# Relativistic Quantum Communication: Energy cost and channel capacities

Ian Bernardes Barcellos<EMAIL_ADDRESS>Centro de Ciências
Naturais e Humanas, Universidade Federal do ABC, Avenida dos Estados, 5001,
09210-580, Bangu, Santo André, São Paulo, Brazil  André G. S. Landulfo
<EMAIL_ADDRESS>Centro de Ciências Naturais e Humanas,
Universidade Federal do ABC, Avenida dos Estados, 5001, 09210-580, Bangu,
Santo André, São Paulo, Brazil

###### Abstract

We consider the communication of classical and quantum information between two
arbitrary observers in asymptotically flat spacetimes (possibly containing
black holes) and investigate what is the energy cost for such information
transmission. By means of localized two-level quantum systems, sender and
receiver can use a quantum scalar field as a communication channel. As we have
already shown in a previous paper, such a channel has non-vanishing classical
capacity as well as entanglement-assisted classical and quantum capacities.
Here we will show that the change in the expectation value of the energy of
the system during the communication process can be separated in: (i) a
contribution coming from the particle creation due to the change of the
spacetime, (ii) a contribution associated with the energy needed to switch-
on/off each qubit, and (iii) a term which comes from the communication process
itself. For the quantum channel considered here, we show that the extra energy
cost needed for communication vanishes. As a result, if one has already
created a system of qubits for some specific task (e.g., quantum computation)
one can also reliably convey information between its parts with no extra
energy cost. We conclude the paper by illustrating the form of the channel
capacities and energy contributions in two paradigmatic cases in Minkowski
spacetime: (1) sender and receiver in inertial motion and (2) sender in
inertial motion while the receiver is uniformly accelerated.

###### pacs:

03.67.-a,03.67.Hk, 04.62.+v

##  I Introduction

Quantify at which rate one can reliably convey information between two or more
parts by means of a quantum communication channel is one of the major goals of
quantum information theory wilde ; GT07 . Although the road to this end has a
rich landscape already in the nonrelativistic context, it is only when
relativity is taken into account that we can fully enjoy all its hues. The
main reason is that relativity opens up the possibility of having non-trivial
structures such as black hole event horizons, causal horizons caused by the
relativistic relative motion of the parts conveying the information (or the
expansion of spacetime itself), or even the presence of Cauchy horizons wald84
. This has led several authors to analyze the communication process in
relativistic settings with particular attention being paid to Minkowski AM03 ;
BHTW10 ; LT13 ; MHM12 ; BHP12 ; CK10 ; JMK14 ; MM15 ; JRMK18 ; J17 ; HLL12 ;
SAKM20 ; YASKM20 , Schwarzschild HBK12 ; BA15 ; JACKM20 , or asymptotically
flat cosmological spacetimes MM ; BGMM16 ; SM17 .

In a recent paper L16 , one of the authors analyzed a communication model
using a bosonic quantum field as a communication channel which is suited to
arbitrary observers communicating in any globally hyperbolic curved spacetime.
In order to convey the information, both sender and receiver interact with the
field, which can be in any quasi-free algebraic quantum state wald94 , by
means of localized two-level quantum systems (qubits). By analyzing such a
quantum communication channel nonperturbatively, it was determined at which
rate one can reliably transfer information between the two parts. It was shown
that the channel has a nonvanishing classical capacity as well as
entanglement-assisted classical and quantum capacities.

However, it is not enough to know how much classical or quantum information
can be transmitted between two parts. A related and equally relevant inquiry
is: how much energy is needed to convey the information? This has important
consequences not only for practical proposes (such as engineering
communication networks) but because it may also shed light in one of the major
open problems in semiclassical/quantum gravity nowadays, namely, what is the
fate of the information which falls into a black hole. For instance, by
studying the energy toll for information transmission one may be able to
address if it is possible for all the information to come back at the end of
the black hole evaporation process, if it needs to come back during its
earlier stages, or even if the information gets destroyed/erased. (For a
review of such an issue, see Ref. UW  and references therein.) Although some
investigation has been performed on the issue of how much energy is needed to
convey information B81 , they are usually restricted to Minkowski spacetime
JMK15 ; HSU15  or 1+1 spacetimes J16 ; Wald19 .

In order to gain a broader perspective on the subject, in the present paper we
will use the communication channel developed in L16  to analyze what is the
energy cost to transmit information in general globally hyperbolic and
asymptotically flat spacetimes (which may contain black holes). By means of
the so-called null-surface quantization and its interplay with the usual
quantization procedure, we will be able to analyze how the total energy of the
system formed by the quantum field, $\phi$, and the two local qubits, $A$ and
$B$, changes as it is evolved from the asymptotic past to the asymptotic
future. It will be shown that such energy change can be separated in three
terms: (i) a contribution coming from the particle creation due to the change
of the spacetime metric, (ii) a contribution accounting for the energy needed
to switch-on/off each qubit, and (iii) a term which measures the extra energy
cost coming from the communication process itself. For the communication
channel being used, it will be proved that the contribution (iii) coming from
the convey of information vanishes. This shows that, once one has already
created the qubits, there is no extra energy cost in reliably transmitting
information between the two parts.

The paper is organized as follows. In Sec. II we will review the quantum
communication model used here. In Sec. III we will develop the so-called null-
surface quantization and relate it to the usual quantization procedure. In
Section IV, we will study how the energy of the total system–field
$\phi$+qubits $AB$–changes when it is evolved from past to future null
infinity and analyze what is the energy cost for communication. Section V will
be used to illustrate the channel capacities and energy cost in two
paradigmatic examples in Minkowski spacetime: two inertial observers and one
inertial and the other uniformly accelerated. Section VI is reserved to our
final remarks. We assume metric signature $(-+++)$ and natural units in which
$c=\hbar=G=1$ unless stated otherwise.

##  II The Communication Channel

The quantum communication model we rely on describes the exchange of
information between two arbitrary observers in any globally hyperbolic curved
spacetime $\left(\mathcal{M},g\right)$ using a quantum scalar field $\phi$ as
a communication channel. Here, $\mathcal{M}$ denotes the four-dimensional
spacetime manifold and $g$ its Lorentzian metric. Let us consider a real free
scalar field $\phi$ propagating in $\left(\mathcal{M},g\right)$. The spacetime
can be foliated by Cauchy surfaces $\Sigma_{t}$ labeled by the real parameter
$t$. The field is described by the action

$S\equiv-\frac{1}{2}\int_{\mathcal{M}}\epsilon_{\mathcal{M}}\,(\nabla_{a}\phi\nabla^{a}\phi+m^{2}\phi^{2}+\xi
R\phi^{2}),$ (1)

where $\epsilon_{\mathcal{M}}=\sqrt{-\mathfrak{g}}dx^{0}\wedge\cdots\wedge
dx^{3}$ is the spacetime volume 4-form, $m$ is the field mass,
$\xi\in\mathbb{R}$, $R$ is the scalar curvature, $\nabla_{a}$ is the torsion-
free covariant derivative compatible with the metric $g$, and
$\mathfrak{g}\equiv\text{det}(g_{\mu\nu})$ in some arbitrary coordinate
system. The extremization of the action (1) gives rise to the Klein-Gordon
equation

$(-\nabla^{a}\nabla_{a}+m^{2}+\xi R)\phi=0.$ (2)

In the canonical quantization procedure, we promote the real field $\phi$ to
an operator111Rigorously, an operator-valued distribution. that satisfies the
“equal-time” canonical commutation relations (CCR)

$\displaystyle\,[\phi(t,{\bf x}),\phi(t,{\bf x}^{\prime})]_{\Sigma_{t}}$
$\displaystyle=$ $\displaystyle[\pi(t,{\bf x}),\pi(t,{\bf
x}^{\prime})]_{\Sigma_{t}}=0,$ (3) $\displaystyle\,[\phi(t,{\bf x}),\pi(t,{\bf
x}^{\prime})]_{\Sigma_{t}}$ $\displaystyle=$ $\displaystyle i\delta^{3}({\bf
x},{\bf x}^{\prime}),$ (4)

where $\mathbf{x}\equiv(x^{1},x^{2},x^{3})$ are spatial coordinates on
$\Sigma_{t}$ and $\pi(x)$ is the conjugate momentum defined as

$\pi\equiv\frac{\delta S}{\delta\dot{\phi}}\,,$ (5)

where $``\;\dot{\;}\;"\equiv\partial_{t}$. In addition, we may formally write
the canonical Hamiltonian of the field as

$H_{\phi}(t)\equiv\int_{\Sigma_{t}}d^{3}{\bf
x}\,\left(\pi(t,\mathbf{x})\dot{\phi}(t,\mathbf{x})-\mathcal{L}[\phi,\nabla_{a}\phi]\right),$
(6)

with

$d^{3}{\bf x}\equiv dx^{1}\wedge dx^{2}\wedge dx^{3}$ (7)

and

$\mathcal{L}[\phi,\nabla_{a}\phi]\equiv-\frac{1}{2}\sqrt{-\mathfrak{g}}\,(\nabla_{a}\phi\nabla^{a}\phi+{m}^{2}\phi^{2}+\xi
R\phi^{2})$ (8)

being the Lagrangian density.

To find a representation of the CCR, Eqs. (3) and (4), we define an
antisymmetric bilinear map $\sigma$ acting on the space
$\mathcal{S}^{\mathbb{C}}$ of complex solutions of Eq. (2) as

$\sigma(\psi_{1},\psi_{2})\equiv\int_{\Sigma_{t}}\epsilon_{\Sigma}\,n^{a}\left[\psi_{2}\nabla_{a}\psi_{1}-\psi_{1}\nabla_{a}\psi_{2}\right],$
(9)

where $\epsilon_{\Sigma}$ represents the proper-volume 3-form on the Cauchy
surface $\Sigma_{t}$ and $n^{a}$ its future-directed normal unit vector. It
allows us to define the Klein-Gordon product as

$\langle\psi_{1},\psi_{2}\rangle\equiv-i\,\sigma(\overline{\psi}_{1},\psi_{2}),$
(10)

and, although this product is not positive-definite on
$\mathcal{S}^{\mathbb{C}}$, we may choose any subspace
$\mathcal{H}\subset\mathcal{S}^{\mathbb{C}}$ (the so-called one-particle
Hilbert space) such that wald94 : (i)
$\mathcal{S}^{\mathbb{C}}\simeq\mathcal{H}\bigoplus\overline{\mathcal{H}}$222For
the sake of mathematical precision, we note that one must first suitably
Cauchy-complete $\mathcal{S}^{\mathbb{C}}$ for this decomposition to be
valid.; (ii) the KG product is positive definite on $\mathcal{H}$, thus making
$(\mathcal{H},\langle,\rangle)$ a Hilbert space333After its completion with
respect to the norm induced by $\langle,\rangle$.; (iii) given any
$u\in\mathcal{H}$ and $v\in\overline{\mathcal{H}}$, $\langle u,v\rangle=0$.
Then, the Hilbert space that comprises the field states is defined as the
symmetric Fock space $\mathfrak{F}_{s}(\mathcal{H})$ and the quantum field
operator is formally defined as

$\phi(t,\mathbf{x})\equiv\sum_{j}\left[u_{j}(t,\mathbf{x})a(\overline{u}_{j})+\overline{u}_{j}(t,\mathbf{x})a^{\dagger}(u_{j})\right],$
(11)

where $\\{u_{j}\\}$ form an orthonormal basis for $\mathcal{H}$ and
$a(\overline{u})$ and $a^{\dagger}(v)$ are the usual annihilation and creation
operators associated with the modes $u$ and $v$, respectively, which satisfy

$\left[a(\overline{u}),a^{\dagger}(v)\right]=\langle u,v\rangle I,$ (12)

with $I$ being the identity operator on $\mathfrak{F}_{s}(\mathcal{H})$. The
vacuum state associated with this representation of the CCR is the normalized
vector $|0\rangle$ that satisfies $a(\overline{u})|0\rangle=0$ for every mode
$u\in\mathcal{H}$.

In order to make it mathematically well-defined, the quantum field operator
must be defined as an operator-valued distribution. To this end, let
$\mathcal{S}\subset\mathcal{S}^{\mathbb{C}}$ be the space of real solutions of
Eq. (2) whose restriction to Cauchy surfaces have compact support; let
$K:\mathcal{S}\rightarrow\mathcal{H}$ be the projection operator that takes
the positive-norm part of any $\psi\in\mathcal{S}$; and define the map
$E:C^{\infty}_{0}(\mathcal{M})\rightarrow\mathcal{S}$ acting on some test
function $f\in C^{\infty}_{0}(\mathcal{M})$, where
$C^{\infty}_{0}(\mathcal{M})$ is the set of all smooth, compactly supported
real functions on $\mathcal{M}$, as

$Ef(x)\equiv Af(x)-Rf(x).$ (13)

Here, $Af$ and $Rf$ are the advanced and retarded solutions, respectively, of
the Klein-Gordon equation with source $f$. Hence, they satisfy

$P(Af)=P(Rf)=f,$ (14)

with $P\equiv(-\nabla^{a}\nabla_{a}+m^{2}+\xi R)$ representing the Klein-
Gordon differential operator. Then, for each test function $f\in
C^{\infty}_{0}(\mathcal{M})$, we define a smeared quantum field operator by

$\phi(f)\equiv i\left[a(\overline{KEf})-a^{\dagger}(KEf)\right],$ (15)

which satisfy the covariant version of the CCR,

$\left[\phi(f_{1}),\phi(f_{2})\right]=-i\Delta(f_{1},f_{2})I,$ (16)

where

$\Delta(f_{1},f_{2})\equiv\int_{\mathcal{M}}\epsilon_{\mathcal{M}}f_{1}(x)Ef_{2}(x)$
(17)

and $f_{1},f_{2}\in C^{\infty}_{0}(\mathcal{M})$. It is easy to see that Eq.
(15) can be obtained by formally integrating Eq. (11) with the test function
$f$, i.e.,

$\phi(f)=\int_{\mathcal{M}}\epsilon_{\mathcal{M}}\,\phi(x)f(x).$ (18)

The above construction has the downside that there are infinitely many choices
of $\mathcal{H}$ satisfying properties (i)-(iii) below Eq. (10) which are, in
general, unitarily inequivalent. This issue can be avoided through the
algebraic approach to quantum field theory (QFT) wald94 ; KM15  in which the
field quantization can be seen as a real linear map $\phi:f\in
C^{\infty}_{0}(\mathcal{M})\rightarrow\phi(f)\in\mathcal{A}(\mathcal{M})$
between the space of test functions and an *-algebra
$\mathcal{A}(\mathcal{M})$ (called algebra of observables) such that

  1. 1.

$\phi(f)^{*}=\phi(f)$ for all $f\in C^{\infty}_{0}(\mathcal{M})$, i.e., the
(smeared) field is Hermitian;

  2. 2.

$\phi(Pf)=0$, for all $f\in C^{\infty}_{0}(\mathcal{M})$, i.e., the field
satisfies the Klein-Gordon equation;

  3. 3.

$\left[\phi(f_{1}),\phi(f_{2})\right]=-i\Delta(f_{1},f_{2})I,$ $f_{1},f_{2}\in
C^{\infty}_{0}(\mathcal{M})$, i.e., the field satisfies the CCR;

  4. 4.

$\mathcal{A}(\mathcal{M})$ is algebraically generated by the identity $I$ and
the $\phi(f)$’s, $f\in C^{\infty}_{0}(\mathcal{M})$.

In the algebraic approach, a quantum state is defined as a complex linear
functional $\omega:\mathcal{A}(\mathcal{M})\rightarrow\mathbb{C}$ which
satisfies $\omega(A^{*}A)\geq 0$ for all $A\in\mathcal{A}(\mathcal{M})$ and
$\omega(I)=1$. The so-called Gelfand-Naimark-Segal (GNS) construction wald94 ;
KM15  ensures that every algebraic state $\omega$ can be realized as a vector
on a Hilbert space together with a representation of the algebra of
observables.

In this work, we will focus on a particular class of states: the quasifree
states, defined as follows. Given a real inner product
$\mu:\mathcal{S}\times\mathcal{S}\rightarrow\mathbb{R}$ satisfying

$|\sigma(\varphi_{1},\varphi_{2})|^{2}\leq
4\mu(\varphi_{1},\varphi_{1})\mu(\varphi_{2},\varphi_{2}),$ (19)

for all $\varphi_{1},\varphi_{2}\in\mathcal{S}$, we define a quasifree state
$\omega_{\mu}$ associated with $\mu$ by the relation

$\omega_{\mu}\left[e^{i\phi(f)}\right]\equiv e^{-\mu(Ef,Ef)/2},$ (20)

for all $f\in C^{\infty}_{0}(\mathcal{M})$. We can extend $\omega_{\mu}$ to
act on all observables of $\mathcal{A}(\mathcal{M})$ by using

$\omega_{\mu}\left[\phi(f_{1})\cdots\phi(f_{n})\right]\\!\equiv\\!\left.(-i)^{n}\partial^{n}_{t_{1}\cdots
t_{n}}\left[e^{it_{1}\phi(f_{1})}\cdots
e^{it_{n}\phi(f_{n})}\right]\right|_{{\bf t}={\bf 0}},$ (21)

where ${\bf t}\equiv(t_{1},\cdots,t_{n})$, together with linearity and
continuity.

Now that the field quantization procedure has been introduced, we present the
communication scheme studied here. Suppose that two observers, Alice and Bob,
want to use the quantum field $\phi$ to communicate with each other. We
consider Alice’s and Bob’s trajectories to be arbitrary and we consider the
field to be initially in some quasifree state $\omega_{\mu}$444Actually, the
results from this section apply to any algebraic state $\omega$ which
satisfies $\omega\left[e^{i\phi(f)}\right]\in\mathbb{R}^{+}$. This condition
includes the vacuum states, n-particle states, as well as KMS (thermal)
states.. Each observer possesses a two-level gapless quantum system that may
interact with the quantum field. The two-dimensional Hilbert spaces associated
with Alice’s and Bob’s qubits are denoted by $\mathcal{H}_{A}$ and
$\mathcal{H}_{B}$, respectively.

The communication setup is as follows: Alice wants to transmit classical or
quantum information to Bob, and for that, she prepares her qubit in some
initial quantum state $\rho_{-\infty}^{A}$ and switches-on its interaction
with the field for a finite time $\Delta t_{A}$ (measured in the parameter
$t$). To measure the information imprinted by Alice on the field’s state, Bob
initially prepares his qubit in a suitable state $\rho_{-\infty}^{B}$. He then
switches-on its interaction with the field for a finite time interval $\Delta
t_{B}$, but only after Alice has switched-off her qubit interaction. Such
communication setup is implemented by means of the Hamiltonian

$H(t)\equiv H_{\phi}(t)+H_{\mathrm{int}}(t),$ (22)

where $H_{\phi}$ is the field Hamiltonian in Eq. (6) and $H_{\mathrm{int}}$ is
the Hamiltonian that describes the interaction between the qubits and the
field which, in the interaction picture, is given by

$H_{\mathrm{int}}^{\mathrm{I}}(t)\equiv\sum_{j}\epsilon_{j}(t)\int_{\Sigma_{t}}d^{3}{\bf
x}\sqrt{-\mathfrak{g}}\;\psi_{j}(t,{\bf x})\phi(t,{\bf x})\otimes\sigma^{\rm
z}_{j},$ (23)

where $j=A,B$ with $A$ and $B$ labelling Alice and Bob qubit, respectively.
Here, $\sigma_{j}^{\mathrm{z}}$ is one of the Pauli matrices
$\left\\{\sigma^{\mathrm{x}}_{j},\sigma^{\mathrm{y}}_{j},\sigma_{j}^{\mathrm{z}}\right\\}$
associated with qubit $j$, $\psi_{j}(t,\mathbf{x})$ is a smooth real function
satisfying $\psi|_{\Sigma_{t}}\in C_{0}^{\infty}\left(\Sigma_{t}\right)$ for
all $t$ which models the range of interaction between the qubit $j$ and the
field (i.e., the interaction occurs only at some vicinity of each qubit
worldline), and $\epsilon_{j}(t)$ is a smooth and compactly-supported real
coupling function modeling the finite-time coupling of the qubit $j$ with the
field. Each coupling function has support

$\mathrm{supp}\;\epsilon_{j}=\left[T_{j}^{\mathrm{i}},T_{j}^{\mathrm{f}}\right],$
(24)

where $T_{j}^{\mathrm{i}}$ and $T_{j}^{\mathrm{f}}$ represent the time (with
respect to the parameter $t$) in which each qubit interaction with the field
is switched-on and -off, respectively. Here, we denote $\Delta
t_{j}\equiv{T_{j}^{\mathrm{f}}-T_{j}^{\mathrm{i}}}$ and assume
$T_{B}^{\mathrm{i}}\geq T_{A}^{\mathrm{f}}$ (i.e., Bob measurement will be
performed after Alice imprinted her information on the field state).

The interaction between each qubit and the field, given by Eq. (23), is very
similar to the Unruh-DeWitt model. However, we assumed that the two levels of
each qubit have the same (zero) energy. This assumption allows us to calculate
the evolution operator of the system and trace out the field degrees of
freedom in a nonperturbative manner, thus making this model interesting to
investigate both the causality in the information exchange process as well as
the communication between parts lying in early and future asymptotic spacetime
regions. We note that one could also have given an energy gap $2\,\delta_{j}$
for each qubit $j$ by adding $H_{j}=\delta_{j}\sigma_{j}^{\mathrm{z}}$ to the
total Hamiltonian in Eq. (22). This would change it to

$H=H_{\phi}+H_{A}+H_{B}+H_{\mathrm{int}},$ (25)

but would keep the interaction Hamiltonian in the interaction picture, Eq.
(23), unchanged. Hence, all the results we have just described would remain
the same.

The interaction-picture time-evolution operator, associated with the foliation
$\Sigma_{t}$, can be written as the time-ordered expression

$U\equiv
T\exp\left[-i\int_{-\infty}^{\infty}dt\,H^{\mathrm{I}}_{\mathrm{int}}(t)\right].$
(26)

As shown in L16 , Eq. (26) can be computed exactly and it is given by

$U=e^{i\Xi}e^{-i\phi(f_{A})\otimes\sigma^{\mathrm{z}}_{A}}e^{-i\phi(f_{B})\otimes\sigma^{\mathrm{z}}_{B}}e^{-i\Delta(f_{A},f_{B})\sigma^{\mathrm{z}}_{A}\otimes\sigma^{\mathrm{z}}_{B}},$
(27)

where $\Xi$ is the c-number

$\Xi\equiv\frac{1}{2}\sum_{j}\int_{-\infty}^{\infty}dt\;\epsilon_{j}(t)\int_{-\infty}^{t}\;dt^{\prime}\epsilon_{j}(t^{\prime})\Delta_{j}(t,t^{\prime}),$

with

$\Delta_{j}(t,t^{\prime})\\!\equiv\\!\\!\\!\int_{\Sigma_{t}}\\!\\!\\!\\!\\!\\!d^{3}{\mathbf{x}}\sqrt{-\mathfrak{g}}\\!\\!\int_{\Sigma_{t^{\prime}}}\\!\\!\\!\\!\\!\\!\\!d^{3}{\mathbf{x^{\prime}}}\sqrt{-\mathfrak{g}^{\prime}}\psi_{j}(t,{\mathbf{x}})\Delta(x,x^{\prime})\psi_{j}(t^{\prime},{\mathbf{x^{\prime}}}),$

and we recall that $[\phi(x),\phi(x^{\prime})]\equiv-i\Delta(x,x^{\prime})I$
is the unsmeared version of Eq. (16). Additionally, we have defined

$f_{j}(t,\mathbf{x})\equiv\epsilon_{j}(t)\psi_{j}(t,\mathbf{x}),$ (28)

which is a compactly supported function on $\mathcal{M}$ carrying all
information about the interaction of qubit $j$ with the field.

The initial state of the two-qubit+field system is
${\rho_{-\infty}\equiv\rho^{A}_{-\infty}\otimes\rho^{B}_{-\infty}\otimes\rho_{\omega}}$,
where $\rho_{\omega}$ is the density operator associated with the initial
quasifree state, $\omega_{\mu}$, of the field. Using the unitary evolution
operator in Eq. (27), we can evolve $\rho_{-\infty}$ to obtain the system’s
state after the communication process,
$\rho_{+\infty}=U\rho_{-\infty}U^{\dagger}$. Additionally, we can trace-out
the field and Alice’s qubit degrees of freedom, obtaining the state of Bob’s
qubit after the communication process has finished:

$\rho^{B}\equiv\mathrm{tr}_{A,\phi}\left(U\,\rho^{A}_{-\infty}\otimes\rho^{B}_{-\infty}\otimes\rho_{\omega}\,U^{\dagger}\right).$
(29)

As shown in L16 , Eq. (29) can be written in the form

$\rho^{B}=\mathcal{E}\left(\rho^{A}_{-\infty}\right),$ (30)

where $\mathcal{E}$ is a linear, completely-positive, and trace-preserving
(CPTP) quantum map that relates the initial state of Alice’s qubit (which has
the information that will be conveyed) to the final state of Bob’s qubit
(which will be measured by him in order to retrieve the message). In other
words, $\mathcal{E}$ is the quantum map that describes the communication
channel used between Alice and Bob. It depends on their trajectories, on the
spacetime geometry, and on both the initial state of the field and of Bob’s
qubit. The explicit form of Eq. (30) as well as the details of the
calculations can be found in L16 . It is worth pointing out, however, that the
initial state $\rho_{-\infty}^{B}$ should not be arbitrarily chosen: since
$\sigma^{\mathrm{z}}_{B}$ commutes with the total Hamiltonian (25),
$\sigma^{\mathrm{z}}_{B}$ is conserved and thus its eigenvalues cannot be used
to recover any information transmitted by Alice. Nevertheless, it can be shown
that some states will maximize the signalling amplitudes between Alice and
Bob, e.g., $\rho^{B}_{-\infty}\equiv|y_{+}\rangle_{B}{}_{B}\langle y_{+}|$,
where $|y_{+}\rangle_{B}$ satisfies
$\sigma_{B}^{\mathrm{y}}|y_{+}\rangle_{B}=|y_{+}\rangle_{B}$. With this choice
of $\rho^{B}_{-\infty}$, it can be shown that this quantum channel has a
classical capacity (i.e., the maximum rate at which classical bits can be
reliably transmitted) given by

$\displaystyle C(\mathcal{E})=$
$\displaystyle\,H\left(\frac{1}{2}+\frac{\nu_{B}}{2}|\cos[2\Delta(f_{A},f_{B})]|\right)$
(31) $\displaystyle-$
$\displaystyle\,H\left(\frac{1}{2}+\frac{\nu_{B}}{2}\right),$

where $H(x)\equiv-x\log_{2}{x}-(1-x)\log_{2}{(1-x)}$ is the Shannon entropy
and

$\nu_{B}\equiv\omega_{\mu}\left[e^{i\phi(2f_{B})}\right]=e^{-2\mu(KEf_{B},KEf_{B})}.$
(32)

On the other hand, since this channel is entanglement-breaking, its quantum
channel capacity (i.e., the rate at which qubits can be reliably transmitted)
is

$Q(\mathcal{E})=0.$ (33)

One could also define protocols for sending both classical and quantum
information when Alice and Bob initially have access to an unlimited supply of
entanglement. In this case, one can define classical (or quantum)
entanglement-assisted channel capacities which measure the maximum rate at
which classical information (or qubits) can be reliably sent through the
channel. As shown in L16 , the classical $C_{ea}(\mathcal{E})$ and quantum
$Q_{ea}(\mathcal{E})$ entanglement-assisted capacities are related to the
classical channel capacity (31) by

$C_{ea}(\mathcal{E})=2Q_{ea}(\mathcal{E})=C(\mathcal{E}).$ (34)

Thus, it is not worth using entanglement in order to try to increase the
classical capacity of this channel. On the other hand, when prior entanglement
is shared between Alice and Bob, it is possible to convey qubits through this
channel at maximum rate $Q_{ea}(\mathcal{E})$, in contrast with the unassisted
case in Eq. (33).

##  III Quantization on Null Surfaces

The so-called quantization on null-surfaces provides the formulation of a QFT
restricted to 3-dimensional null submanifolds such as black hole horizons MP03
, asymptotic infinities M06 ; DMP06 , and cosmological horizons DMP09 . In
this section, we will build a quantum field theory restricted to a special
class of null hypersurfaces. Then, under a few assumptions, we show how one
can relate the ordinary QFT presented in Sec. II to the algebra of operators
defined in the asymptotic past and future null infinities as well as causal
horizons.

Let $\mathfrak{h}$ be a 3-dimensional null hypersurface satisfying:

  1. (1)

$\mathfrak{h}$ is diffeomorphic to $\mathbb{R}\times\Gamma$, where $\Gamma$ is
a two-dimensional spacelike submanifold of $\mathcal{M}$;

  2. (2)

there exists coordinates $(\Omega,\lambda,s^{1},s^{2})$ on $\mathcal{M}$ such
that:

    1. (a)

$s\equiv(s^{1},s^{2})$ are coordinates of $\Gamma$;

    2. (b)

$\mathfrak{h}=\\{p\in\mathcal{M}\,|\,\Omega(p)=0\\}$ and $d\Omega\neq 0$ at
$\mathfrak{h}$;

    3. (c)

the restriction of the metric to $\mathfrak{h}$ takes the form

$g\big{|}_{\mathfrak{h}}=-\gamma^{2}\,\left(d\Omega\otimes
d\lambda+d\lambda\otimes d\Omega\right)+h_{\Gamma}$ (35)

where $h_{\Gamma}$ is the metric induced by $g$ on $\Gamma$ and
$\gamma\in\mathbb{R}$.

It follows from conditions (1) and (2) above that $(\lambda,s)$ defines a
coordinate system for $\mathfrak{h}$ and that the curves
$\lambda\rightarrow(\lambda,s)$, defined for fixed $s$, are the null
generators of $\mathfrak{h}$. From now on, we will refer to $\mathfrak{h}$
generically as “the horizon” (although it can also be describing past or
future infinity).

To parallel the usual QFT construction we have presented in Sec. II, let us
define the “solution space” on $\mathfrak{h}$ as

$S^{\mathbb{C}}_{\mathfrak{h}}\equiv\left\\{\mathrm{smooth}\;\psi:\mathfrak{h}\rightarrow\mathbb{C}\,\big{|}\,\psi,\partial_{\lambda}\psi\in
L^{2}(\mathfrak{h},d\lambda\wedge\epsilon_{\Gamma})\right\\},$ (36)

where $\epsilon_{\Gamma}$ is the natural volume element on $\Gamma$ and
$L^{2}(\mathfrak{h},d\lambda\wedge\epsilon_{\Gamma})$ is the space of square-
integrable functions on $\mathfrak{h}$ with respect to the measure
$d\lambda\wedge\epsilon_{\Gamma}$. Similarly to Eq. (9), we define the
sympletic product on $\mathcal{S}^{\mathbb{C}}_{\mathfrak{h}}$ as

$\sigma_{\mathfrak{h}}(\psi_{1},\psi_{2})\equiv\int_{\mathfrak{h}}d\lambda\wedge\epsilon_{\Gamma}\left[\psi_{2}\partial_{\lambda}\psi_{1}-\psi_{1}\partial_{\lambda}\psi_{2}\right],$
(37)

which again allows us to define the Klein-Gordon inner-product on
$\mathcal{S}^{\mathbb{C}}_{\mathfrak{h}}$ by

$\langle\psi_{1},\psi_{2}\rangle_{\mathfrak{h}}\equiv-i\,\sigma_{\mathfrak{h}}(\overline{\psi_{1}},\psi_{2}).$
(38)

Now, let us note that Eq. (35) is invariant under translations
$\lambda\rightarrow\lambda+a$, with $a\in\mathbb{R}$. We can explore this
translation symmetry on $\lambda$ to choose a preferable representation of the
CCR. For this purpose, let us first define the positive-frequency projection
operator $K$ acting on $\psi\in\mathcal{S}^{\mathbb{C}}_{\mathfrak{h}}$ by

$K\psi(\lambda,s)\equiv\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}^{+}}dE\,e^{-iE\lambda}\,\widetilde{\psi}(E,s),$
(39)

where

$\widetilde{\psi}(E,s)\equiv\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}d\lambda\,e^{iE\lambda}\,\psi(\lambda,s)$
(40)

is the Fourier transform of $\psi$ with respect to $\lambda$. Then, the one-
particle Hilbert space is defined as

$\mathcal{H}_{\mathfrak{h}}\equiv\overline{\left\\{K\psi\,|\,\psi\in\mathcal{S}^{\mathbb{C}}_{\mathfrak{h}}\right\\}},$
(41)

where the closure is with respect to the norm induced by the product in Eq.
(38) (which is positive definite on $\mathcal{H}_{\mathfrak{h}}$). It is easy
to see that the horizon one-particle Hilbert space
$\mathcal{H}_{\mathfrak{h}}$ in Eq. (41) satisfies properties (i)-(iii) below
Eq. (10).

Having established the one-particle Hilbert space $H_{\mathfrak{h}}$, we can
now follow the same procedure described in Sec. II to build the bosonic Fock
space $\mathfrak{F}_{s}(\mathcal{H_{\mathfrak{h}}})$ and the
creation/annihilation operators associated with each mode
$u\in\mathcal{H}_{\mathfrak{h}}$. Then, the horizon smeared quantum field can
be defined as

$\phi^{(\mathfrak{h})}(\psi)\equiv
i\left[a(\overline{K\psi})-a^{\dagger}(K\psi)\right],$ (42)

for every
$\psi\in\mathcal{S}_{\mathfrak{h}}\subset\mathcal{S}_{\mathfrak{h}}^{\mathbb{C}}$
with

$\mathcal{S}_{\mathfrak{h}}\equiv\\{\psi\in\mathcal{S}^{\mathbb{C}}_{\mathfrak{h}}\,|\,\psi\mathrm{\;is\;real}\\}.$
(43)

The horizon algebra of observables, $\mathcal{A}(\mathfrak{h})$, is generated
by the identity operator
${I:\mathfrak{F}_{s}(\mathcal{H}_{\mathfrak{h}})\rightarrow\mathfrak{F}_{s}(\mathcal{H}_{\mathfrak{h}})}$
and the set of field operators
${\\{\phi(\psi)\,|\,\psi\in\mathcal{S}_{\mathfrak{h}}\\}}$.

It will be useful to write a basis for the one-particle space
$\mathcal{H}_{\mathfrak{h}}$. To this end, let
$\\{\varphi_{\alpha}\\}_{\alpha\in\Lambda}\subset
L^{2}(\Gamma,\epsilon_{\Gamma})$ be an orthonormal basis for
$L^{2}(\Gamma,\epsilon_{\Gamma})$ with respect to some measure $d\mu(\alpha)$
on the set $\Lambda$ of quantum numbers $\alpha$. Hence, every ${\psi\in
L^{2}(\Gamma,\epsilon_{\Gamma})}$ can be written as

$\psi(s)=\int_{\Lambda}d\mu(\alpha)\,\tilde{\psi}(\alpha)\,\varphi_{\alpha}(s)$
(44)

for some function $\tilde{\psi}(\alpha)$, with
$\varphi_{\alpha},\varphi_{\beta}$ satisfying

$\int_{\Gamma}\epsilon_{\Gamma}\,\overline{\varphi_{\alpha}(s)}\,\varphi_{\beta}(s)=\delta_{\mu}(\alpha,\beta),$
(45)

where $\delta_{\mu}$ is the Dirac distribution relative to the measure
$d\mu(\alpha)$. Then, define the set of modes
$\\{u_{E\alpha}\\}\subset\mathcal{H}_{\mathfrak{h}}$ as

$u_{E\alpha}(\lambda,s)\equiv\frac{1}{\sqrt{4\pi
E}}e^{-iE\lambda}\varphi_{\alpha}(s),\;\;\;E>0,$ (46)

which satisfy

$\langle
u_{E\alpha},u_{E^{\prime},\alpha^{\prime}}\rangle_{\mathfrak{h}}=\delta(E-E^{\prime})\delta_{\mu}(\alpha,\alpha^{\prime})$
(47)

and thus, form a orthonormal basis for the one-particle Hilbert space
$\mathcal{H}_{\mathfrak{h}}$. They allow us to define annihilation operators,
$a_{E\alpha}\equiv a\left(\overline{u_{E\alpha}}\right)$, and write the
horizon unsmeared quantum field operator as

$\phi^{(\mathfrak{h})}(\lambda,s)\equiv\int_{\mathbb{R^{+}}}dE\int_{\Lambda}d\mu(\alpha)\left[\,u_{E\alpha}a_{E\alpha}+\mathrm{H.c.}\,\right],$
(48)

which satisfy the commutation relation

$\left[\phi^{(\mathfrak{h})}(\lambda,s),\partial_{\lambda}\phi^{(\mathfrak{h})}(\lambda^{\prime},s^{\prime})\right]=\frac{i}{2}\delta(\lambda-\lambda^{\prime})\delta_{\Gamma}(s-s^{\prime}).$
(49)

It’s worth noting that the smeared and unsmeared quantum fields, Eqs. (42) and
(48), are related by

$\displaystyle\phi^{(\mathfrak{h})}(\psi)$
$\displaystyle=\sigma_{\mathfrak{h}}(\psi,\phi^{\mathfrak{h}})$ (50)
$\displaystyle=2\int_{\mathfrak{h}}d\lambda\wedge\epsilon_{\Gamma}\,\partial_{\lambda}\psi(\lambda,s)\phi^{\mathfrak{h}}(\lambda,s)$
$\displaystyle=\int_{\mathfrak{h}}2d\psi\wedge\epsilon_{\Gamma}\,\phi^{\mathfrak{h}}(\lambda,s),$

from which we see that the correct way to smear this field is with forms. This
is because there is no natural volume element on $\mathfrak{h}$ (the induced
metric is degenerate).

Let us now discuss the application of the null-surface quantization and its
relation to the ordinary QFT presented in Sec. II. Suppose that the spacetime
$(\mathcal{M},g)$ is asymptotically flat with future null infinity
$\mathcal{I}^{+}$, possibly containing a future causal horizon
$\mathfrak{h}^{+}$ (e.g., the event horizon of a black hole). The future null
infinity $\mathcal{I}^{+}$ is a 3-dimensional null hypersurface which
satisfies the properties (1)-(2) defined at the beginning of this section with
$\Gamma=\mathbb{S}^{2}$ and $\lambda\equiv u$ being the so-called “retarded
time”. Similarly, the causal horizon $\mathfrak{h}^{+}$ is a 3-dimensional
null hypersurface which satisfies the same properties but with $\lambda\equiv
v$ being the so-called “advanced time”. Thus, we can apply the quantization
procedure introduced in this section to both surfaces $\mathcal{I}^{+}$ and
$\mathfrak{h}^{+}$ (if present) and build the field algebras
$\mathcal{A}(\mathcal{I}^{+})$ and $\mathcal{A}(\mathfrak{h}^{+})$,
respectively.

Let us take now $\mathfrak{h}\equiv\mathfrak{h}^{+}\cup\mathcal{I}^{+}$ as the
union of future null infinity and the future causal horizon and let
$\widetilde{\mathcal{M}}\equiv I^{-}(\mathfrak{h})$ be the asymptotically flat
region outside the horizon, where $I^{-}(A)$ indicates the chronological past
of a subset $A\subset\mathcal{M}$. For the sake of simplicity, from now on, we
will restrict our analysis to minimally-coupled massless fields. Suppose we
have constructed a quantum field theory in $\widetilde{\mathcal{M}}$ following
the steps of Sec. II, obtaining an algebra of observables
$\mathcal{A}(\widetilde{\mathcal{M}})$ (called the bulk algebra). Since we are
dealing with a massless field, all the information carried by the field will
be “imprinted” on $\mathfrak{h}$. Thus, we expect that: (BB1) every solution
$\psi\in\mathcal{S}$ of the Klein-Gordon equation in $\widetilde{\mathcal{M}}$
which has compact support on Cauchy surfaces can be extended by continuity to
(unique) functions
$\psi^{\mathcal{I}^{+}}\in\mathcal{S}^{\mathbb{C}}_{\mathcal{I}^{+}}$ on
$\mathcal{I}^{+}$ and
$\psi^{\mathfrak{h}^{+}}\in\mathcal{S}^{\mathbb{C}}_{\mathfrak{h}^{+}}$ on
$\mathfrak{h}^{+}$. Moreover, $\psi,\psi^{\mathcal{I}^{+}},$ and
$\psi^{\mathfrak{h}^{+}}$ should satisfy

$\mathbf{(BB2)}\;\;\sigma(\psi_{1},\psi_{2})=\sigma_{\mathcal{I}^{+}}(\psi_{1}^{\mathcal{I}^{+}},\psi_{2}^{\mathcal{I}^{+}})+\sigma_{\mathfrak{h}^{+}}(\psi_{1}^{\mathfrak{h}^{+}},\psi_{2}^{\mathfrak{h}^{+}}),$
(51)

where the LHS is defined in Eq. (9) and the horizon bilinear products in the
RHS are defined in Eq. (37). By using (BB1) and (BB2) above, each operator in
$A(\widetilde{\mathcal{M}})$ can then be mapped with an operator in
${\mathcal{A}(\mathfrak{h})\equiv\mathcal{A}(\mathcal{I}^{+})\otimes\mathcal{A}(\mathfrak{h}^{+})}$
by the identification

$\phi(f)\rightarrow\phi^{(\mathfrak{h})}(Ef^{\mathfrak{h}}),\;\;\;\;\forall
f\in C^{\infty}_{0}\left(\widetilde{\mathcal{M}}\right).$ (52)

An algebraic state
${\omega:\mathcal{A}(\widetilde{M})\rightarrow\mathbb{R}^{+}}$ induces the
state
${\omega_{\mathfrak{h}}:\mathcal{A}(\mathcal{\mathfrak{h})}\rightarrow\mathbb{R}^{+}}$
on $\mathfrak{h}$ through the identification

$\omega_{\mathfrak{h}}\left[\phi^{(\mathfrak{h})}(Ef^{\mathfrak{h}})\right]\equiv\omega\left[\phi(f)\right]\;,\;\;\;\;\forall\phi^{(\mathfrak{h})}(Ef^{\mathfrak{h}})\in\mathcal{A}(\mathfrak{h}).$
(53)

By following a completely analogous procedure, one can also relate the bulk
algebra to the algebra defined at past null infinity, $\mathcal{I}^{-}$.

It is important to note that, although we will restrict ourselves to massless
and minimally-coupled real scalar fields, the above relation between between
$\mathcal{A}(\widetilde{M})$ and $\mathcal{A}(\mathfrak{h})$ will always
exists provided that the field in question satisfies conditions (BB1) and
(BB2) DMP09 .

##  IV Energy Cost for the Transmission of Information

In Sec. II, we have discussed a communication channel that allows the
transmission of information between two arbitrary observers in a globally
hyperbolic spacetime $(\mathcal{M},g)$. Now, we turn our attention to
investigate the energy cost involved in this communication process when
$(\mathcal{M},g)$ is asymptotically flat with past and future null infinities
given by $\mathcal{I}^{-}$ and $\mathcal{I}^{+},$ respectively. Our goal will
be to analyze the total energy variation of the two-qubit+field system between
early and late times. We recall that the initial state of the system is given
by

$\rho_{-\infty}\equiv\rho^{A}_{-\infty}\otimes\rho^{B}_{-\infty}\otimes\rho_{\omega},$
(54)

where $\rho_{\omega}$ is the density operator associated with some initial
quasi-free field state $\omega_{\mu}$ and $\rho^{j}_{-\infty}$ is the initial
state of qubit $j=A,B$. When the communication process finishes, the final
state of the two-qubits+field system is

$\rho_{+\infty}\equiv U\rho_{-\infty}U^{\dagger},$ (55)

where $U$ is the evolution operator given by Eq. (27). As a result, the total
energy variation of the system is formally written as

$\Delta E\equiv\langle H(+\infty)\rangle_{\rho_{+\infty}}-\langle
H(-\infty)\rangle_{\rho_{-\infty}},$ (56)

with $H(t)$ defined in Eq. (22). As the interaction time of each qubit with
the field is finite, the interaction Hamiltonian vanishes for
$t\rightarrow\pm\infty$ and thus, Eq. (56) can be cast as

$\displaystyle\Delta
E=\mathrm{tr}\left(H_{\phi}(+\infty)U\rho_{-\infty}U^{\dagger}\right)-\mathrm{tr}\left(H_{\phi}(-\infty)\rho_{-\infty}\right).$
(57)

As in Sec. II, define $\mathfrak{h}\equiv\mathfrak{h}^{+}\cup\mathcal{I}^{+}$
or $\mathfrak{h}\equiv\mathcal{I}^{+}$, depending on whether there is a future
causal horizon $\mathfrak{h}^{+}$ or not. Now, let us restrict our attention
to the globally-hyperbolic region
$\widetilde{\mathcal{M}}=I^{-}\left(\mathfrak{h}\right)$ outside the horizon
and let us foliate it with Cauchy surfaces $\Sigma_{t}$ such that
$\Sigma_{t\rightarrow-\infty}=\mathcal{I}^{-}$ and
$\Sigma_{t\rightarrow\infty}=\mathfrak{h}$. By using the identification
between the algebra $\mathcal{A}\left(\widetilde{\mathcal{M}}\right)$ with the
algebras $\mathcal{A}\left(\mathfrak{h}\right)$ and
$\mathcal{A}\left(\mathcal{I}^{-}\right)$, we can cast Eq. (57) as

$\displaystyle\Delta E$ $\displaystyle=$
$\displaystyle\mathrm{tr}\left(H^{(\mathfrak{h})}_{\phi}\,U^{(\mathfrak{h})}\rho_{-\infty}^{\mathfrak{h}}U^{(\mathfrak{h})\,\dagger}\right)-\mathrm{tr}\left(H_{\phi}^{(\mathcal{I}^{-})}\rho_{-\infty}^{(\mathcal{I}^{-})}\right)$
$\displaystyle=$
$\displaystyle\mathrm{tr}\left(U^{(\mathfrak{h})\,\dagger}H^{(\mathfrak{h})}_{\phi}\,U^{(\mathfrak{h})}\rho_{-\infty}^{\mathfrak{h}}\right)-\mathrm{tr}\left(H_{\phi}^{(\mathcal{I}^{-})}\rho_{-\infty}^{(\mathcal{I}^{-})}\right),$

with $H^{(\mathfrak{h})}_{\phi}$ $\left(H_{\phi}^{(\mathcal{I}^{-})}\right)$
and $\rho_{-\infty}^{\left(\mathfrak{h}\right)}$
$\left(\rho_{-\infty}^{(\mathcal{I}^{-})}\right)$ being the horizon field
Hamiltonian and the state induced by $\rho_{-\infty}$ at $\mathfrak{h}$
$\left(\mathcal{I}^{-}\right)$, respectively. Similarly, $U^{(\mathfrak{h})}$
is the evolution operator (27) written using the algebra
$\mathcal{A}\left(\mathfrak{h}\right)$, i.e., we have used the identification
$\phi(f_{j})\rightarrow\phi^{(\mathfrak{h})}(Ef_{j}^{\mathfrak{h}})$.

The field Hamiltonian at $X=\mathcal{I}^{-},\mathfrak{h}$ can be written as

$\displaystyle H^{X}_{\phi}$
$\displaystyle=\int_{X}d\lambda_{X}\wedge\epsilon_{\Gamma_{X}}\;T^{X}_{ab}k^{a}k^{b}$
(59)
$\displaystyle=\int_{X}d\lambda_{X}\wedge\epsilon_{\Gamma_{X}}\left[\partial_{\lambda_{X}}\phi^{X}\right]^{2}.$

where $\Gamma_{X}$ is the spacelike 2-surface transverse to the null
generators of X,

$T^{X}_{ab}\equiv\nabla_{(a}\phi^{X}\nabla_{b)}\phi^{X}-\frac{1}{2}g_{ab}\nabla_{c}\phi^{X}\nabla^{c}\phi^{X}$
(60)

is the stress-energy-momentum tensor at $X$ of the massless KG field, and
$k^{a}\equiv(\partial_{\lambda_{X}})^{a}$ is the vector field tangent to the
affinely-parametrized null generators of $X$ whose affine parameter is given
by $\lambda_{\mathcal{I}^{-}}=v$ or $\lambda_{\mathfrak{h}}=\lambda$ whenever
$X=\mathcal{I}^{-}$ or $X=\mathfrak{h}$, respectively.

Let us evaluate Eq. (LABEL:eq:delta-E-horizon) by steps. For this purpose, we
first define

$U^{(\mathfrak{h})}_{j}\equiv
e^{-i\phi^{(\mathfrak{h})}(Ef_{j}^{\mathfrak{h}})\otimes\sigma_{j}^{\mathrm{z}}},\;\;j=A,B,$
(61)

and use Eq. (27), together with the identification between field algebras in
$\mathfrak{h}$ and $\widetilde{\mathcal{M}}$, to write

${U^{(\mathfrak{h})}}^{\dagger}H_{\phi}^{(\mathfrak{h})}U^{(\mathfrak{h})}=\\\
{U^{(\mathfrak{h})}_{B}}^{\dagger}{U^{(\mathfrak{h})}_{A}}^{\dagger}\,H_{\phi}^{(\mathfrak{h})}U^{(\mathfrak{h})}_{A}U^{(\mathfrak{h})}_{B}.$
(62)

Next, by using Eqs. (49) and (61) together with the relation

$e^{\mathfrak{a}}\mathfrak{b}e^{-\mathfrak{a}}=\mathfrak{b}+[\mathfrak{a},\mathfrak{b}],$
(63)

valid when
$\left[[\mathfrak{a},\mathfrak{b}],\mathfrak{a}\right]=\left[[\mathfrak{a},\mathfrak{b}],\mathfrak{b}\right]=0$,
we can write

${U^{(\mathfrak{h})}_{j}}^{\dagger}\,\partial_{\lambda}\phi^{\mathfrak{h}}\,U_{j}^{(\mathfrak{h})}=\partial_{\lambda}\phi^{\mathfrak{h}}-\partial_{\lambda}Ef_{j}^{\mathfrak{h}}\,\sigma^{\mathrm{z}}_{j},$
(64)

where we recall that $j=A,B$. Now, using Eqs. (61) and (64) in Eq. (62), we
obtain

${U^{(\mathfrak{h})}}^{\dagger}\,\partial_{\lambda}\phi^{\mathfrak{h}}\,U^{(\mathfrak{h})}=\partial_{\lambda}\phi^{\mathfrak{h}}-\sum_{j=A,B}\partial_{\lambda}Ef_{j}^{\mathfrak{h}}\,\sigma^{\mathrm{z}}_{j}.$
(65)

By using Eqs. (59) and (65) we can cast the evolved Hamiltonian on
$\mathfrak{h}$ as

$\displaystyle
U^{(\mathfrak{h})\dagger}H_{\phi}^{(\mathfrak{h})}U^{(\mathfrak{h})}=\;H_{\phi}^{(\mathfrak{h})}$
(66) $\displaystyle-$ $\displaystyle
2\sum_{j=A,B}\int_{\mathfrak{h}}d\lambda\wedge\epsilon_{\Gamma}\partial_{\lambda}Ef_{j}^{\mathfrak{h}}\partial_{\lambda}\phi^{\mathfrak{h}}\otimes\sigma_{j}^{\mathrm{z}}$
$\displaystyle+$
$\displaystyle\sum_{i,j=A,B}\int_{\mathfrak{h}}d\lambda\wedge\epsilon_{\Gamma}\partial_{\lambda}Ef_{i}^{\mathfrak{h}}\partial_{\lambda}Ef_{j}^{\mathfrak{h}}\sigma^{\mathrm{z}}_{i}\otimes\sigma^{\mathrm{z}}_{j}.$

Finally, by substituting Eq. (66) in Eq. (LABEL:eq:delta-E-horizon) we can
write the energy variation as

$\Delta E=W_{\phi}+W_{A}+W_{B}+W_{AB},$ (67)

where

$W_{\phi}\equiv\mathrm{tr}\left(H^{(\mathfrak{h})}_{\phi}\,\rho^{(\mathfrak{h})}_{\omega}\right)-\mathrm{tr}\left(H^{(\mathcal{I}^{-})}_{\phi}\,\rho^{(\mathcal{I}^{-})}_{\omega}\right),$
(68)
$W_{j}\equiv\int_{\mathfrak{h}}d\lambda\wedge\epsilon_{\Gamma}\left(\partial_{\lambda}Ef^{\mathfrak{h}}_{j}\right)^{2},$
(69) $W_{AB}\equiv
2\left[\int_{\mathfrak{h}}d\lambda\wedge\epsilon_{\Gamma}\,\left(\partial_{\lambda}Ef^{\mathfrak{h}}_{A}\right)\left(\partial_{\lambda}Ef^{\mathfrak{h}}_{B}\right)\right]\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\langle\sigma_{B}^{\mathrm{z}}\rangle_{\rho^{B}_{-\infty}},$
(70)

and we have used that, for any quasi-free state $\omega$,

$\langle\partial_{\lambda}\phi^{\mathfrak{h}}\rangle_{\omega}\equiv{\rm
tr}\left(\rho^{\mathfrak{h}}_{\omega}\partial_{\lambda}\phi^{\mathfrak{h}}\right)=0.$

Note that we have separated the energy variation into three parts. The first
one, $W_{\phi}$, is the contribution to the energy that arises from the
particle creation due to the change in the spacetime metric. It depends only
on the field state and spacetime metric and has nothing to do with the
presence of Alice and Bob. A difficulty we face now is that some sort of
renormalization of the field energy operator $H_{\phi}$ is needed. For this
purpose, we will restrict ourselves to the so-called Hadamard states, for
which a general renormalization procedure is possible wald94 . By noting that
any state that is Hadamard in some open neighborhood of a Cauchy surface is
Hadamard everywhere FSW78 , we can see that the spacetime evolution preserves
the renormalizability of the state. As a result, we can see that, for Hadamard
states, Eq. (68) (and, thus, Eq. (57) as $W_{\phi}$ is the only contribution
to $\Delta E$ where divergences appear) is well-defined and gives finite
results.

The second contribution, $W_{A}+W_{B}$, depends independently on each qubit
interaction with the field. This contribution is due to the work necessary to
switch-on/off each qubit, and it depends on their trajectories, coupling
constants, as well as on spacetime parameters.

The third contribution, $W_{AB}$, measures the extra energy cost arising from
the communication process itself. It depends on the initial state of each
qubit, on the spacetime metric, and on the relative motion between Alice and
Bob. We note that, by integrating by parts and using Eq. (37), we can write

$2\int_{\mathfrak{h}}d\lambda\wedge\epsilon_{\Gamma}\,\left(\partial_{\lambda}Ef^{\mathfrak{h}}_{A}\right)\left(\partial_{\lambda}Ef^{\mathfrak{h}}_{B}\right)=\sigma_{\mathfrak{h}}\left(Ef^{\mathfrak{h}}_{A},\partial_{\lambda}Ef^{\mathfrak{h}}_{B}\right),$
(71)

which, by using Eqs. (49) and (50), enable us to cast Eq. (70) as

$W_{AB}=\left\langle
i\left[\phi^{(\mathfrak{h})}(Ef^{\mathfrak{h}}_{A}),\phi^{(\mathfrak{h})}(\partial_{\lambda}Ef_{B})\right]\right\rangle_{\omega}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\langle\sigma_{B}^{\mathrm{z}}\rangle_{\rho^{B}_{-\infty}}$
(72)

As expected, we can see that $W_{AB}$ vanishes if Alice’s and Bob’s qubits
interact with the field in causally disconnected regions of the spacetime.
More interestingly, we can make the $W_{AB}$ contribution to identically
vanish with a convenient choice of $\rho^{B}_{-\infty}$. Recall that Alice
encodes the information she wants to convey in her qubit’s initial state
$\rho^{A}_{-\infty}$. On the other hand, we are free to choose the initial
state of Bob’s qubit. The choice
$\rho^{B}_{-\infty}\equiv|y+\rangle_{B}{}_{B}\langle y_{+}|$, for example,
leads to $W_{AB}=0$ while it maximizes the channel capacities. This shows that
one can convey arbitrary amounts of information through this quantum channel
without extra energy costs.

##  V Two Paradigmatic Examples

We now illustrate the results presented in the previous sections with two
paradigmatic examples in Minkowski spacetime. Let us begin with the field
quantization, following the steps presented in Sec II. Consider a free and
massless scalar field $\phi$ propagating in the Minkowski spacetime
$(\mathbb{R}^{4},\eta)$. Let $(t,x,y,z)\in\mathbb{R}^{4}$ denote global
inertial Cartesian coordinates and let us denote the spatial coordinates as
$\mathbf{x}\equiv(x,y,z)$. The Klein-Gordon equation is simply

$\Box\phi=0,$ (73)

with $\Box\equiv-\sum_{\mu,\nu}\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$,
$\eta\equiv\sum_{\mu,\nu}\eta_{\mu\nu}dx^{\mu}\otimes dx^{\nu}$, and

$\eta=-dt\otimes dt+dx\otimes dx+dy\otimes dy+dz\otimes dz.$ (74)

Let $\mathcal{S}^{\mathbb{C}}$ be the space of complex solutions of Eq. (73)
with compact-support initial data and consider the antisymmetric bilinear map
(9), which takes the form

$\sigma(\psi_{1},\psi_{2})=\int_{\Sigma_{t=0}}d^{3}\mathbf{x}\left[\psi_{2}\partial_{t}\psi_{1}-\psi_{1}\partial_{t}\psi_{2}\right],$
(75)

where

$\Sigma_{t={\rm cte}}\equiv\left\\{{(t,{\bf x})\in\mathbb{R}^{4}|t={\rm
cte}}\right\\}.$ (76)

We choose as the one-particle Hilbert space $\mathcal{H}$ the space spanned by
the positive frequency parts, with respect to the inertial time $t$, of
solutions in $\mathcal{S}^{\mathbb{C}}$ Cauchy-completed with the norm induced
by the Klein-Gordon inner product (10). One then builds the bosonic Fock space
$\mathfrak{F}_{s}(\mathcal{H})$ as usual, to represent the space of field
states and define the field operators via Eq. (15). This is the standard CCR
representation in Minkowski spacetime associated with inertial observers and
we will refer to its vacuum state, $|0_{M}\rangle$, as the inertial (or
Minkowski) vacuum state.

Using the Green functions of the D’alambertian operator $\Box$, one can show
that the map $E:C^{\infty}_{0}(\mathcal{M})\rightarrow\mathcal{S}$ defined in
Eq. (13) takes the form F89

$Ef(x^{\prime})=\int\epsilon_{\mathcal{M}}\,f(x)\,E(x^{\prime},x)$ (77)

with

$E(x,x^{\prime})\equiv\frac{1}{4\pi|\mathbf{x}-\mathbf{x^{\prime}}|}\left[\phantom{\big{|}}\delta\left(t-t^{\prime}-|\mathbf{x}-\mathbf{x^{\prime}}|\right)\right.\\\
\left.-\,\delta\left(t-t^{\prime}+|\mathbf{x}-\mathbf{x^{\prime}}|\right)\phantom{\big{|}}\right].$
(78)

For later use, it will be useful to consider the standard positive-frequency
modes

$u_{\mathbf{k}}(t,\mathbf{x})\equiv\frac{1}{4\pi^{\frac{3}{2}}|\mathbf{k}|^{\frac{1}{2}}}\,e^{-i|\mathbf{k}|t}e^{i\mathbf{k\cdot
x}}\;,\;\;\;\mathbf{k}\in\mathbb{R}^{3},$ (79)

which comprises a complete basis for the one-particle Hilbert space
$\mathcal{H}$.

Now that we have chosen a representation for the CCR in Minkowski spacetime,
let us analyze the effects of the field state as well as the state of motion
of both Alice and Bob in the communication process.

###  V.1 Inertial sender and receiver

We consider first the following scenario: suppose Alice is at rest at the
origin of our inertial coordinate system and wants to transmit some
information to Bob, which is at rest at the spatial position
$\mathbf{x}=(L,0,0)$ (thus at rest relative to Alice and separated by a
spatial distance $L$). For simplicity, we consider that both are equipped with
pointlike detectors. To avoid divergences, we consider that the interactions
of each qubit with the field are switched-on/off continuously. We have seen
that Eq. (28) carries all the information about the qubit interaction with the
field. Applying it to Alice’s qubit+field interaction gives

$f_{A}(t,\mathbf{x})=\epsilon_{A}c_{A}(t)\delta^{3}(\mathbf{x}),$ (80)

where $\epsilon_{A}$ is a dimensionless coupling constant and

$c_{A}(t)=\begin{cases}e^{\alpha_{A}(t-T_{A}^{\mathrm{i}})},&t<T_{A}^{\mathrm{i}}\\\
1,&T_{A}^{\mathrm{i}}\leq t\leq T_{A}^{\mathrm{f}}\\\
e^{-\alpha_{A}(t-T_{A}^{\mathrm{f}})},&t>T_{A}^{\mathrm{f}}\end{cases}$ (81)

models the switching function. Similarly, the function modeling Bob’s
qubit+field interaction is

$f_{B}(t,\mathbf{x})=\epsilon_{B}c_{B}(t)\delta^{3}(\mathbf{x}-L\mathbf{\hat{x}}),$
(82)

where $\epsilon_{B}$ is a dimensionless coupling constant and $c_{B}(t)$ is
defined as $c_{A}(t)$ but replacing the $A$’s by $B$’s in Eq. (81).

Our goal is to explicitly evaluate the classical channel capacity in Eq. (31)
and analyze its dependence on the various parameters involved in this
communication process. To this end, we first substitute Eq. (82) in Eq. (77)
to write

$Ef_{B}(t,\mathbf{x})=\frac{\epsilon_{B}}{4\pi|\mathbf{x}-L\mathbf{\hat{x}}|}\left[c_{B}\left(t-|\mathbf{x}-L\mathbf{\hat{x}}|\right)\right.\\\
\left.-c_{B}\left(t+|\mathbf{x}-L\mathbf{\hat{x}}|\right)\right].$ (83)

Then, by using Eqs. (80) and (83) in Eq. (17), we can cast the smeared
propagator as

$\Delta(f_{A},f_{B})=\frac{\epsilon_{A}\epsilon_{B}}{4\pi
L}\int_{\mathbb{R}}dt\,c_{A}(t)\left[c_{B}(t-L)-c_{B}(t+L)\right].$ (84)

Now, note that $\nu_{B}$ defined in Eq. (32) depends on Bob’s state of motion
as well as the quantum field state. Let us consider two cases: if the field is
initially in the inertial vacuum state $|0_{M}\rangle$, $\nu_{B}$ is simply

$\nu_{B}=\exp\left[-2\langle KEf_{B},KEf_{B}\rangle\right],$ (85)

where $\langle\;,\;\rangle$ is the Klein-Gordon inner product (10) and we
recall that $K:\mathcal{S}^{\mathbb{C}}\rightarrow\mathcal{H}$ takes the
positive-frequency part of any solution of Eq. (73). Now, if the field is in a
KMS (thermal) state at temperature $\Theta$, then kay1985a

$\nu_{B}=\exp\left[-2\left\langle
KEf_{B},\coth\left(\frac{\beta\hat{h}}{2}\right)KEf_{B}\right\rangle\right],$
(86)

where $\beta\equiv\Theta^{-1}$ is the inverse temperature and
${\hat{h}:\mathcal{H}\rightarrow\mathcal{H}}$ is the one-particle Hamiltonian,
which is given by $\hat{h}=i\partial_{t}$ and satisfies

${H_{\phi}=d\Gamma(\hat{h})\equiv
1\oplus\hat{h}\oplus\left(\hat{h}\otimes\hat{h}\right)\oplus\cdots}\;.$

We note that, in the zero-temperature limit (i.e., $\beta\rightarrow\infty$),
Eq. (86) reduces to Eq. (85), as it should be.

Since the modes $u_{\mathbf{k}}$ defined in Eq. (79) form a complete basis for
$\mathcal{H}$, we can decompose $Ef_{B}$ as

$KEf_{B}=\int d^{3}{\bf k}\;\langle u_{\bf k},Ef_{B}\rangle\;u_{\bf k}$ (87)

and thus, as $\hat{h}$ is diagonal in this basis, we can write

$\left\langle
KEf_{B},\coth\left(\frac{\beta\hat{h}}{2}\right)KEf_{B}\right\rangle\\\ =\int
d^{3}\mathbf{k}\;\coth\left(\frac{\beta|\mathbf{k}|}{2}\right)\left|\langle
u_{\mathbf{k}},Ef_{B}\rangle\right|^{2}.$ (88)

By making use of Eq. (10) and Lemma 3.2.1 of wald94  we can cast the Klein-
Gordon inner product in Eq. (88) as

$\displaystyle\langle u_{\mathbf{k}},Ef_{B}\rangle$
$\displaystyle=i\int_{\mathcal{M}}\epsilon_{\mathcal{M}}\,\overline{u_{\mathbf{k}}(x)}f_{B}(x)$
(89)

which, by using Eq. (79), can be put in the form

$\displaystyle\langle u_{\mathbf{k}},Ef_{B}\rangle$
$\displaystyle=\frac{i\epsilon_{B}}{2^{\frac{3}{2}}\pi|\mathbf{k}|^{\frac{1}{2}}}\,\widetilde{c_{B}}(|\mathbf{k}|)e^{-ik_{x}L},$
(90)

where

$\widetilde{c_{B}}(\omega)\equiv\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}dt\,e^{i\omega
t}c_{B}(t)$ (91)

is the Fourier transform of $c_{B}(t)$. Putting together Eqs. (86), (88), and
(90) we obtain

$\nu_{B}(\Theta)=\exp\left[-\frac{2\epsilon_{B}^{2}}{\pi}\int_{0}^{\infty}dk\,k\,\coth{\left(\frac{k}{2\Theta}\right)}|\tilde{c_{B}}(k)|^{2}\right].$
(92)

We can now use Eqs. (31), (84), and (92) to investigate the classical channel
capacity when sender and receiver are inertial observers at rest relative to
each other. Let us consider that Alice and Bob let their qubits interact with
the quantum field for the same amount of time

$\Delta T\equiv
T_{A}^{\mathrm{f}}-T_{A}^{\mathrm{i}}=T_{B}^{\mathrm{f}}-T_{B}^{\mathrm{i}},$

where, for the sake of simplicity, we have set $T_{A}^{\mathrm{i}}=0$. We note
that by choosing large values of $\alpha_{A},\alpha_{B}$ (i.e.,
$\alpha_{A},\alpha_{B}\gg 1/\Delta T$) in the switching functions
$c_{A}(t),c_{B}(t)$, we can model the case where qubit+field interactions take
place at finite time intervals $\Delta T$. In Fig. 1 we plot Alice and Bob
worldlines for a spatial separation $L$ as well as the regions where the
emission and detection events take place. Note that the emission and detection
events are spacelike separated whenever $T_{B}^{\mathrm{i}}<L-\Delta T$ or
time-like separated whenever $T_{B}^{\mathrm{i}}>L+\Delta T$. As the field is
massless, the channel capacity is expected to be zero in such cases since Bob
cannot intercept any signal emitted by Alice.

$x$$t$$T^{\mathrm{i}}_{B}$$T^{\mathrm{f}}_{B}$$L$ Figure 1: Spacetime diagram
representing Alice’s worldline (red dashed line) and Bob’s worldline (blue
dashed line). The red and blue rectangles represent the regions where their
respective qubits interact with the quantum field. As we are considering a
massless field, the gray region represents the region where signals emitted by
Alice should be present.

Let us begin by analyzing how the coupling constants influence the channel
capacity $C\left(\mathcal{E}\right)$. For this purpose we consider the case
where the quantum field is initially in the inertial vacuum state and, thus,
we are considering the $\beta\rightarrow\infty$ limit in Eq. (92). In Fig. 2,
we plot how $C\left(\mathcal{E}\right)$ varies when one changes the couplings
$\epsilon_{A}$ and $\epsilon_{B}$. For the sake of illustration, we have
considered the case where $T_{B}^{\mathrm{i}}=L=4\,\Delta T$. This guarantees
that Bob’s detection process takes place entirely in the gray region of the
plot. We can see that the channel capacity increases very close to 1 (maximum
efficiency) for large values of $\epsilon_{A}$ but decreases rapidly with the
increase of Bob’s coupling constant $\epsilon_{B}$. This happens, for a fixed
value of $\Delta T,$ because as Alice wants to imprint some information on the
field state, the stronger her interaction with the field, the more efficient
this process will be. On the other hand, Bob’s qubit state is altered when it
is allowed to interact with the field. If the interaction is too strong (or if
it is switched-on for a long period of time), the information encoded in his
qubit state can be lost due to quantum decoherence.

Figure 2: Classical channel capacity as a function of Bob’s and Alice’s
coupling constants, $\epsilon_{A}$ and $\epsilon_{B}$, respectively. Here,
$\alpha_{A}=\alpha_{B}=100\,\Delta T^{-1}$ and $T_{B}^{\mathrm{i}}=L=4\,\Delta
T$. The channel capacity and coupling constants are dimensionless.

Having established how one can tune $\epsilon_{A}$ and $\epsilon_{B}$ to
maximize the channel capacity, let us choose suitable values for the coupling
constants and investigate the communication process for different choices of
the field initial thermal state as well as different causal relations between
Alice emission and Bob measurement events. The results are shown in Fig. 3,
where we plot the channel capacity, $C\left(\mathcal{E}\right),$ as a function
of the time $T^{i}_{B}$ where Bob begins his measurement. In view of our
previous results, we have chosen $\epsilon_{A}=800,$ $\epsilon_{B}=0.05$, and
$L=4\Delta T$. We can see that the channel capacity vanishes if Bob’s qubit
interacts with the field too soon ($T_{B}^{\mathrm{i}}<L-\Delta T=3\,\Delta
T$) or too late ($T_{B}^{\mathrm{i}}>L+\Delta T=5\,\Delta T$), regardless of
the initial field state. One may observe in Fig. 1 that these are the cases
where emission and detection events are spacelike and timelike separated,
respectively, and thus Bob cannot intercept any signal emitted by Alice. On
the other hand, the maximum communication efficiency is reached when
$T_{B}^{\mathrm{i}}=L=4\,\Delta T$, since now Bob is able to intercept every
signal emitted by Alice. Additionally, note how the temperature of the field
state limits the maximum channel capacity. The higher the temperature
$\Theta$, the greater the noise in the quantum channel. This increases the
quantum decoherence in Bob’s qubit state (as the decoherence time decreases)
and, thus, it becomes impossible to achieve high efficiency in the
communication process.

Figure 3: Classical channel capacity as a function of the time
$T^{\mathrm{i}}_{B}$ when Bob starts the measurement process for different
initial thermal states of the field. Each curve represents a different
temperature $\Theta$ of the initial quantum field state (with $\Theta=0$
representing the inertial vacuum state). Here,
$\alpha_{A}=\alpha_{B}=100\,\Delta T^{-1}$, $\epsilon_{A}=800$,
$\epsilon_{B}=0.05$, and $L=4\,\Delta T$.

###  V.2 Inertial sender, accelerated receiver

$x$$t$$x_{0}$ (a)

$x$$t$$x_{0}$ (b)

Figure 4: Spacetime diagram, in Cartesian coordinates $(t,x)$, representing
Alice (red dashed line) and Bob worldline with different accelerations (blue
and orange dashed lines). The solid regions represent emission and detection
events that maximize the channel capacity. The detectors remain switched-on
for the same proper-time interval.

Let us consider now the following scenario: suppose Alice is at rest at the
origin of some inertial Cartesian coordinate system $(t,x,y,z)$ and wants to
transmit information to Bob, which travels uniformly accelerated following the
worldline

$\displaystyle t_{B}(\tau)$ $\displaystyle=a^{-1}\sinh\left(a\tau\right),$
(93) $\displaystyle x_{B}(\tau)$
$\displaystyle=x_{0}+a^{-1}\left[\cosh\left(a\tau\right)-1\right],$
$\displaystyle y_{B}(\tau)$ $\displaystyle=z_{B}(\tau)=0.$

Here, $a$ is Bob proper acceleration, $\tau$ is his proper-time (synchronized
as $\tau=0$ when $t=0$), and $x_{0}$ is the spatial distance between Bob and
Alice (as measured by Alice) at the point of maximum approximation. Both
worldlines are shown in Fig. 4 for two different values of Bob proper
acceleration $a$. The quantum field is supposed to be initially in the
inertial vacuum state $|0_{M}\rangle$ and we consider again that both
observers are equipped with pointlike detectors which are continuously
switched-on/off. The function modeling Alice’s qubit+field interaction remains
the one in Eq. (80).

To discuss Bob’s qubit interaction with the field, let us first introduce
Rindler coordinates $(\tau,\xi,y,z)$, with $\tau,\xi\in\mathbb{R}$ implicitly
defined by

$t=a^{-1}e^{a\xi}\sinh{a\tau}\;,\;\;x=\left(x_{0}-\frac{1}{a}\right)+a^{-1}e^{a\xi}\cosh{a\tau}.\\\
$ (94)

These coordinates cover the right Rindler wedge (RRW), i.e., the region
defined by $\left[x-(x_{0}-1/a)\right]>|t|$, in which the metric takes the
form

$g=e^{2a\xi}(-d\tau\otimes d\tau+d\xi\otimes d\xi)+dy\otimes dy+dz\otimes dz.$
(95)

In such coordinates, Bob remains static at $\xi=y=z=0$ and its qubit+field
interaction is simply described by the function

$f_{B}(\tau,\xi,y,z)=\epsilon_{B}\,c_{B}(\tau)\,\delta(\xi)\,\delta(y)\,\delta(z),$
(96)

where

$c_{B}(\tau)=\begin{cases}e^{\alpha_{B}(\tau-\tau_{B}^{\mathrm{i}})},&\tau<\tau_{B}^{\mathrm{i}}\\\
1,&\tau_{B}^{\mathrm{i}}\leq\tau\leq\tau_{B}^{\mathrm{f}}\;.\\\
e^{-\alpha_{B}(\tau-\tau_{B}^{\mathrm{f}})},&\tau>\tau_{B}^{\mathrm{f}}\end{cases}$
(97)

In order to analyze the channel capacity, we need to evaluate again
$\Delta(f_{A},f_{B})$. By an analogous procedure to the one leading to Eq.
(83), we obtain

$Ef_{A}(t,\mathbf{x})=\frac{\epsilon_{A}}{4\pi|\mathbf{x}|}\left[c_{A}\left(t-|\mathbf{x}|\right)-c_{A}\left(t+|\mathbf{x}|\right)\right].$
(98)

Using Eq. (16), we have

$\displaystyle\Delta(f_{A},f_{B})$ $\displaystyle=-\Delta(f_{B},f_{A})$ (99)
$\displaystyle=-\int_{\mathcal{M}}\epsilon_{\mathcal{M}}\;f_{B}(x)Ef_{A}(x)$

which, in Rindler coordinates (94), can be straightforwardly evaluated giving

$\Delta(f_{A},f_{B})=\frac{\epsilon_{A}\epsilon_{B}}{4\pi}\int_{\mathbb{R}}d\tau\;\frac{c_{B}(\tau)}{|x_{B}(\tau)|}\left\\{c_{A}\left[t_{B}(\tau)+x_{B}(\tau)\right]\right.\\\
\left.-c_{A}\left[t_{B}(\tau)-x_{B}(\tau)\right]\right\\}.$ (100)

Now, we proceed to calculate the quantity $\nu_{B}$ defined in Eq. (32), which
depends on Bob’s state of motion and the initial state of the quantum field.
In order to do so, it will be useful to introduce the so-called Right Rindler
modes defined by

$v_{\omega\mathbf{k_{\perp}}}^{R}\equiv\left[\frac{\sinh(\pi\omega/a)}{4\pi^{4}a}\right]^{1/2}K_{i\omega/a}\left(\frac{|\mathbf{k_{\perp}}|e^{a\xi}}{a}\right)e^{i\mathbf{k_{\perp}\cdot
x_{\perp}}}e^{-i\omega\tau}$ (101)

in the RRW and vanishing in the left Rindler wedge (LRW), which is the region
where $\left[x-(x_{0}-1/a)\right]>-|t|$. Here,
$\mathbf{x_{\perp}}\equiv(y,z)$, $\mathbf{k_{\perp}}\in\mathbb{R}^{2}$,
$\omega>0$, and $K_{\nu}(x)$ is the modified Bessel function. The left Rindler
modes, $v^{L}_{\omega{\bf k}_{\perp}}$, are defined by $v^{L}_{\omega{\bf
k}_{\perp}}(t,x,{\bf x}_{\perp})\equiv\overline{v^{R}_{\omega{\bf
k}_{\perp}}}(-t,-x,{\bf x}_{\perp})$. Hence, they vanish in the RRW and take
the form (101) in Rindler coordinates covering the LRW.

By using $v^{R}_{\omega{\bf k}_{\perp}}$ and $v^{L}_{\omega{\bf k}_{\perp}}$
we can define the so-called Unruh modes

$w^{1}_{\omega\mathbf{k_{\perp}}}\equiv\frac{v_{\omega\mathbf{k_{\perp}}}^{R}+e^{-\pi\omega/a}\,\overline{v_{\omega-\mathbf{k_{\perp}}}^{L}}}{\sqrt{1-e^{-2\pi\omega/a}}},$
(102)
$w^{2}_{\omega\mathbf{k_{\perp}}}\equiv\frac{v_{\omega\mathbf{k_{\perp}}}^{L}+e^{-\pi\omega/a}\,\overline{v_{\omega-\mathbf{k_{\perp}}}^{R}}}{\sqrt{1-e^{-2\pi\omega/a}}},$
(103)

which have purely positive frequency relative to inertial time and comprise a
complete basis for the one-particle space $\mathcal{H}$ defined below Eq.
(76). Therefore, we can write

$\displaystyle\langle
KEf_{B},KEf_{B}\rangle=\int_{\mathbb{R}}d\omega\int_{\mathbb{R}^{2}}d^{2}\mathbf{k_{\perp}}\;\left|\langle
w^{1}_{\omega\mathbf{k_{\perp}}},Ef_{B}\rangle\right|^{2},$ (104)

where we have used that
$w^{1}_{-\omega\mathbf{k_{\perp}}}=w^{2}_{\omega\mathbf{k_{\perp}}}$.
Analogously to Eq. (90), we can write

$\langle
w^{1}_{\omega\mathbf{k_{\perp}}},Ef_{B}\rangle=i\int\epsilon_{\mathcal{M}}\;\overline{w^{1}_{\omega\mathbf{k_{\perp}}}}(x)f_{B}(x)$
(105)

which, by using Eq. (102), gives

$\displaystyle\langle w^{1}_{\omega\mathbf{k_{\perp}}},Ef_{B}\rangle$
$\displaystyle=$
$\displaystyle\frac{i\epsilon_{B}}{\sqrt{1-e^{-2\pi\omega/a}}}\left[\frac{\sinh(\pi\omega/a)}{2\pi^{3}a}\right]^{1/2}$
(106) $\displaystyle\times$ $\displaystyle
e^{-\pi\omega/a}K_{i\omega/a}\left(|\mathbf{k_{\perp}}|/a\right)\widetilde{c_{B}}(\omega),$

Substituting Eqs. (106) and (104) in Eq. (85) and evaluating the transverse,
$\mathbf{k_{\perp}}$, integral by means of the identity

$\int_{0}^{\infty}dxx|K_{i\omega/a}(x)|^{2}=\frac{\omega}{4a\sinh{(\pi\omega/a)}}$
(107)

we obtain

$\nu_{B}=\exp\left[-\frac{2\epsilon_{B}^{2}}{\pi}\int_{0}^{\infty}d\omega\,\omega\coth{\left(\frac{\pi\omega}{a}\right)}|\widetilde{c_{B}}(\omega)|^{2}\right].$
(108)

Now, we can use Eqs. (31), (100), and (108) to investigate the classical
channel capacity $C\left(\mathcal{E}\right)$ for inertial sender and
accelerated receiver. We consider that Alice and Bob let their qubits interact
with the quantum field for the same amount of their respective proper-time,
hence

$\Delta T\equiv
T_{A}^{\mathrm{f}}-T_{A}^{\mathrm{i}}=\tau_{B}^{\mathrm{f}}-\tau_{B}^{\mathrm{i}}.$
(109)

In Fig. 4, we plot their spacetime trajectories as well as regions where
emission and detection events may take place.

Figure 5: Classical channel capacity as a function of the proper time
$\tau_{B}^{\mathrm{i}}$ when Bob starts the measuring process, for different
proper accelerations. Each curve represents one of the situations schematized
in Fig. 4. Here, $\alpha_{A}=\alpha_{B}=100\,\Delta T^{-1}$,
$\epsilon_{A}=420$, $\epsilon_{B}=0.05$ and $x_{0}=2\,\Delta T$. The field is
initially in the inertial vacuum state.

In Fig. 5 we show the behavior of the classical channel capacity as a function
of the (proper) time $\tau^{i}_{B}$ in which bob begins the measurement
process. Let us first consider the case where that Bob’s worldline is the one
in Fig. 4a, where $x_{0}=2\,\Delta T$ and $a=0.1\,\Delta T^{-1}$. In this
case, Bob intercepts the first and final signal emitted by Alice at proper
times $\tau_{1}\simeq 1.05\,\Delta T$ and $\tau_{2}\simeq 2.23\,\Delta T$,
respectively. As it can be seen in Fig. 5, information exchange is possible
only if Bob starts the measurement process at proper-time
$\tau_{B}^{\mathrm{i}}$ satisfying $\tau_{1}-\Delta
T<\tau_{B}^{\mathrm{i}}<\tau_{2}$. When $\tau_{B}^{\mathrm{i}}<\tau_{1}-\Delta
T$ or $\tau_{B}^{\mathrm{i}}>\tau_{2}$, the classical channel capacity
vanishes since the emission and detection events will be spacelike or timelike
separated, respectively.

Let us consider now that Bob’s trajectory is the one depicted in Fig. 4b,
where $x_{0}=2\,\Delta T$ and $a=0.5\,\Delta T^{-1}$. In this case, Bob’s
worldline intercepts the first signal emitted by Alice at proper time
$\tau_{1}^{\prime}\simeq 1.37\,\Delta T$. In Fig. 5, we see that it is exactly
when $\tau_{B}^{\mathrm{i}}=\tau_{1}^{\prime}$ that the maximum channel
capacity is attained. However, since the surface $t=x$ is Bob’s causal
horizon, he never leaves the lightcone section where the information emitted
by Alice is traveling. Thus, the channel capacity never vanishes. It only
decreases as Bob accelerates away from Alice.

###  V.3 Work for switching on/off the detectors

We have shown in Sec. IV that one can use the quantum channel presented here
to convey arbitrary amounts of information without any extra energy cost.
However, some work is necessary to switch-on/off the interaction of each qubit
with the quantum field, which is given by Eq. (69). Let us now estimate this
energy cost for by means of the inertial detectors previously discussed.

For this purpose, let us first relate the ordinary Minkowski QFT construction
presented at the beginning of this section to the null-surface construction
introduced in Sec. III. By recalling that $(t,x,y,z)$ are inertial Cartesian
coordinates where Minkowski metric takes the form given in Eq. (74), let

$u\equiv t-|\mathbf{x}|\;{\rm and}\;v\equiv t+|\mathbf{x}|$ (110)

be the retarded and advanced null coordinates, respectively, where
$\mathbf{x}\equiv(x,y,z)$. By defining

$\tan{V}\equiv v,$ (111)

one can cast Eq. (74) using $u$ and $V$ coordinates as

$\eta=-\frac{1}{2}\sec^{2}V\left(du\otimes dV+dV\otimes du\right)+\frac{(\tan
V-u)^{2}}{4}g_{\mathbb{S}^{2}},$ (112)

where $g_{\mathbb{S}^{2}}$ is the standard metric on a unit 2-sphere.

Now, it is easy to see that Minkowski spacetime is future null asymptotically
flat in the following sense HE : there exists a second spacetime
$(\widehat{\mathcal{M}},\hat{g})$ with metric

$\hat{g}=-\frac{2}{(\sin V-u\cos V)^{2}}\left(du\otimes dV+dV\otimes
du\right)+g_{\mathbb{S}^{2}}$ (113)

such that: (1) $\mathcal{M}$ is conformally embedded in
$\widehat{\mathcal{M}}$ satisfying $\hat{g}=\Omega^{2}\eta$ with a conformal
factor

$\Omega\equiv\frac{2}{\tan V-u};$ (114)

(2) the future null infinity $\mathcal{I}^{+}$ is the 3-dimensional null
hypersurface defined by the set

$\left\\{p\in\widehat{\mathcal{M}}|V=\frac{\pi}{2}\right\\}$

in which the conformal factor satisfies
$\left.\Omega\right|_{\mathcal{I}^{+}}=0$ and
$\left.d\Omega\right|_{\mathcal{I}^{+}}\neq 0$.

The conformal metric $\hat{g}$ restricted to $\mathcal{I}^{+}$ takes the form

$\hat{g}|_{\mathcal{I}^{+}}=-\frac{1}{2}\left(du\otimes d\Omega+d\Omega\otimes
du\right)+g_{\mathbb{S}^{2}},$ (115)

which has the form given in Eq. (35). Thus, we can perform the null
quantization procedure at future null infinity $\mathcal{I}^{+}$ described in
Sec. III with $\lambda=u$.

Now, let us consider the inertial qubit carried by Alice whose interaction
with the field is described by the function
$f_{A}=\epsilon_{A}c_{A}(t)\delta^{3}({\bf x})$ with $c_{A}$ given in Eq.
(80). To compute the energy cost $W_{A}$, given in Eq. (69), to switch-on/off
the qubit, we need to first compute $Ef_{A}^{\mathfrak{h}}$. To this end, we
can use the explicit form of $Ef_{A}$ given in Eq. (98) together with Eq.
(110) to write

$Ef_{A}=\frac{\epsilon_{A}}{4\pi|\mathbf{x}|}\left[c_{A}(u)-c_{A}(v)\right].$
(116)

By using the above equation, the extension $Ef_{A}^{\mathfrak{h}}$ of $Ef_{A}$
to $\mathcal{I}^{+}$ is simply computed using

$Ef_{A}^{\mathfrak{h}}\equiv\lim_{v\rightarrow\infty}\;\Omega^{-1}Ef_{A}$
(117)

and

$|{\bf x}|=\frac{v-u}{2}=\Omega^{-1}$

yielding

$Ef_{A}^{\mathfrak{h}}=\frac{\epsilon_{A}}{4\pi}\,c_{A}(u).$ (118)

By using Eqs. (118) and (80) in Eq. (69) leads to

$\displaystyle W_{A}$
$\displaystyle=\int_{\mathcal{I}^{+}}du\wedge\epsilon_{\mathbb{S}^{2}}\,\left(\frac{\epsilon_{A}}{4\pi}\right)^{2}\left[\partial_{u}c_{A}(u)\right]^{2}$
(119) $\displaystyle=\frac{\epsilon_{A}^{2}\alpha_{A}}{4\pi}.$

We can see that the work necessary to switch-on/off the detector increases
with the coupling strength and it is inversely proportional to the time-scale
$\tau_{\alpha_{A}}\equiv\alpha_{A}^{-1}$ characterizing the switching process.
This gives the usual trade-off between the energy $W_{A}$ of the process and
its characteristic time $\tau_{\alpha_{A}}$:

$W_{A}\tau_{\alpha_{A}}=\frac{\epsilon_{A}^{2}}{4\pi},$ (120)

which shows that the more energy is needed the more rapid is the switching-
on/off of the qubit interaction.

##  VI Conclusions

In the present paper, we have analyzed the energy cost in conveying classical
and quantum information between two arbitrary observers in asymptotically flat
and globally hyperbolic spacetimes (possibly containing black holes) when they
use a quantum scalar field as a communication channel. We have shown that the
energy variation of the total 2-qubits+field system, $\Delta E$, can be cast
as $\Delta E=W_{\phi}+W_{A}+W_{B}+W_{AB}$. This shows that such energy
variations has three may contributions: (1) $W_{\phi}$ which accounts for the
particle creation due to the change of the spacetime geometry; (2)
$W_{A}+W_{B}$ which gives the energy needed to switch-on/off qubits $A$ and
$B$ used by Alice and Bob, respectively, to communicate; (3) $W_{AB}$ which
describes the extra energy cost needed for the communication process. We have
shown that, by suitably choosing Bob’s initial (ready-to-measure) state, the
term $W_{AB}$ vanishes. Such condition is satisfied by the channel
$\mathcal{E}$ considered here.

We have then illustrated the communication process and analyzed how the
classical channel capacity $C\left(\mathcal{E}\right)$ (and, as a result, the
entanglement-assisted classical and quantum capacities as well) behaves in two
paradigmatic examples in Minkowski spacetime: (A) when sender and receiver are
inertial observers and (B) when the sender is inertial and the receiver is
uniformly accelerated. By using example (A), we were able to analyze how the
coupling constants as well as the initial field state influence
$C\left(\mathcal{E}\right)$. Example (B) enabled us to analyze how causal
horizons affects the communication process when the field state is the
inertial vacuum (which, by the Unruh effect, is perceived as a thermal state
with temperature $T_{U}=a/2\pi$ by the uniformly accelerated receiver).

We have ended the paper using the behaviour of Alice inertial qubit in
Minkowski spacetime to estimate the energy cost in switching-on/off the
interaction, i.e., $W_{A}$. We have shown that, as one would expect, $W_{A}$
satisfies the energy-time relation:
$W_{A}\tau_{\alpha_{A}}=\epsilon_{A}^{2}/4\pi$, where $\tau_{\alpha_{A}}$ is
the characteristic time of the switching-on/off process and $\epsilon_{A}$ is
the coupling constant describing Alice’s qubit interaction with the field
$\phi$. Hence, one would expect to spend an amount
$W_{X}\sim\epsilon_{X}^{2}\tau_{\alpha_{X}}^{-1}$, $X=A,B$, of energy in order
to create the qubits and switch them on/off to perform some task. However, if
one has already created the qubits for some purpose (and the energy cost for
it is accounted by $W_{A}+W_{B}$), there is no extra energy cost in using them
to convey information.

###### Acknowledgements.

I. B. and A. L. were fully and partially supported by São Paulo Research
Foundation under Grants 2018/23355-2 and 2017/15084-6, respectively.

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