{n}=(1-\gamma)\gamma^{2n}$, it is straight forward to compute the von
Neumann entropy

$S(\rho_{k})=-(1-\gamma)\sum_{n}\gamma^{2n}\log_{2}((1-\gamma)\gamma^{2n})=\log_{2}\left[\frac{\gamma^{\gamma/(\gamma-1)}}{1-\gamma}\right].$
(5fqtwaaam)

Entanglement has been created in the remote future due to the expansion of the
universe. The entanglement depends on the cosmological constants since the
coefficient $\gamma$ depends on the expansion rate $\sigma$, the expansion
volume $\epsilon$ and the frequency of the modes involved through

$\gamma=\frac{\sinh^{2}(\pi\omega_{-}/\sigma)}{\sinh^{2}(\pi\omega_{+}/\sigma)},$
(5fqtwaaan)

with $\omega_{\pm}=1/2(\omega_{out}\pm\omega_{in})$,
$\omega_{in}=[k^{2}+m^{2}]^{\frac{1}{2}}$ and
$\omega_{out}=[k^{2}+m^{2}(1+2\epsilon)]^{\frac{1}{2}}.$ In the case of light
particles, the equations can be inverted and we can show that we can estimate
the expansion parameters from the entanglement. For a calculation in the
covariant matrix formalism see [20]. The analysis of entanglement in an
expanding universe has been extended to Dirac fields [34] and entanglement in
particle creation has also been studied in [93].

##  4 Extracting global mode entanglement using point-like detectors

Although early investigations on relativistic entanglement allowed us to
understand some aspects of the observer dependent nature of entanglement, the
results are purely academic. Global mode entanglement cannot be measured nor
manipulated by local observers. Alice would not be able to take an entangled
state and apply, for example, a CNOT gate on it since the states live in the
whole spacetime. This would require infinite time and energy making impossible
any practical quantum information task. However, it has been shown that global
mode entanglement can be accessed by local observers employing Unruh-Dewitt
detectors [7, 8, 9, 10, 12]. Two Unruh-deWitt detectors at rest can become
entangled by interacting with the Minkowski vacuum even if they are space-like
separated. This shows that the vacuum state is entangled. The vacuum is not
mode-wise entangled however, it contains entanglement in spatial degrees of
freedom. This entanglement is extracted, or transferred (swapped in the
quantum information language) from the field to the detector’s degrees of
freedom. In principle, it should be possible to use Unruh-Dewitt detectors to
study some aspects of the global mode entanglement analysis within a more
practical approach [94].

Since Unruh-Dewitt detectors are local, one can consider them as suitable
systems for the implementation of quantum information tasks. For example,
Alice and Bob could be spacelike separated and entangle thier detectors
through the interaction with the vacuum. This entanglement could be exploited
for quantum information tasks. The entanglement extracted is currently too
small to be useful, but in the future, enhancement schemes may be developed.
At the moment, the main limitation in using Unruh-Dewitt detectors for quantum
information processing is that the mathematical techniques involved already
become un-manageable with a few systems. For example, for two detectors one
must consider the interaction Hamiltonian

$\displaystyle\hat{H}_{I}(\tau)$ $\displaystyle=$ $\displaystyle\int
dk\lambda(\tau_{1})(d_{1}\,e^{i\Omega_{1}\tau_{1}}+d_{1}^{\dagger}\,e^{-i\Omega_{1}\tau_{1}})\left(a_{k}\,e^{i(kx_{1}(\tau_{1})-\omega
t_{1}(\tau_{1}))}+a_{k}^{\dagger}\,e^{-i(kx_{1}(\tau_{1})-t_{1}(\tau_{1}))}\right),$
$\displaystyle+$ $\displaystyle\int
dk\lambda(\tau_{2})(d_{2}\,e^{i\Omega_{2}\tau_{2}}+d_{2}^{\dagger}\,e^{-i\Omega_{2}\tau_{2}})\left(a_{k}\,e^{i(kx_{2}(\tau_{2})-\omega
t_{2}(\tau_{2}))}+a_{k}^{\dagger}\,e^{-i(kx_{2}(\tau_{2})-t_{2}(\tau_{2}))}\right),$

for which entanglement calculations involve $4th$-order perturbation theory
[8, 9, 10]. Current research is focused on finding ways to modify the
detectors in order to make them more mathematically accessible [51].

In spite of the mathematical difficulties involved, interesting results have
been found in relativistic quantum information using point-like detectors. For
example, the entanglement between an Unruh-DeWitt detector and a massless
bosonic field has been considered in [22] showing that the field and detector
remain entangled at late times. Information stored in the point-like system
flows into the field propagating with the radiation into null infinity. The
information lost by the detector is never restored (non-Markovian regime).
Another analysis considers two detectors, one inertial belonging to Alice, and
one in uniform acceleration belonging to Rob, which do not interact directly
but are coupled to the same field [24]. The detectors are initially entangled
and their entanglement vanishes after a finite time. The time at which the
entanglement vanishes is shorter for higher accelerations, as expected. This
setting has been considered recently for teleportation of a coherent state
[11]. When both detectors are at rest and initially entangled, the detectors
also disentangle and the dependence with their separation is studied in [23].
The case of Rob having a detector in non-uniform acceleration has been studied
in [24]. The authors show that the disentangling time slows down if the motion
is non-uniform. Two uniformly accelerated detectors have also been considered,
each on of them in a different Rindler wedge [7, 9]. The authors show that
entanglement is generated through their interaction with the Minkowski vacuum
even if they are always causally disconnected. This shows again that the
vacuum contains quantum correlations. Another interesting analysis is that of
the creation and loss of entanglement between detectors moving in circular
paths [25].

Recently, Olson and Ralph [10] found that by independently quantizing the past
and future spacetime regions of a massless quantum field it is also possible
to extract vacuum entanglement from timelike separated regions. To extract the
entanglement they considered two timelike separated point-like detectors at
rest characterized by a strongly time-dependent energy gap. Time-like
entanglement has been further analyzed in [95] and exploited in a
teleportation-type protocol [13, 14] which can be implemented in a circuit QED
experiment [14].

In curved spacetimes the entanglement between point-like detectors can been
used to distinguish between two universes. Two inertial detectors in flat
spacetime would become entangled if they interacted with a Minkowski thermal
field however, they would not become entangled in an exponentially expanding
de Sitter spacetime [96].

##  5 Accelerated cavities and localized wave-packet for quantum information

Work on the use of Unruh-Dewitt detectors for relativistic quantum information
processing has inspired efforts to find suitable systems to store information
in the framework of quantum field theory, with the hope of developing
mathematical techniques which render the calculations accessible. At the
moment moving cavities and localized wave-packets seem plausible systems for
this purpose.

###  5.1 Entanglement creation and degradation in moving cavities

The idea of using cavities moving in spacetime for relativistic quantum
information processing was proposed in [5] where two observers Alice and Rob
share an entangled state of two cavities as a resource for quantum
teleportation in non-inertial frames. The mathematical techniques to implement
this idea were introduced in [15] for inertial and uniformly accelerated
cavities and in [16] for non-uniform motion. The authors in [15] showed that
it is possible to entangle two cavities, one inertial and the second in
uniform acceleration, by letting a single two level atom interact with the
cavities. The scheme is a generalization of [97] in which the modes of two
cavities at rest positioned next to each other become maximally entangled
after an excited atom moving through them emits an excitation. Since it is not
possible to know in which cavity the atom emitted, the cavities become
entangled. When one of the cavities is in uniform acceleration the cavity
modes become detuned affecting the atom’s emission probabilities. The modes of
an inertial and accelerated cavity are given in (5fo) and (2.1.3),
respectively. The interaction Hamiltonian considered by the authors is of the
Unruh-Dewitt type given in (4) only in the sense that the atom is a two-level
ponit-like system interacting with discrete cavity modes. The authors show
that the ability to maximally entangle the cavities is reduced with
acceleration. However, by appropriately modifying the cavity length it is
possible to alter the modes and compensate for the effect. An advantage of
this scheme is that the quantum information stored in cavities is protected
from the Unruh effect (unlike for information stored in point-like systems) by
the mirrors. As long as the cavity walls are perfectly reflecting the field
inside the cavity is independent from the field outside. While the cavity
undergoes uniform acceleration (again we consider the case $aL/c^{2}\ll 1$ so
that the acceleration is essentially constant across the proper length of the
cavity), the states within the cavity however, remain pure since the Unruh
horizon is not contained in this spacetime region [52]. The quantum
information stored in field modes remains constant at all times as long as the
cavities are inertial or in uniform acceleration. However the situation
changes when the motion is non-uniform.

Consider a single inertial cavity with all modes in the vacuum state. If the
cavity accelerates for a finite time the modes will become populated as shown
is section 2.1.3. Working in the low acceleration regime $h<1$ it is possible
to find analytical expressions for the Bogoliubov coefficients for bosons [16]
and fermions [18] and calculate the entanglement produced between two modes
$k$ and $k^{\prime}$ using the covariant matrix formalism [98]. The
entanglement quantified by the logarithmic negativity is given by
$\mathcal{N}=|\beta^{(1)}_{k,k^{\prime}}|$ which is the first order correction
to the Bogoliubov coefficient $\beta_{k,k^{\prime}}$ [20]. The entanglement
between the modes can be calculated for any trajectory composed of segments of
inertial and uniformly accelerated segments. The entanglement in a basic
building block composed of initial and final inertial segments with a single
period of uniform acceleration is periodic in the acceleration period. A
consequence of this is that if one maximally entangles two modes each in a
different cavity as proposed in [15] and then accelerates for a finite time,
the entanglement between those modes will be degraded since the initial
information will be transferred to other modes. Since the evolution of
entanglement is periodic, Rob can plan his travel such that at the end of it
his cavity will still be maximally entangled with Alice’s.

Interesting conclusions can be drawn from comparing the analysis of
entanglement generation in fermionic and bosonic cavities [98]. In both cases
entanglement is always generated under non-uniform motion however, the
entanglement generated is different due to particle statistics. In the bosonic
case motion can populate modes with the characteristic that any number of
excitations can be produced in a given mode. However, fermionic statistics
constrains the number of particles in each mode giving rise to differences in
the entanglement produced.

Another interesting property of motion generated entanglement is that for any
trajectory segment constructed from periods of inertial and uniformly
accelerated motion, there is an entanglement resonance when the segment is
repeated $N$ times [19]. It can be shown analytically that when the frequency
associated with the segment period is equal to the sum of the frequencies of
the two modes then the entanglement between those two modes grows linearly
with $N$. The presence of the resonance is independent of the segment travel
details however the degree of entanglement does depend on the specific
trajectory. Using the covariant matrix formalisms it is possible to calculate
the entanglement at resonance of any trajectory analytically. Interestingly,
through these resonances it is possible to produce motion generated quantum
gates [19, 21, 29]. Recent results also show that genuine multipartite
entanglement is generated through the periodic motion of the cavity [21].

The general case of a cavity undergoing an arbitrary acceleration and the
possibility of performing actual experiments with mechanically oscillating
optical cavities are under study [29]. In the cases of a linear sinusoidal or
a uniform circular motion a resonance appears at much lower frequencies for
which no new photons are generated. The resonance leads to the generation of
entanglement between existing and previously non-entangled cavity modes [29].

Understanding the effects of gravity on quantum properties and quantum
information is of great interest. The analysis for entanglement in moving
cavities already allows us to draw some interesting conclusions: via the
equivalence principle [50] the results suggest that gravity can create
entanglement in the field modes of a cavity (as long as the cavity is small).
Consider a cavity in the vacuum held at a fixed position in a uniform
gravitational field. The modes of the cavity would become entangled by
changing the cavity’s position and letting the cavity undergo periods of free
fall.

###  5.2 Localized wave packets

An alternative for constructing localized field states is achieved by
superimposing global plane wave modes $U_{k}(x)=e^{ik_{\mu}x^{\mu}}$ localized
by a complete set of shape functions $f_{m}(k)$ via $U_{mk}(x)=\int
dkf_{m}(k)U_{k}(x)$ [41, 99]. A common example involves “box normalization” in
which one quantizes in a box of length $L$ and imposes periodic boundary
conditions, leading to normalized states $f_{m}^{box}=L^{-1/2}e^{i2\pi mx/L}$.
One can also normalize in both momentum and space degrees of freedom by using
functions [41] $f_{ml}(k)=\chi_{m}(k)e^{-i2\pi lk/\epsilon}$ where
$\chi_{m}(k)=\epsilon^{-1/2}$ for $|k-m\epsilon|<\epsilon/2$. Here $m,l$ are
integers and $\epsilon$ is a small positive constant with dimension of inverse
length. The additional degree of freedom represented by the index $l$
expresses the location of the box. In box normalization one takes the limit as
$L\rightarrow\infty$, while in the latter procedure one effectively uses and
infinite number of boxes of length $\sim 2\pi\epsilon$.

###  5.3 Quantum communication with moving wave-packets

As one recently proposed example of a relativistic quantum information
protocol, Downes et al. [26] consider the case of Alice sending coherent
states to Rob’s accelerated detector, who then performs homodyne detection.
Here the authors bypass the intermediate Unruh modes and consider the full
integral Bogoliubov transformation (5fg) from which they form localized modes
in frequency, as in the first equality in (5fqtwaaawaybe) in the appendix
(section 8). In addition, they eschew the Schrodinger state transformation
approach and instead consider the transformation of observables in the
Heisenberg picture. With Alice stationary at an $x$-coordinate position to the
right of focus of Rob’s hyperbola at $x=1/a$ ($\chi=0$), they consider only
left moving $L$ modes that travel along geodesics with $x+t$ constant. Rob’s
homodyne operator is given by
$\hat{O}(\tau)=a_{i,I}^{S}\,a_{i,I}^{L\dagger}\,e^{i\phi}+h.c.$ where the
superscripts $K=S,L$ denote the signal and local oscillator modes (both of
which are sent to Rob by Alice as localized Minkowski frequency modes) and the
subscript $i$ denotes modes (labeled by components of the wavevector), both
parallel (longitudinal) and perpendicular (transverse) to Rob’s acceleration.
Rob integrates the photocurrent from his detector over a long time compared to
the pulse length of Alice’s signal. Thus the average value of the signal
received by Rob is given by the expectation value $X=\langle\int
d\tau\hat{O}_{i}(\tau)\rangle$ with variance $V=\langle(\int
d\tau\hat{O}_{i}(\tau))^{2}\rangle-X^{2}$, with the Minkowski coherent state
$|\alpha,\beta,t\rangle_{i}=D^{S}_{i}(\alpha)D^{S}_{i}(\beta)|0\rangle_{M}$.
Here $D^{K}_{i}(\gamma)=\exp(\gamma a^{K}_{i}-\gamma^{*}a^{K\dagger}_{i})$ is
the Minkowski displacement operator with $\gamma=\alpha,\beta$ the amplitude
of the signal and local oscillator respectively. The authors make reasonable
approximations for matching the transverse spatial profile of the signal and
local oscillator to Rob’s detector (“beam-like” communication between Alice
and Rob), and further assume that the longitudinal source wavevector is
strongly peaked about a central frequency $\omega_{so}$. They obtain the
result that the normalized variance, defined as the variance $V$ divided by
the amplitude of the local oscillator $\beta_{R}^{2}$ as seen by Rob is given
by the following Planck factor
$\bar{V}=V/\beta^{2}_{R}=1/(e^{2\pi\omega_{so}(x+t)}-1)$.

The “novel” dependence of the Planck factor $\bar{V}$ on $x+t$ is correctly
attributed by the authors to the time dependent Doppler shift of the signal
frequency $\omega_{so}$ sent by Alice as measured by Rob [42, 100, 101]. This
should not be confused with the usual Planck factor
$1/(e^{2\pi\omega_{d,R}}-1)$ associated with the spectrum of frequencies
$\omega_{d,R}$ Rob’s detector measures when intercepting a Minkowski plane
wave $e^{i\omega_{so}(x+t)}$ from Alice. The two are closely related as
follows. The signal sent by Alice is centered on the 4-wavevector
$k^{\mu}_{so}=(\omega_{so},-\omega_{so},0,0)$, where we have taken
$k_{x,so}=-\omega_{so}$ for a left moving photon. For Rob’s trajectory
$x^{\mu}=1/a\,(\sinh(a\tau),\cosh(a\tau),0,0)$ his 4-velocity is given by
$u^{\mu}_{R}=(\cosh(a\tau),\sinh(a\tau),0,0)$. By the previous discussion on
Killing vectors in section 2.1.1, the frequency measured by Rob of this photon
is given by $\omega_{R}(\tau)=k^{\mu}_{so}\,u_{\mu,R}=\omega_{so}e^{a\tau}$.
By substituting in $x+t$, the variance can be written as
$\bar{V}=1/(e^{2\pi\omega_{R}(\tau)/a}-1)$, that is, the variance depends on
the time dependent Doppler shifted frequency $\omega_{R}(\tau)$ that Rob
measures when he detects a Minkowski plane wave
$e^{i\omega_{so}(x+t)}=e^{i\omega_{R}(\tau)/a}$ of fixed Minkowski frequency
$\omega_{so}$. This Minkowski plane wave is not detected as a positive
frequency Rindler plane wave $e^{-i\omega_{d,R}\tau}$ by Rob, but rather as a
combination of positive and negative frequency Rindler plane waves $e^{\mp
i\omega_{d,R}\tau}$. The square of the Fourier transform $S$ of
$e^{i\omega_{so}(x+t)}$ as analyzed by Rob
$S=\int_{-\infty}^{\infty}d\tau\,e^{-i\omega_{d,R}\tau}\,e^{i\omega_{R}(\tau)/a}$,
yields the Planck factor $1/(e^{2\pi\omega_{d,R}}-1)$ [100, 101]. In fact, for
finite integration times
$S(\tau)=\int_{0}^{\tau}d\tau\,e^{-i\omega_{d,R}\tau}\,e^{i\omega_{R}(\tau)/a}\equiv\int_{0}^{\tau}d\tau\,e^{i\varphi(\tau)}$
the phase of the integrand has a stationary point [42, 102] at
$d\varphi(\tau)/d\tau=0$ yielding the condition
$\omega_{d,R}=\omega_{so}\,e^{a\tau}$ which is the frequency that appears in
$\bar{V}$. Previously wave packets for relativistic quantum information where
considered in [103]. Recently a model on how to detect localized field modes
has been introduced [27]. This model can also be used to study entanglement
between localized wavepackets in non-inertal frames [28].

##  6 Change of state under Lorentz transformations

While the main body of this work has been concerned with observer dependent
entanglement for one or more observers undergoing constant acceleration, there
is a large body of research on closely related effects for purely inertial
observers (the zero acceleration case). This work is loosely classified under
the heading of the change in quantum states under Lorentz transformations
(LTs), and is intimately tied to the concept of Wigner rotation and the
unitary transformation of massive and massless particles under the Poincare
group (translations, rotations and boosts). In the following, we review this
research, which continues to be very active today, especially in connection to
the transformation of entangled quantum states in curved spacetime.

####  6.0.1 Preliminaries: Wigner rotation

In flat (Minkowski) spacetime, the positive energy, single particle states of
a massive particle forms a spinor representation of the inhomogeneous Lorentz
(Poincare) group [104, 105] consisting of ten generators: four translations
$P^{\mu}=(P^{0},P^{i})$, three generators $\\{J^{i}\\}$ of angular momentum
that generate spatial rotations, and three generators $\\{K^{i}\\}$ of pure
boosts. The quantum states, denoted by
$|\mbox{{\boldmath$p$}},\lambda\rangle$, are labeled by their spatial
3-momentum $p$ (where $p^{\mu}=(p^{0},\mbox{{\boldmath$p$}})$ with
$p^{0}=E=\sqrt{\mbox{{\boldmath$p$}}^{\hskip 1.75pt2}+m^{2}}\,$), and
$\lambda$ the component of angular momentum along a quantization axis in its
rest frame (typically taken to be along the third ($z$) spatial direction).
Under a Lorentz transformation $\Lambda$ the one-particle state transforms
under the unitary transformation $U(\Lambda)$ as [104, 105]

$U(\Lambda)|\mbox{{\boldmath$p$}},\lambda\rangle=\sum_{\lambda^{\prime}}\,D^{(j)}_{\lambda^{\prime}\lambda}(W(\Lambda,p))\,|\mbox{{\boldmath$p$}}_{\Lambda},\lambda^{\prime}\rangle,$
(5fqtwaaao)

where $j$ is total angular momentum of the particle (equal to the spin of the
particle in its rest frame), the summation is over
$\lambda^{\prime}=(-j,-j+1,\ldots,j)$, and $\mbox{{\boldmath$p$}}_{\Lambda}$
are the spatial components of the Lorentz transformed 4-momentum, i.e.
$\mbox{{\boldmath$p$}}^{\prime}$ where $p^{\prime\mu}=\Lambda^{\mu}_{\hskip
3.5pt\nu}\,p^{\nu}$. Most research has been primarily concerned with massive
spin-$\frac{1}{2}$ particles $(j=\frac{1}{2})$, and massless spin-1 particles
($j=1$, photons). While we will primarily discuss spin $\frac{1}{2}$
particles, many related issues carry over to the case of photons, with
important technical distinctions due to their absence of mass. In (5fqtwaaao),
$D^{(j)}_{\lambda^{\prime}\lambda}(W(\Lambda,p))$ is a $(2j+1)\times(2j+1)$
matrix spinor representation of the rotation group $SU(2)$. The quantity
$W(\Lambda,p)$ is called the Wigner rotation (matrix), and depends on both the
Lorentz transformation $\Lambda$ describing the (constant velocity) inertial
reference frame of the observer and the 4-momentum $p$ of the particle
observed. The important point to note about (5fqtwaaao) is that a single basis
state $|\mbox{{\boldmath$p$}},\lambda\rangle$ is transformed by a unitary
representation of a momentum dependent rotation $W(\Lambda,p)$ into a
superposition of all $\lambda$ (evaluated at the transformed momentum
$\mbox{{\boldmath$p$}}_{\Lambda}$).

The explicit form of the Wigner rotation matrix is given by

$W(\Lambda,p)=L^{-1}(\Lambda p)\,\Lambda\,L(p),$ (5fqtwaaap)

where $L(p)$ is a standard boost taking the standard rest frame 4-momentum
$k\equiv(m,0,0,0)$ to an arbitrary 4-momentum $p$, $\Lambda$ is an arbitrary
LT taking $p\to\Lambda p$ , and $L^{-1}(\Lambda p)$ is an inverse standard
boost taking the final 4-momentum $\Lambda p$ back to the particle’s rest
frame. Because of the form of the standard rest 4-momentum $k$, this final
rest momentum $k^{\prime}$ can at most be a spatial rotation of the initial
standard 4-momentum $k$, i.e. $k^{{}^{\prime}}=W(\Lambda,p)\,k$. The rotation
group $SO(3)$ is then said to form (Wigner’s) little group for massive
particles, i.e. the invariance group of the particle’s rest 4-momentum. The
explicit form of the standard boost is given by [104]

$\displaystyle L^{0}_{\hskip 1.75pt0}(\xi)$ $\displaystyle=$
$\displaystyle\frac{p^{0}}{m}\equiv\cosh\xi,$ $\displaystyle L^{i}_{\hskip
1.75pt0}(\xi)$ $\displaystyle=$
$\displaystyle\frac{p^{i}}{m}\equiv\sinh\xi\,\hat{p}^{i},\quad L^{0}_{\hskip
1.75pti}=-\sinh\xi\,\hat{p}_{i},$ $\displaystyle L^{i}_{\hskip 1.75ptj}(\xi)$
$\displaystyle=$ $\displaystyle\delta^{i}_{\hskip
3.5ptj}-(\cosh\xi-1)\,\hat{p}^{i}\hat{p}_{j},\qquad i,j=(1,2,3),$ (5fqtwaaaq)

where $\gamma=p^{0}/m=E/m$ is the particle’s energy per unit rest mass. Note
that for the flat spacetime metric $\eta_{\alpha\beta}=$diag$(1,-1,-1,-1)$,
$p_{0}=p^{0}$ and $p_{i}=-p^{i}$, and
$\hat{\mbox{{\boldmath$p$}}}=\mbox{{\boldmath$p$}}/|\mbox{{\boldmath$p$}}|=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)=\hat{p}^{i}$
is the direction of the 3-momentum $p$.

In the rest frame of the particle, with (standard) 4-momentum
$k=(m,\mbox{{\boldmath$k$}}=\mbox{{\boldmath$0$}})$ the basis vectors
$|\mbox{{\boldmath$0$}},\lambda\rangle$ are eigenstates of the four momentum
$P^{\mu}$, total angular momentum $\mbox{{\boldmath$J$}}^{2}=j(j+1)$, and
third component of angular momentum $J^{3}$ as (in units $\hbar=c=1$)

$\displaystyle P^{\mu}|\mbox{{\boldmath$0$}},\lambda\rangle$ $\displaystyle=$
$\displaystyle k^{\mu}|\mbox{{\boldmath$0$}},\lambda\rangle,$
$\displaystyle\mbox{{\boldmath$J$}}^{2}|\mbox{{\boldmath$0$}},\lambda\rangle$
$\displaystyle=$ $\displaystyle j(j+1)|\mbox{{\boldmath$0$}},\lambda\rangle,$
$\displaystyle J^{3}|\mbox{{\boldmath$0$}},\lambda\rangle$ $\displaystyle=$
$\displaystyle\lambda|\mbox{{\boldmath$0$}},\lambda\rangle.$ (5fqtwaaar)

The standard boost (6.0.1) can be decomposed [104, 105] as
$L(p)=R(\hat{\mbox{{\boldmath$p$}}})\,B_{3}(|\mbox{{\boldmath$p$}}|)\,R^{-1}(\hat{\mbox{{\boldmath$p$}}})$
where $R(\hat{\mbox{{\boldmath$p$}}})$ is a rotation that takes the
$\hat{\mbox{{\boldmath$z$}}}$-axis to $\hat{\mbox{{\boldmath$p$}}}$, and
$B_{3}(|\mbox{{\boldmath$p$}}|)$ is a pure boost in the
$\hat{\mbox{{\boldmath$z$}}}$ direction such that
$||\mbox{{\boldmath$p$}}|\hat{\mbox{{\boldmath$z$}}},\sigma\rangle=B_{3}(|\mbox{{\boldmath$p$}}|)\,|\mbox{{\boldmath$0$}},\sigma\rangle$
with $|\mbox{{\boldmath$p$}}|=m\,\sinh\xi$. The unitary representation of
$R(\hat{\mbox{{\boldmath$p$}}})$ on the Hilbert space of states is given by
$U(R(\hat{\mbox{{\boldmath$p$}}}))=e^{i\phi J_{3}}\,e^{i\theta J_{2}}$. Since
the rightmost rotation in $L(p)$ has no effect on
$|\mbox{{\boldmath$0$}},\lambda\rangle$ we have [105]

$|\mbox{{\boldmath$p$}},\lambda\rangle=L(p)\,|\mbox{{\boldmath$0$}},\lambda\rangle=R(\hat{\mbox{{\boldmath$p$}}})\,B_{3}(|\mbox{{\boldmath$p$}}|)\,|\mbox{{\boldmath$0$}},\lambda\rangle\equiv
H(p)\,|\mbox{{\boldmath$0$}},\lambda\rangle.$ (5fqtwaaas)

The states $|\mbox{{\boldmath$p$}},\lambda\rangle$ are eigenstates of
$P^{\mu}$, $\mbox{{\boldmath$J$}}^{2}$, and the helicity operator
$\mbox{{\boldmath$J$}}\cdot\hat{\mbox{{\boldmath$P$}}}/|\hat{\mbox{{\boldmath$P$}}}|$
with eigenvalue $\lambda$, the component of the particle’s angular momentum
along the direction $\hat{\mbox{{\boldmath$p$}}}$ of it’s linear momentum
[105],

$\displaystyle P^{\mu}|\mbox{{\boldmath$p$}},\lambda\rangle$ $\displaystyle=$
$\displaystyle p^{\mu}|\mbox{{\boldmath$p$}},\lambda\rangle,$
$\displaystyle\mbox{{\boldmath$J$}}^{2}|\mbox{{\boldmath$p$}},\lambda\rangle$
$\displaystyle=$ $\displaystyle j(j+1)|\mbox{{\boldmath$p$}},\lambda\rangle,$
$\displaystyle\frac{\mbox{{\boldmath$J$}}\cdot\mbox{{\boldmath$P$}}}{|\mbox{{\boldmath$P$}}|}|\mbox{{\boldmath$p$}},\lambda\rangle$
$\displaystyle=$ $\displaystyle\lambda|\mbox{{\boldmath$p$}},\lambda\rangle.$
(5fqtwaaat)

(Note that
$J^{3}|\mbox{{\boldmath$p$}},\lambda\rangle\neq\lambda|\mbox{{\boldmath$p$}},\lambda\rangle$
since $J^{3}$ now contains a contribution from the orbital motion). The Wigner
rotation in (5fqtwaaap) is given by $W(\Lambda,p)=H^{-1}(\Lambda
p)\,\Lambda\,H(p)$. The $|\mbox{{\boldmath$p$}},\lambda\rangle$ are called
helicity states. One can also define spin states denoted by
$|\mbox{{\boldmath$p$}},\sigma\rangle$ by suitable rotation
$|\mbox{{\boldmath$p$}},\sigma\rangle=\sum_{\lambda}{D^{(j)}_{\lambda\sigma}(R^{-1}(\hat{\mbox{{\boldmath$p$}}}))|\mbox{{\boldmath$p$}},\lambda\rangle}$
[106], which are commonly used in the literature. Here
$R^{-1}(\hat{\mbox{{\boldmath$p$}}})$ rotates the 3-momentum direction
$\hat{\mbox{{\boldmath$p$}}}$ to the $\hat{\mbox{{\boldmath$z$}}}$-axis.

The formalism above carries through for massless particles, but with important
physical differences. Since there is no rest frame for a massless particle,
the standard 4-momentum is chosen to be $k\equiv(1,0,0,1)$ with
$\mbox{{\boldmath$k$}}=\hat{\mbox{{\boldmath$z$}}}$, and basis state
$|\mbox{{\boldmath$k$}},\lambda\rangle$ labeled by $J^{3}$ as in (6.0.1)
(which hold with substitution $\mbox{{\boldmath$0$}}\to\mbox{{\boldmath$k$}}$
). The general state
$|\mbox{{\boldmath$p$}},\lambda\rangle=H(p)\,|\mbox{{\boldmath$k$}},\lambda\rangle$
is built up in exactly the same fashion as the massive case discussed in
(5fqtwaaas), where now the “energy” in $B_{3}(|\mbox{{\boldmath$p$}}|)$ is
given by $p^{0}=|\mbox{{\boldmath$p$}}|=e^{\xi}$. The helicity states
$|\mbox{{\boldmath$p$}},\lambda\rangle$ again satisfy (6.0.1) with $\lambda$
now restricted to the two values $\pm j$. The new feature for massless states
is that the helicity $\lambda$ is a Lorentz invariant since only the phase of
the state $|\mbox{{\boldmath$p$}},\lambda\rangle$ changes in a momentum
dependent fashion under a LT $\Lambda$

$U(\Lambda)|\mbox{{\boldmath$p$}},\lambda\rangle=e^{-i\,\lambda\,\theta(\Lambda,\hat{\bf
p})}\,|\mbox{{\boldmath$p$}},\lambda\rangle.$ (5fqtwaaau)

Here, the notation indicates that
$\theta(\Lambda,\hat{\mbox{{\boldmath$p$}}})$ depends only upon the LT
$\Lambda$ and the direction $\hat{\mbox{{\boldmath$p$}}}$ of the particle’s
momentum, but not upon its frequency $p^{0}$. The Wigner rotation angle
$\theta(\Lambda,\hat{\mbox{{\boldmath$p$}}})$ can be computed as
$\langle\mbox{{\boldmath$k$}},\lambda|W(\Lambda,p)|\mbox{{\boldmath$k$}},\lambda\rangle$,
where again $W(\Lambda,p)=H^{-1}(\Lambda p)\,\Lambda\,H(p)$ leaves the
standard momentum $k$ invariant $k^{{}^{\prime}}=W(\Lambda,p)\,k$ (Wigner’s
little group is now $ISO(2)$, rotations in plane perpendicular to $k$). For
$j=1$, $\theta(\Lambda,\hat{\mbox{{\boldmath$p$}}})$ describes the amount of
rotation experienced by the plane of polarization of a linearly polarized
photon (for example calculations of
$\theta(\Lambda,\hat{\mbox{{\boldmath$p$}}})$ see [104, 107, 108]).

Returning to the massive case, the spin states
$|\mbox{{\boldmath$p$}},\sigma\rangle$ (and similarly for the helicity states)
are normalized as
$\langle{\mbox{{\boldmath$p$}}^{{}^{\prime}},\sigma^{{}^{\prime}}}|\mbox{{\boldmath$p$}},\sigma\rangle=(2\pi)^{3}\,(2p^{0})\delta_{\sigma^{\prime}\sigma}\delta^{3}(\mbox{{\boldmath$p$}}^{\prime}-\mbox{{\boldmath$p$}})$.
One-particle states (wavepackets) are given by [105, 109]

$|\Psi\rangle=\sum_{\sigma}\,\int^{\infty}_{-\infty}d\mu(p)\,\psi_{\sigma}(p)\,|\mbox{{\boldmath$p$}},\sigma\rangle,\,\,\psi_{\sigma}(p)=\langle\mbox{{\boldmath$p$}},\sigma|\Psi\rangle,\,\,\sum_{\sigma}\,\int^{\infty}_{-\infty}d\mu(p)\,|\psi_{\sigma}(p)|^{2}=1,$
(5fqtwaaav)

with respect to the Lorentz invariant measure
$d\mu(p)=d^{3}\mbox{{\boldmath$p$}}/(2\pi)^{3}\,(2p^{0})$. Under a LT
$\Lambda$, the same state $|\Psi\rangle$ described in the boosted frame is
given by

$\displaystyle|\Psi^{{}^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\,\int^{\infty}_{-\infty}d\mu(p)\,\psi^{{}^{\prime}}_{\sigma}(p)\,|\mbox{{\boldmath$p$}},\sigma\rangle,$
(5fqtwaaawa) $\displaystyle\psi^{{}^{\prime}}_{\sigma}(p)$ $\displaystyle=$
$\displaystyle\sum_{\eta}D^{(1/2)}_{\sigma\eta}(W(\Lambda,\Lambda^{-1}p))\,\psi_{\sigma}(\Lambda^{-1}p)=\langle\mbox{{\boldmath$p$}},\sigma|U(\Lambda)|\Psi\rangle,$
(5fqtwaaawb)

which can be obtained from a straight forward calculation using (5fqtwaaav)
and (5fqtwaaao), the basis state normalization, a change of variables
$p\to\Lambda^{-1}\,p$ and utilizing $d\mu(\Lambda^{-1}p)=d\mu(p)$, along with
some index relabeling.

####  6.0.2 Early investigations

Quantum information was first considered in a relativistic setting by Czachor
in 1997 who analyzed a relativistic version of the Einstein-Podolsky-Rosen-
Bohm experiment [107, 108, 109, 110, 111, 112, 113, 114]. Czachor’s work
showed that relativistic effects are relevant to the experiment where the
degree of violation of Bell’s inequalities depends on the velocity of the
entangled particles. However, it was not until 2002 that the importance of
investigating quantum information in the presence of spacetime was recognized.
A. Peres & D. Terno pointed out that most concepts in quantum information
theory may require a reassessment [109, 115]. Further interesting results on
entanglement in flat spacetime were obtained by Alsing & Milburn [107], Adami,
Bergou, & Gingrich [116, 117], and Solano & Pachos [118]. Their work showed
that although entanglement is conserved overall under a change of inertial
frame, it may swap between spin and position degrees of freedom. (For a
comprehensive account of this early work up to 2004, with indications of
future work on accelerated observers, see the review by A. Peres and D.R.
Terno [109]).

The insightful and provoking work of Peres and Terno [109, 115] considered
single particle wavepackets as in (5fqtwaaav) with density matrix
$\rho=|\Psi\rangle\langle\Psi|$ having components
$\rho_{\sigma\sigma^{{}^{\prime}}}(\mbox{{\boldmath$p$}},\mbox{{\boldmath$p$}}^{{}^{\prime}})=\psi^{*}_{\sigma}(p)\,\psi_{\sigma^{{}^{\prime}}}(p^{{}^{\prime}})$,
and traced out over the momentum to produce the reduced spin density matrix

$\rho^{red}_{\sigma\sigma^{{}^{\prime}}}=\int^{\infty}_{-\infty}d\mu(p)\,\psi^{*}_{\sigma}(p)\,\psi_{\sigma^{{}^{\prime}}}(p^{{}^{\prime}}).$
(5fqtwaaawax)

They showed that owing to the momentum dependent Wigner rotations that arise
in the full state (5fqtwaaawa), that are induced by the LT $\Lambda$
describing the reference frame of a new observer, the reduced quantity
$\rho^{red}$ has no covariant transformation law, except in the limiting case
of sharp momentum. Only the complete density $\rho$ has a covariant
transformation law. They concluded that the “spin state” of a single particle
is meaningless if one does not specify its complete state, including momentum
variables. Even though it may be possible to formally define spin in any
Lorentz frame, there will be no relationship between the observable
expectation values in different Lorentz frames [115].

Alsing and Milburn [107] and Terashima and Ueda [108] considered an EPR Bell
state with factorable pure momentum eigenstates. In a second rest frame of
another pair of observers $A,B$ traveling perpendicularly to the initial rest
frame, described by the pure boost $\Lambda$, the initial state $|\Psi\rangle$
is described as
$|\Psi^{{}^{\prime}}\rangle=U_{AB}(\Lambda)\,|\Psi\rangle=U_{A}(\Lambda)\otimes
U_{B}(\Lambda)\,|\Psi\rangle$. While the state $|\Psi^{{}^{\prime}}\rangle$
remains factorable between momentum and spin, the initial Bell state is
coherently superposed with other Bell states. The reason is that
$U_{A}(\Lambda)$ and $U_{B}(\Lambda)$ give Wigner rotation angles of opposite
signs for particles traveling in opposite directions. Maximal EPR correlations
are recovered if the detectors in the boosted frame are rotated to the new
rest frame direction of spin.

Gingrich and Adami [116] considered wavepackets for two massive spin $1/2$
particles, with similar form and transformation properties
$(U_{AB}(\Lambda)=U_{A}(\Lambda)\otimes U_{B}(\Lambda))$ as (5fqtwaaav) and
(5fqtwaaawa), respectively

$\displaystyle|\Psi_{AB}\rangle$ $\displaystyle=$
$\displaystyle\sum_{\sigma\tau}\,\int\int
d\mu(p)\,d\mu(q)\,g_{\sigma\tau}(\mbox{{\boldmath$p$}},\mbox{{\boldmath$q$}})\,|\mbox{{\boldmath$p$}},\sigma\rangle\,|\mbox{{\boldmath$q$}},\tau\rangle,$
(5fqtwaaawaya) $\displaystyle
g_{\sigma\tau}(\mbox{{\boldmath$p$}},\mbox{{\boldmath$q$}})$
$\displaystyle\to$
$\displaystyle\sum_{\sigma^{{}^{\prime}}\tau^{{}^{\prime}}}\,D^{(1/2)}_{\sigma\sigma^{{}^{\prime}}}(W(\Lambda,\Lambda^{-1}p))D^{(1/2)}_{\tau\tau^{{}^{\prime}}}(W(\Lambda,\Lambda^{-1}q))\,g_{\sigma^{{}^{\prime}}\tau^{{}^{\prime}}}(\Lambda^{-1}\mbox{{\boldmath$p$}},\Lambda^{-1}\mbox{{\boldmath$q$}}).$
(5fqtwaaawayb)

Since LTs entangle the spin and momentum degrees of freedom within a single
particle, entanglement can be transferred between them. Their work showed that
this is also the case for pairs of particles, and that LTs affects the
entanglement between the spins of different particles, as measured by the
Wootters concurrence [119] on the reduced density matrix formed by tracing out
over the momentum of the complete state
$\rho_{AB}=|\Psi_{AB}\rangle\langle\Psi_{AB}|$. Thus, spin-spin entanglement
is not a Lorentz invariant.

These works generated an intense study of relativistic EPR correlations for
both spin $1/2$ particles(see [109, 120] and references therein) and photons
[107, 109, 117, 121] that continues today. Much of the work for spin $1/2$
particles (which we will primarily discuss henceforth) has focused on the
relevant covariant observable(s) for spin such that the expectation values
obtained from measurements are the same in all inertial frames. Also in
question is the meaning and validity of the reduced density matrix, especially
in the case of tracing out the momentum from the complete quantum state. Terno
et al. [115, 109] argued that in the relativistic setting momentum is a
primary variable that has relativistic transformation laws that depends only
on the LT matrix $\Lambda$ that acts on the spacetime coordinates. Quantities
such as spin and polarization are secondary variables that depend both on
$\Lambda$ and the primary variable momentum. Thus, even though the reduced
density matrix for the secondary variable may be well defined in any
coordinate system, it has no transformation law relating values in different
Lorentz frames. Further, the unambiguous and seemingly natural construction of
a reduced density matrix by means of tracing out over the primary variables is
possible only if the secondary variables are unconstrained. In the absence of
a general description, a case-by-case treatment is required.

Shortly afterward, Czachor [122], Caban and Rembielinski [123] (see also [124]
and references therein) showed that it is possible to define a Lorentz-
covariant reduced spin density matrix for single massive particles. Such an
object [123] contains information about the average polarization of the
particle, as well as information about its average kinematical state. For
sharp momentum, the reduced density matrix does not change under LTs. However,
in the case of an arbitrary momentum distribution (5fqtwaaav) the entropy of
the reduced density matrix is in general not a Lorentz invariant. The
conclusion of these studies is that while one can define a Lorentz-covariant
finite dimensional matrix describing the polarization of a massive particle,
in the relativistic case one cannot completely separate out kinematical
degrees of freedom. These results ultimately stem from the momentum dependent
spin transformations (5fqtwaaao) induced by a LT. Extensions of this line of
research have been carried out for states of two spin $1/2$ particles (see
e.g. [120] and references therein). Note that as pointed out by Peres and
Terno [109] an invariant definition of entanglement of a pair of spin $1/2$
particles typically utilizes the center-of-mass “rest frame” where
$\langle\sum\mbox{{\boldmath$p$}}\rangle=0$. However, due to the problem of
cluster decomposition [104] this definition is not adequate for more than two
constituent particles since subsets of particles may have different rest
frames. This is a difficult and still unresolved problem that has relevance to
investigations of relativistic multipartite entanglement for more than two
particles.

####  6.0.3 Spin observable issues

It is worth noting that complications in defining an appropriate reduced spin
density matrix arise from the long appreciated non-unique definition of the
“spin” operator for massive particles [125, 126, 127]. Ultimately, this is
traced back to the fact that spin is only defined in the rest frame of the
particle, where it coincides with the total angular momentum (since for
$\mbox{{\boldmath$p$}}=\mbox{{\boldmath$0$}}$ the orbital angular momentum
contribution vanishes). For studies of relativistic EPR correlations, most
works have settled on some variation of the the spin operator defined from the
Paul-Lubanski (PL) vector [127]
$W_{\mu}=-1/2\epsilon_{\mu\nu\rho\sigma}\,J^{\nu\rho}\,P^{\sigma}$ where
$J_{\mu\nu}$ are generators of the proper orthochronous Lorentz group with
$J_{ij}=-J_{ji}=\epsilon_{ijk}\,J_{k}$ and $J_{i0}=-J_{0i}=K_{i}$, the
generators of rotations and boosts, respectively. The two Casimir operators of
the Poincare group are given by $C_{1}=P^{\mu}P_{\mu}=m^{2}$ and
$C_{2}=W^{\mu}W_{\mu}=-m^{2}\,j(j+1)$ that commute with all the generators
$\\{P^{\mu},J^{\mu\nu}\\}$, and label the basis states
$|\mbox{{\boldmath$p$}},\sigma\rangle$ by their mass $m$ and spin $j$. The PL
vector
$W^{\mu}=(\mbox{{\boldmath$J$}}\cdot\mbox{{\boldmath$P$}},\,P^{0}\mbox{{\boldmath$J$}}+\mbox{{\boldmath$P$}}\times\mbox{{\boldmath$K$}})$
transforms as a 4-vector under LTs and is proportional to the total angular
momentum in the rest frame of the particle
$W^{\mu}_{rest}=(0,m\mbox{{\boldmath$J$}})$. Hence, spin is well defined in
the rest frame of the particle. In an arbitrary frame, the time component of
the PL vector is proportional to the helicity operator
$\mbox{{\boldmath$J$}}\cdot\mbox{{\boldmath$P$}}/|\mbox{{\boldmath$P$}}|$.

For observers in an arbitrary inertial frame the spin
$\mbox{{\boldmath$S$}}\equiv\mbox{{\boldmath$J$}}-\mbox{{\boldmath$L$}}$ is
defined as the difference between the total angular momentum, which is well
defined as a generator of the Poincare group, and the orbital angular momentum
$\mbox{{\boldmath$L$}}=\mbox{{\boldmath$R$}}\times\mbox{{\boldmath$P$}}$.
While the momentum $P$ is a well defined generator of the Poincare group,
there is no well defined notion of a position operator (and hence the concept
of localizability) in relativistic quantum mechanics. Hence $L$ and $S$ are
not uniquely defined. Popular choices for the position operator [126] include
the center-of-mass operator $\mbox{{\boldmath$R$}}_{c.m.}$ and Newton-Wigner
position operator $\mbox{{\boldmath$R$}}_{NW}$ defined by

$\mbox{{\boldmath$R$}}_{c.m.}=-\frac{1}{2}\left[\frac{1}{P^{0}}\,\mbox{{\boldmath$K$}}+\mbox{{\boldmath$K$}}\,\frac{1}{P^{0}}\right],\quad\mbox{{\boldmath$R$}}_{NW}=\mbox{{\boldmath$R$}}_{c.m.}-\displaystyle\frac{\mbox{{\boldmath$P$}}\times\mbox{{\boldmath$K$}}}{mP^{0}(m+P^{0})}.$
(5fqtwaaawayaz)

This leads to the center-of-mass and Newton-Wigner spin operators

$\mbox{{\boldmath$S$}}_{c.m.}=\displaystyle\frac{\mbox{{\boldmath$W$}}}{P^{0}},\quad\mbox{{\boldmath$S$}}_{NW}=\frac{1}{m}\left(\mbox{{\boldmath$W$}}-\displaystyle\frac{W^{0}\mbox{{\boldmath$P$}}}{P^{0}+m}\right).$
(5fqtwaaawayba)

The operator $\mbox{{\boldmath$S$}}_{NW}$ (favored by researchers such as
Terno et al [115, 128] and Caban et al [123, 120]) satisfies the usual $su(2)$
commutation relations and is the only axial-vector operator that is a linear
combination of the PL vector. Using (5fqtwaaawayaz), a spin observable is
defined by $\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{NW}$ as the
projection of spin along the unit spatial direction
$\hat{\mbox{{\boldmath$n$}}}$. The downside of $\mbox{{\boldmath$S$}}_{NW}$ is
that by itself, it is not part of any known 4-vector, nor second rank tensor
with well defined transformation properties under LTs. (Note that for both
operators
$\mbox{{\boldmath$S$}}\cdot\mbox{{\boldmath$P$}}=\mbox{{\boldmath$J$}}\cdot\mbox{{\boldmath$P$}}$
so that we can also write
$W^{0}=\mbox{{\boldmath$S$}}\cdot\mbox{{\boldmath$P$}}$). The center-of-mass
spin operator $\mbox{{\boldmath$S$}}_{c.m.}$ (favored by researchers such as
Czachor [110, 111, 115, 129, 128, 130]) does not generate the $su(2)$ spin
algebra and its eigenvalues are momentum dependent. Further, in contrast to
the operator $\mbox{{\boldmath$S$}}^{2}_{NW}$, the operator
$\mbox{{\boldmath$S$}}^{2}_{c.m.}$ does not reduce to the second Casimir
operator $C_{2}=j(j+1)$. Despite these flaws, the spin observable
$\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{c.m.}$, has many
favorable properties and is often used in the literature, as well as a
normalized (nonlinear) version (introduced by Czachor [110])
$\mbox{{\boldmath$S$}}_{CZ}(\hat{\mbox{{\boldmath$n$}}})=\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$W$}}/(m^{2}+(\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$p$}})^{2})^{1/2}$.
A comparison of EPR correlations using both the Newton-Wigner and center-of-
mass of spin observables [126] reveals that for fixed measurement directions,
the EPR correlations do not necessarily decrease monotonically with velocity,
and may exhibit local extrema for certain configurations (which also holds
true for spin 1 particles). Furthermore, this effect occurs for both types of
spin observables, and hence appears to be a generic feature of relativistic
correlations, and in some cases strongly depends on which definition of spin
observable is utilized.

Recent work by Friis et al. [130] utilized $\mbox{{\boldmath$S$}}_{c.m.}$ for
the investigation of a parameterized set of pure states of two spin $1/2$
particles, with various configurations of entanglement between the spin and
momentum degrees of freedom. As indicated by earlier works [107, 108, 110,
116] the maximum EPR correlation can be recovered in any inertial frame if
both the state and the spin observable are Lorentz transformed. In an inertial
frame in which the particle has 4-momentum
$p^{\mu}=(p^{0},\mbox{{\boldmath$p$}})$, the authors show the observable
defined by
$\hat{n}(p)\equiv(\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{c.m.})/|\lambda(\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{c.m.})|$,
with $\lambda(\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{c.m.})$
the eigenvector of the operator
$\hat{\mbox{{\boldmath$n$}}}\cdot\mbox{{\boldmath$S$}}_{c.m.}$, can also be
written as
$\hat{n}_{local}(p)=\hat{\mbox{{\boldmath$n$}}}_{local}(p)\cdot\mbox{{\boldmath$\sigma$}}$.
Here $\sigma$ is the usual 3-vector of Pauli spin matrices, and
$\hat{\mbox{{\boldmath$n$}}}_{local}(p)=(\sqrt{1-\beta^{2}}\hat{\mbox{{\boldmath$n$}}}_{\perp}+\hat{\mbox{{\boldmath$n$}}}_{\parallel})/(1-\beta^{2}(1-\hat{\mbox{{\boldmath$n$}}}^{2}_{\parallel}))^{1/2}$
can be interpreted as the detector orientation
$\hat{\mbox{{\boldmath$n$}}}_{local}$ as seen from the rest frame of the
particle (where $\beta=v/c$ is the velocity of the inertial frame). This later
expression can also be written as
$\hat{n}_{local}(p)=[L^{-1}(p)n]^{i}\,\sigma_{i}/|[L^{-1}(p)n]^{j}|$ where
$|[L^{-1}(p)n]^{j}|$ is the norm of the spatial portion of the Lorentz
transformed orientation vector $L^{-1}(p)\,n$. Here
$n=(n^{0},\hat{\mbox{{\boldmath$n$}}})$ is the 4-vector whose spatial portion
$\hat{\mbox{{\boldmath$n$}}}$ is the orientation of the detector in the
inertial frame with 4-momentum $p$. Thus, $L^{-1}(p)$ transforms the
orientation 4-vector $n$ back to the rest frame $n_{local}$, whose spatial
portion $\hat{\mbox{{\boldmath$n$}}}_{local}$ indicates the local direction of
the detector. Under a general LT $\Lambda$ the orientation 4-vector is given
by $n^{\prime\prime}=\Lambda\,n=\Lambda\,L(p)n_{local}$ so that
$\hat{n}^{\prime\prime}(p)=[L^{-1}(\Lambda
p)n^{\prime\prime}]^{i}\,\sigma_{i}/|[L^{-1}(\Lambda
p)n^{\prime\prime}]^{j}|=[W(\Lambda,p)n_{local}]^{i}\,\sigma_{i}/|[W(\Lambda,p)n_{local}]^{j}|$.
Since the Wigner rotation $W(\Lambda,p)$ does not change the norm of
$\hat{\mbox{{\boldmath$n$}}}_{local}$ we have that the observable
$\hat{n}^{\prime\prime}(p)$ transforms under LTs as
$\hat{n}^{\prime\prime}(p)=\big{(}W(\Lambda,p)\hat{\mbox{{\boldmath$n$}}}_{local}\big{)}\cdot\mbox{{\boldmath$\sigma$}}=U(\Lambda,p)\,\big{(}\hat{\mbox{{\boldmath$n$}}}_{local}\cdot\mbox{{\boldmath$\sigma$}}\big{)}\,U^{\dagger}(\Lambda,p)$.
Thus, the expectation values of observable $\hat{n}^{\prime\prime}(p)$ are the
same as that in the rest frame of the particle $\hat{n}_{local}(p)$. Most
importantly, while the spin observable $\hat{n}^{\prime\prime}(p)$ depends on
the momentum of the particle, the measurement direction