# A relativistic quantum broadcast channel

Ian Bernardes Barcellos<EMAIL_ADDRESS>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 investigate the transmission of classical and quantum information between
three observers in a general globally hyperbolic spacetime using a quantum
scalar field as a communication channel. We build a model for a quantum
broadcast channel in which one observer (sender) wishes to transmit (classical
and quantum) information to two other observers (receivers). They possess some
localized two-level quantum system (a qubit) that can interact with the
quantum field in order to prepare an input or receive the output of this
channel. The field is supposed to be in an arbitrary quasifree state, the
three observers may be in arbitrary states of motion, and no choice of
representation of the field canonical commutation relations is made. The
interaction of the field and qubits is such that it allows us to obtain the
map that describes this channel in a non-perturbative manner. We conclude by
analyzing the rates at which information can be transmitted through this
channel and by investigating relativistic causality effects on such rates.

###### pacs:

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

##  I Introduction

Network information theory is the area of knowledge that studies classical
communication problems involving multiple parts. Here, the word “classical”
stands not only for the fact that the information being transmitted is classic
(bits) but also for the physical systems in which such information is encoded,
i.e., systems that can be described by some area of classical physics (such as
Electromagnetism). One particular case of interest is the broadcast channel,
where typically one sender wishes to transmit information to multiple
receivers (like radio and TV stations broadcasting their signals, for
example).

Nowadays, one of the main goals of quantum information theory is to extend
several results of information theory to the quantum world [1, 2],
investigating any new features or advantages that can arise when one uses
quantum systems to encode, process, and transmit information. The quantum
network information theory comprises the studies of communication protocols
using quantum systems to convey classical (bits) or quantum (qubits)
information. In particular, the classical broadcast channels can be extended
to the so-called quantum broadcast channels, where one sender transmits
classical or quantum input information to many receivers using a quantum
system as a communication channel with quantum outputs [3, 4, 5].

Such communication scenarios are very suitable for analyzing how relativistic
effects can influence one’s ability to communicate using quantum channels.
This could be due to the existence of nontrivial spacetime structures such as
black hole event horizons, Cauchy horizons, and causal horizons arising from
the relativistic relative motion between senders and receivers or even due to
the expansion of spacetime [6].

In order to consistently analyze quantum information theory in general
spacetimes, one should use quantum field theory in curved spacetimes (QFTCS)
[7]. This approach was used by several authors to analyze the communication
process in relativistic settings, with particular attention being paid to
Minkowski [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], Schwarzschild
[21, 22, 23, 24], or asymptotically flat cosmological spacetimes [25, 26, 27].
However, only recently [28] a communication model valid in general globally
hyperbolic spacetimes and in which the parts that convey information can move
in arbitrary worldlines and interact with the quantum field (used as
communication channel) only in the vicinity of its worldlines was developed
(and, since then, other works in this context have emerged as, for instance,
Ref. [29]). This is interesting for two reasons: firstly, it allows the
analysis of information exchange between more general observers, not only
observers following orbits of some Killing field (which does not even exist in
spacetimes lacking timelike symmetries). Secondly, the model studied in [28]
allows one to investigate the outputs of the quantum communication in a
nonperturbative manner and thereby is suitable to investigate both the
causality as well as the communication between parts lying in early and future
asymptotic regions (limits that would invalidate results obtained by
perturbative methods).

In the present paper, we generalize the analysis of [28]. This is done by
constructing a model for a classical-quantum as well as entanglement-assisted
classical-quantum and quantum-quantum broadcast channels. We consider an
arbitrary globally hyperbolic spacetime in which one observer (Alice) wants to
convey classical (or quantum) information to two receivers (Bob and Charlie)
using a quantum scalar field as a communication channel. The three observers
will use two-level quantum systems (qubits) to locally interact with the
quantum field in order to send or receive information. The observers may be in
arbitrary states of motion, the interaction between the detectors and the
field is similar to the one given by the Unruh-DeWitt model [30], and the
field may initially be in an arbitrary quasifree state [7]. We suppose,
however, that the two levels of each qubit have the same energy. This model is
interesting because the evolution of the system can be computed exactly, and
therefore we will obtain nonperturbative results for the communication rates
associated with such a broadcast channel. As we will see, causality in the
information exchange is explicitly manifest in our results.

This work is organized as follows. In Sec. II we will present the quantization
procedure of a free scalar field on a globally hyperbolic spacetime as well as
the class of states we will be using. In Sec. III we describe the interaction
between the qubits and the field and determine the quantum map that relates
the information Alice wants to convey to the final joint state of Bob’s and
Charlie’s qubits. In Sec. IV we investigate the rates at which information can
be transmitted using this broadcast channel, as well as the influence of the
spacetime curvature or relative motion of observers in the communication
process. In Sec. V we give our final remarks. We assume metric signature
$(-+++)$ and natural units in which $c=\hbar=G=k_{B}=1$, unless stated
otherwise.

##  II Field Quantization

Let us consider a free, real scalar field $\phi$ propagating in an arbitrary
four-dimensional globally hyperbolic spacetime $\left(\mathcal{M},g\right)$,
where $\mathcal{M}$ denotes the four-dimensional spacetime manifold and $g$
its Lorentzian metric. Let the spacetime 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
$\\{x^{\mu}\\}$. 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)

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

with the notation $\dot{\phi}\equiv\partial_{t}\phi$. 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: (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 space;333After its completion with
respect to the norm induced by $\langle,\rangle$. (iii) given any
$u\in\mathcal{H}$ and $v\in\overline{\mathcal{H}}$, $\langle u,v\rangle=0$.
(See [7] for details.) 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 $\\{u_{j}\\}$ comprise an orthonormal basis for $\mathcal{H}$ and
$a(\overline{u})$/$a^{\dagger}(v)$ are the usual annihilation/creation
operators associated with the modes $u$/$v$, respectively. They satisfy the
commutation relations

$\left[a(\overline{u}),a^{\dagger}(v)\right]=\langle u,v\rangle I,$ (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 and
$K:\mathcal{S}\rightarrow\mathcal{H}$ be the projection operator that takes
the positive-norm part of any $\psi\in\mathcal{S}$. If
$C^{\infty}_{0}(\mathcal{M})$ denote the set of all smooth compactly-supported
real functions on $\mathcal{M}$, we define the map
$E:C^{\infty}_{0}(\mathcal{M})\rightarrow\mathcal{S}$ acting on some test
function $f\in C^{\infty}_{0}(\mathcal{M})$ as

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

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

Now, 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 satisfies 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)

for all $f_{1},f_{2}\in C^{\infty}_{0}(\mathcal{M})$. As shown in [7], Eq.
(15) can be obtained by formally integrating Eq. (11) weighed by 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) listed below Eq. (10) and
their respective Fock spaces are, in general, unitarily inequivalent. As
discussed in [28], this issue can be avoided through the algebraic approach to
quantum field theory (QFT). For more details, see Refs. [7, 31].

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

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

for all $f\in C^{\infty}_{0}(\mathcal{M})$, where the so-called Weyl operators
$W(Ef)$ are defined by

$W(Ef)\equiv e^{i\phi(f)}\,,\,\,f\in C^{\infty}_{0}(\mathcal{M}).$ (21)

The vacuum, n-particle, and thermal states are examples of quasifree states.

##  III The quantum broadcast channel

A typical broadcast communication scenario involves the transmission of
information between one station (sender) and several receivers who will decode
the information independently. Let us consider a model in which one observer,
Alice, wants to transmit separate information to two other observers, Bob and
Charlie, using the quantum field $\phi$ as a broadcast channel. Suppose that
the field is initially in some quasifree state $\omega_{\mu}$444We note,
however, that the results from this section apply to any algebraic state
$\omega$ which satisfies $\omega\left[W(Ef)\right]\in\mathbb{R}^{+}$ for all
$f\in C^{\infty}_{0}(\mathcal{M})$.. Suppose also that the three observers
follow arbitrary trajectories in the curved spacetime and that each one of
them possesses a two-level quantum system that may interact with the quantum
field at their will. The two-dimensional Hilbert spaces associated with
Alice’s, Bob’s, and Charlie’s qubits are denoted by $\mathcal{H}_{A}$,
$\mathcal{H}_{B}$, and $\mathcal{H}_{C}$, respectively.

Figure 1: The Figure depicts the quantum broadcast protocol being used. The
dashed lines display the worldlines of the sender, Alice (A, Red), and
receivers, Bob and Charlie (B and C, blue). The solid lines in each worldline
depict the interaction interval of each observer’s qubit with the quantum
field. Here, $\Sigma_{t_{1}}$ and $\Sigma_{t_{2}}$ represent two Cauchy
surfaces of the spacetime.

The communication setup, illustrated by Fig. 1, is as follows: In order to
transmit information to Bob and Charlie, Alice prepares her qubit in some
initial quantum state $\rho_{-\infty}^{A}$ and switches on its interaction
with the field for a finite time interval $\Delta t_{A}$ (measured by the
parameter $t$). To measure the information imprinted by Alice on the field’s
state, Bob and Charlie initially prepare their qubits in suitable states
$\rho_{-\infty}^{B}$ and $\rho_{-\infty}^{C}$ and then they switch on each of
their qubit interaction with the field for finite time intervals $\Delta
t_{B}$ and $\Delta t_{C}$, respectively. For the sake of simplicity, we will
consider here the case where

  * (QB1)

Bob lets his qubit interact with the field only after Alice finishes her
transmission;

  * (QB2)

Charlie lets his qubit interact with the field only after Bob finishes his
measurement process.

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 each qubit 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 $j\in\\{A,B,C\\}$, with $A$, $B$, and $C$ labeling Alice’s, Bob’s, and
Charlie’s 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 $\psi_{j}|_{\Sigma_{t}}\in C_{0}^{\infty}\left(\Sigma_{t}\right)$
for all $t$, which models the finite range of interaction between 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 qubit $j$ with the
field. Each coupling function has support

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

where $T_{j}^{i}$ and $T_{j}^{f}$ 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 $\Delta
t_{j}\equiv{T_{j}^{f}-T_{j}^{i}}$. Thus, the hypotheses (QB1) and (QB2)
previously listed can be summarized as

$T^{i}_{C}\geq T^{f}_{B}\geq T^{i}_{B}\geq T^{f}_{A}.$ (25)

The interaction between each qubit and the field given by Eq. (23) is very
similar to the Unruh-DeWitt model [30]. However, we assumed that the two
levels of each qubit have the same (zero) energy. As we shall see, 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 $j$ in $z$-direction
by adding $H_{j}=\delta_{j}\sigma_{j}^{\mathrm{z}}$ to the total Hamiltonian
in Eq. (22) and still keep the model exactly solvable. This would change it to

$H=H_{\phi}+H_{A}+H_{B}+H_{C}+H_{\mathrm{int}},$ (26)

but would keep the interaction Hamiltonian in the interaction picture, Eq.
(23), unchanged. Hence, all the results we will describe below would remain
the same.

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

$U\equiv
T\exp\left[-i\int_{-\infty}^{\infty}dt\,H^{\mathrm{I}}_{\mathrm{int}}(t)\right].$
(27)

It can be computed nonperturbatively by using the Magnus expansion [32]

$U=\exp{\left[\sum_{n=1}^{\infty}\Omega_{n}\right]},$ (28)

where

$\Omega_{1}=-i\int_{-\infty}^{\infty}dt\,H_{\mathrm{int}}^{\mathrm{I}}(t)\;,$
(29)
$\Omega_{2}=-\frac{1}{2}\int_{-\infty}^{\infty}dt\int_{-\infty}^{t}dt^{\prime}[H_{\mathrm{int}}^{\mathrm{I}}(t)\,,\,H_{\mathrm{int}}^{\mathrm{I}}(t^{\prime})],$
(30) $\displaystyle\\!\\!\\!\Omega_{3}=$
$\displaystyle\frac{i}{6}\int_{-\infty}^{\infty}\\!\\!dt\int_{-\infty}^{t}\\!\\!dt^{\prime}\int_{-\infty}^{t^{\prime}}\\!\\!dt^{\prime\prime}\\{[H_{\mathrm{int}}^{\mathrm{I}}(t),[H_{\mathrm{int}}^{\mathrm{I}}(t^{\prime}),H_{\mathrm{int}}^{\mathrm{I}}(t^{\prime\prime})]]$
$\displaystyle+[H_{\mathrm{int}}^{\mathrm{I}}(t^{\prime\prime}),[H_{\mathrm{int}}^{\mathrm{I}}(t^{\prime}),H_{\mathrm{int}}^{\mathrm{I}}(t)]]\\},$
(31)

and so on. By using Eqs. (18), (23), and (29), we get

$\Omega_{1}=-i\sum_{j}\phi(f_{j})\otimes\sigma_{j}^{\mathrm{z}},$ (32)

where we have defined

$f_{j}(t,\mathbf{x})\equiv\epsilon_{j}(t)\psi_{j}(t,\mathbf{x}).$ (33)

Now, by making use of Eqs. (18) and (23) together with Eqs. (16), (25), and
(30) we can cast $\Omega_{2}$ as

$\displaystyle\Omega_{2}=$ $\displaystyle\,i\Xi
I-\frac{i}{2}\Delta(f_{A},f_{B})\sigma_{A}^{\mathrm{z}}\otimes\sigma_{B}^{\mathrm{z}}-\frac{i}{2}\Delta(f_{A},f_{C})\sigma_{A}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}$
$\displaystyle-\frac{i}{2}\Delta(f_{B},f_{C})\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}},$
(34)

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). Finally, since
$[H_{\mathrm{int}}^{\mathrm{I}}(t),H_{\mathrm{int}}^{\mathrm{I}}(t^{\prime})]$
is proportional to the identity, we get

$\Omega_{k}=0\;\;\;\mathrm{for}\;k\geq 3.$ (35)

Using the Zassenhaus formula

$e^{A+B}=e^{A}e^{B}e^{-\frac{1}{2}[A,B]},$ (36)

valid whenever $[A,B]$ is a proportional to the identity, together with Eqs.
(28), (32), (III), and (35) we obtain the following unitary evolution
operator:

$U=e^{i\Xi}e^{-i\phi(f_{C})\otimes\sigma_{C}^{\mathrm{z}}}e^{-i\phi(f_{B})\otimes\sigma_{B}^{\mathrm{z}}}e^{-i\phi(f_{A})\otimes\sigma_{A}^{\mathrm{z}}}.$
(37)

Now that we have the exact evolution operator $U$, we can use it to evolve the
initial state of the 3 qubit + field system and then trace out the field and
Alice’s qubit degrees of freedom. This procedure allows us to obtain the final
state of Bob’s and Charlie’s qubits after the communication protocol has
ended. This is the state that they will measure to recover the information
that Alice has sent. Explicitly, the final Bob+Charlie state is given by

$\rho^{BC}\equiv\mathrm{tr}_{\phi,A}\left(U\rho^{A}_{-\infty}\otimes\rho^{B}_{-\infty}\otimes\rho^{C}_{-\infty}\otimes\rho_{\omega}U^{\dagger}\right),$
(38)

where $\rho^{j}_{-\infty}$ and $\rho_{\omega}$ are the initial states of qubit
$j$ and the field, respectively.

To compute the trace in Eq. (38), let us cast the operators in Eq. (37) as

$e^{-i\phi(f_{j})\otimes\sigma_{j}^{\mathrm{z}}}=\cos{[\phi(f_{j})]}-i\sin{[\phi(f_{j})]}\otimes\sigma_{j}^{\mathrm{z}},$
(39)

where

$\cos{[\phi(f_{j})]}\equiv\frac{1}{2}\left[W(Ef_{j})+W(-Ef_{j})\right]$ (40)

and

$\sin{[\phi(f_{j})]}\equiv\frac{1}{2i}\left[W(Ef_{j})-W(-Ef_{j})\right],$ (41)

where $W(Ef)$ is defined in Eq. (21). By plugging Eqs. (37) and (39) into Eq.
(38) and then taking the partial traces on $\phi$ and $A$, a direct
calculation yields

$\displaystyle\rho^{BC}$
$\displaystyle=(\Gamma_{cccccc}+\Gamma_{sccccs})\rho^{BC}_{-\infty}$
$\displaystyle+(\Gamma_{csccsc}+\Gamma_{ssccss})\sigma_{B}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}$
$\displaystyle+(\Gamma_{ccsscc}+\Gamma_{scsscs})\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{C}^{\mathrm{z}}$
$\displaystyle+(\Gamma_{cssssc}+\Gamma_{ssssss})\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}$
$\displaystyle+[(\Gamma_{ccscsc}+\Gamma_{scscss})\sigma_{B}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{C}^{\mathrm{z}}+\mathrm{h.c.}]$
$\displaystyle-[(\Gamma_{cssccc}+\Gamma_{sssccs})\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}+\mathrm{h.c.}]$
(42)
$\displaystyle+[(\Gamma_{cscccs}-\Gamma_{sscccc})\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}+\mathrm{h.c.}]$
$\displaystyle+[(\Gamma_{ccsccs}-\Gamma_{scsccc})\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\rho^{BC}_{-\infty}\sigma_{C}^{\mathrm{z}}+\mathrm{h.c.}]$
$\displaystyle+[(\Gamma_{csscss}-\Gamma_{ssscsc})\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\sigma_{B}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}+\mathrm{h.c.}]$
$\displaystyle+[(\Gamma_{cssscs}-\Gamma_{sssscc})\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}+\mathrm{h.c.}],$

where $\mathrm{h.c.}$ stands for Hermitian conjugation, and we have defined

$\rho^{BC}_{-\infty}\equiv\rho_{-\infty}^{B}\otimes\rho_{-\infty}^{C},$ (43)
$\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\equiv\mathrm{tr}\left(\sigma^{\mathrm{z}}_{A}\rho^{A}_{-\infty}\right),$
(44)

and

$\displaystyle\Gamma_{\alpha\beta\gamma\delta\epsilon\zeta}\equiv\omega_{\mu}\big{(}$
$\displaystyle\mathcal{F}_{\alpha}[\phi(f_{A})]\mathcal{F}_{\beta}[\phi(f_{B})]\mathcal{F}_{\gamma}[\phi(f_{C})]$
$\displaystyle\times\mathcal{F}_{\delta}[\phi(f_{C})]\mathcal{F}_{\epsilon}[\phi(f_{B})]\mathcal{F}_{\zeta}[\phi(f_{A})]\big{)},$
(45)

with $\alpha,\beta,\gamma,\delta,\epsilon,\zeta\in\\{c,s\\}$,
$\mathcal{F}_{c}(x)\equiv\cos x,$ and $\mathcal{F}_{s}(x)\equiv\sin x$. We
note that we have written the algebraic field state $\omega_{\mu}$ as a
density matrix with
$\mathrm{tr}\left[\rho_{\omega}W(Ef)\right]\equiv\omega_{\mu}\left[W(Ef)\right]$.
Furthermore, we have used the fact that the expected value of odd functions of
the field operator vanishes since we are assuming that $\omega_{\mu}$ is a
quasifree state (a consequence of Wick’s theorem).

Now, each $\Gamma_{\alpha\beta\gamma\delta\epsilon\zeta}$ in Eq. (III) can be
evaluated by substituting Eqs. (40) and (41) in Eq. (III) and then using the
identity

$W(Ef_{1})W(Ef_{2})=e^{\frac{i}{2}\Delta(f_{1},f_{2})}W[E(f_{1}+f_{2})],$ (46)

for all $f,f_{1},f_{2}\in C^{\infty}_{0}(\mathcal{M})$, to simplify the
product of the Weyl operators. By substituting these coefficients in Eq.(III)
one finds the explicit form of the state $\rho^{BC}$, which is given in Eq.
(A) of Appendix A. The expression in Eq. (A) allows one to write the final
joint state for Bob’s and Charlie’s qubits given any initial state
configuration for the 3 qubits+field.

To define a quantum broadcast channel, we must choose suitable initial states
for Bob and Charlie qubits in order to obtain a quantum map relating the
initial state of Alice’s qubit $\rho^{A}_{-\infty}$ (which encodes the
messages) to the final states that will be probed by them (to decode the
messages). Since Bob only performs measurements in his own two-level system,
we calculate the expression for the reduced state of his qubit, i.e.,

$\rho^{B}\equiv\mathrm{tr}_{C}\left(\rho^{BC}\right).$ (47)

Taking the trace in Eq. (A) relative to Charlie’s degrees of freedom, we
obtain

$\displaystyle\rho^{B}$
$\displaystyle=\frac{1}{2}\left(1+\nu_{B}\cos{[2\Delta(f_{A},f_{B})]}\right)\rho^{B}_{-\infty}$
$\displaystyle+\frac{1}{2}\left(1-\nu_{B}\cos{[2\Delta(f_{A},f_{B})]}\right)\sigma_{B}^{\mathrm{z}}\rho^{B}_{-\infty}\sigma_{B}^{\mathrm{z}}$
(48)
$\displaystyle+\frac{i}{2}\nu_{B}\sin{[2\Delta(f_{A},f_{B})]}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho_{-\infty}^{A}}\left[\rho^{B}_{-\infty},\sigma_{B}^{\mathrm{z}}\right],$

where

$\nu_{B}\equiv\omega_{\mu}\left(W[E(2f_{B})]\right)=e^{-2\mu(KEf_{B},KEf_{B})},$
(49)

with $\mu$ be the inner product associated with the field quasifree state
$\omega_{\mu}$ as in Eq. (20). Note that it is the last term in Eq. (III) that
contains the information encoded by Alice, and thus it will be useless for Bob
to choose the eigenstates $|0\rangle_{B}$ and $|1\rangle_{B}$ of
$\sigma^{\mathrm{z}}_{B}$ as his initial state $\rho^{B}_{-\infty}$ since this
term would vanish. Furthermore, since $\sigma^{\mathrm{z}}_{B}$ commutes with
the interaction Hamiltonian, he won’t recover any information either if he
performs projective measurements on this basis. To choose a suitable state
$\rho^{B}_{-\infty}$ that maximizes the chances of success in their
communication, suppose for simplicity that Alice encodes a pair of messages in
states $\rho^{A}_{-\infty+}$ and $\rho^{A}_{-\infty-}$ which will be decoded
by Bob using a set of projective measurements in the $x$-direction,

$\\{F^{B}_{+}\equiv|+\rangle_{B}{}_{B}\langle+|,F^{B}_{-}\equiv|-\rangle_{B}{}_{B}\langle-|\\},$
(50)

where $\sigma^{\mathrm{x}}_{B}|\pm\rangle_{B}=\pm|\pm\rangle_{B}$. From Eq.
(III), we conclude that the probability that Bob measures $l=\pm$ given that
Alice has encoded the message $k=\pm$ in $\rho^{A}_{-\infty k}$ is

$p(l|k)\equiv\mathrm{tr}\left(F^{B}_{l}\rho^{B}_{k}\right)=\frac{1}{2}(1+l\nu_{B}\Lambda_{k}),$
(51)

where

$\\!\\!\\!\Lambda_{k}\\!\equiv\\!2\mathfrak{R}\\{\beta_{B}(\cos[2\Delta(f_{A},f_{B})]-i\langle\sigma^{\mathrm{z}}_{A}\rangle_{\rho^{A}_{-\infty
k}}\\!\\!\\!\sin[2\Delta(f_{A},f_{B})])\\}$

and $\beta_{B}\equiv{}_{B}\langle 0|\rho^{B}_{-\infty}|1\rangle_{B}$. From
these two equations, we see that it is the second term $\Lambda_{k}$ that
contains the information encoded by Alice on her qubit state, and thus we are
motivated to choose a state $\rho^{B}_{-\infty}$ that makes $\beta_{B}$ a pure
imaginary number, which will make the first term of $\Lambda_{k}$ vanish while
maximizing the amplitude of the second term. This motivates us to choose

$\rho^{B}_{-\infty}\equiv|y_{+}\rangle_{B}{}_{B}\langle y_{+}|,$ (52)

where

$|y_{+}\rangle_{B}\equiv\frac{1}{\sqrt{2}}\left(|0\rangle_{B}+i|1\rangle_{B}\right)$
(53)

is an eigenvalue of $\sigma^{\mathrm{y}}_{B}$ (in this case,
$\beta_{B}=-i/2$). With this choice, we can write Eq. (51) as

$p(l|k)=\frac{1}{2}(1-l\nu_{B}\langle\sigma^{\mathrm{z}}_{A}\rangle_{\rho^{A}_{-\infty
k}}\sin[2\Delta(f_{A},f_{B})]).$ (54)

Now we turn our attention to Charlie. The final reduced state for his qubit is

$\rho^{C}\equiv\mathrm{tr}_{B}\left(\rho^{BC}\right).$ (55)

Taking the trace in Eq. (A) relative to Bob’s degrees of freedom and using Eq.
(52) we obtain

$\displaystyle\rho^{C}$
$\displaystyle=\frac{1}{2}(1+\nu_{C}\cos{[2\Delta(f_{A},f_{C})]}\\!\cos{[2\Delta(f_{B},f_{C})]})\rho^{C}_{-\infty}$
$\displaystyle+\frac{1}{2}$
$\displaystyle(1-\nu_{C}\cos{[2\Delta(f_{A},f_{C})]}\\!\cos{[2\Delta(f_{B},f_{C})]})\sigma_{C}^{\mathrm{z}}\rho^{C}_{-\infty}\sigma_{C}^{\mathrm{z}}$
$\displaystyle+\frac{i}{2}$
$\displaystyle\nu_{C}\\!\sin{[2\Delta(f_{A},f_{C})]}\\!\cos{[2\Delta(f_{B},f_{C})]}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho_{-\infty}^{A}}\\![\rho^{C}_{-\infty},\sigma_{C}^{\mathrm{z}}],$
(56)

where

$\nu_{C}\equiv\omega_{\mu}\left(W[E(2f_{C})]\right)=e^{-2\mu(KEf_{C},KEf_{C})}.$
(57)

To obtain Eq. (III), we explicitly used the choice in Eq. (52), which implies
that
$\langle\sigma_{B}^{\mathrm{z}}\rangle_{\rho^{B}_{-\infty}}\equiv\mathrm{tr}\left(\sigma^{\mathrm{z}}_{B}\rho^{B}_{-\infty}\right)=0$.
By a completely similar reasoning as the one used to choose Bob’s initial
state, we are motivated to choose Charlie’s initial qubit state as

$\rho^{C}_{-\infty}\equiv|y_{+}\rangle_{C}{}_{C}\langle y_{+}|,$ (58)

where $\sigma^{\mathrm{y}}_{C}|y_{+}\rangle_{C}=|y_{+}\rangle_{C}$.

Now, the quantum broadcast channel is completely characterized by a linear,
completely positive and trace-preserving (CPTP) quantum map $\mathcal{E}$
which takes $\rho^{A}_{-\infty}$ into a final state $\rho^{BC}$, i.e.,

$\rho^{BC}=\mathcal{E}(\rho^{A}_{-\infty}).$ (59)

By substituting the initial states of Bob’s and Charlie’s qubits given in Eqs.
(52) and (58) into Eq. (A), we find the explicit expression for the quantum
broadcast channel $\mathcal{E}$. For the sake of clarity, due to its lengthy
expression, we write its explicit form in Eq. (A) of Appendix A.

For later use, we will denote the reduced channels
$\mathcal{E}_{B}:A\rightarrow B$, $\mathcal{E}_{C}:A\rightarrow C$ by

$\displaystyle\mathcal{E}_{B}(\rho^{A}_{-\infty})$
$\displaystyle\equiv\mathrm{tr}_{C}\left[\mathcal{E}(\rho^{A}_{-\infty})\right],$
(60) $\displaystyle\mathcal{E}_{C}(\rho^{A}_{-\infty})$
$\displaystyle\equiv\mathrm{tr}_{B}\left[\mathcal{E}(\rho^{A}_{-\infty})\right],$
(61)

respectively. It then follows from Eqs. (A), (60), and (61) that they can be
explicitly written as

$\displaystyle\mathcal{E}_{B}(\rho^{A}_{-\infty})$
$\displaystyle=\frac{1}{2}I_{B}+\frac{\nu_{B}}{2}\cos{[2\Delta(f_{A},f_{B})]}\sigma^{\mathrm{y}}_{B}$
$\displaystyle-\frac{\nu_{B}}{2}\sin{[2\Delta(f_{A},f_{B})]}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho_{-\infty}^{A}}\sigma^{\mathrm{x}}_{B}$
(62)

and

$\displaystyle\mathcal{E}_{C}($
$\displaystyle\rho^{A}_{-\infty})=\frac{1}{2}I_{C}$
$\displaystyle+\frac{\nu_{C}}{2}\cos{[2\Delta(f_{A},f_{C})]}\\!\cos{[2\Delta(f_{B},f_{C})]}\sigma^{\mathrm{y}}_{C}$
(63)
$\displaystyle-\frac{\nu_{C}}{2}\sin{[2\Delta(f_{A},f_{C})]}\\!\cos{[2\Delta(f_{B},f_{C})]}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho_{-\infty}^{A}}\\!\sigma^{\mathrm{x}}_{C}.$

Given an initial state $\rho^{A}_{-\infty}$ prepared by Alice on her qubit,
these expressions for $\mathcal{E}_{B}$ and $\mathcal{E}_{C}$ determine the
final local states of Bob’s and Charlie’s qubit, respectively.

##  IV Achievable communication rates

Now that we have constructed a model for a relativistic quantum broadcast
channel, we can investigate at which rates classical and quantum information
can be reliably transmitted by Alice to Bob and Charlie. We first review a few
protocols for quantum broadcast communication published in the literature and
then we investigate the achievable rates for our quantum broadcast channel
$\mathcal{E}$ defined in Eq. (59).

###  IV.1 Unassisted classical communication

Let us begin with the investigation of unassisted transmission of classical
information. We follow the protocol present in [3], where more details can be
found. We evaluate achievable rates for our model and then we discuss how
causality is explicitly manifest in our results.

Suppose Alice wishes to transmit a common message $m\in M$ intended for both
receivers while sending additional personal messages $m_{B}\in M_{B}$ and
$m_{C}\in M_{C}$ intended for Bob and Charlie, respectively. Each message is
chosen from one of the following sets,

$M=\\{1,\cdots,|M|\\}\;,\;\;M_{j}=\\{1,\cdots,|M_{j}|\\},$ (64)

with $j\in\\{B,C\\}$ and $|M|$ denoting the cardinality of $M$. Since the
broadcast channel $\mathcal{E}$ is noisy, Alice needs to do a suitable block
coding on the possible messages and then make $n$ independent uses of the
channel in order to be able to reliably convey the information. More
precisely, Alice maps each message triple $(m_{B},m,m_{C})$ to a codeword
$x^{n}(m_{B},m,m_{C})$ which is then associated with a quantum state
$\rho_{x^{n}(m_{B},m,m_{C})}^{A_{n}}$ defined in the space
$\mathcal{H}_{A}^{\otimes n}$. Then, she transmits
$\rho_{x^{n}(m_{B},m,m_{C})}^{A_{n}}$ by making $n$ independent uses of the
channel $\mathcal{E}$. The output of the channel is the state

$\rho_{x^{n}(m_{B},m,m_{C})}^{B_{n}C_{n}}\equiv\mathcal{E}^{\otimes
n}\left(\rho_{x^{n}(m_{B},m,m_{C})}^{A_{n}}\right)$ (65)

defined on $\mathcal{H}_{B}^{\otimes n}\otimes\mathcal{H}_{C}^{\otimes n}$. To
decode the message, Bob chooses a positive-operator valued measure (POVM)
$\\{F^{B_{n}}_{m_{B},m}\,|\,(m_{B},m)\in M_{B}\times M\\}$ which acts on the
system $B_{n}$. Similarly, Charlie chooses a POVM
$\\{G^{C_{n}}_{m,m_{C}}\,|\,(m,m_{C})\in M\times M_{C}\\}$ which acts on the
system $C_{n}$. We say that an error has occurred when at least one message is
incorrectly decoded. Hence, the error probability associated with the
transmission of the triple $(m_{B},m,m_{C})$ is

$p_{e}(m_{B},m,m_{C})\equiv 1-\mathrm{tr}\left[\\!(F^{B_{n}}_{m_{B},m}\otimes
G^{C_{n}}_{m,m_{C}})\rho_{x^{n}(m_{B},m,m_{C})}^{B_{n}C_{n}}\\!\right].$

The transmission rates associated with each message are defined as

$R\equiv\frac{1}{n}\log_{2}|M|\;,\;\;R_{j}\equiv\frac{1}{n}\log_{2}|M_{j}|.$
(66)

These rates essentially measure how many bits of classical information are
sent per channel use. If, given an $\epsilon>0$, the average probability of
error $\overline{p}_{e}$ is bounded by $\epsilon$, i.e.,

$\overline{p}_{e}\equiv\frac{1}{|M_{B}||M||M_{C}|}\sum_{m_{B},m,m_{C}}p_{e}(m_{B},m,m_{C})\leq\epsilon,$
(67)

the classical-quantum broadcast channel coding protocol described above is
said to be a $(n,R_{B},R,R_{C},\epsilon)$ code. We say that a rate triple
$(R_{B},R,R_{C})$ is achievable if given $\epsilon,\delta>0$ there exists a
$(n,R_{B}-\delta,R-\delta,R_{C}-\delta,\epsilon)$ code for sufficiently large
$n$. Hence, saying that a rate triple is achievable means that classical
information can be reliably transmitted at rates arbitrarily close to them.

The achievable rates depend highly on the coding and decoding techniques
chosen by the sender and receivers. The best known achievable rate region for
general broadcast channels is attained through the so-called Marton coding
scheme. Following [3], we investigate here the quantum version of this
protocol.

Suppose for simplicity that no common message is meant to be sent, i.e., let
us consider a $(R_{B},0,R_{C})$ quantum broadcast channel. In this scenario,
one strategy they can use is the Marton coding scheme, where one chooses two
correlated random variables $U$ and $V$, with joint probability distribution
denoted by $p$ and reduced probability distributions denoted by $p_{U}$ and
$p_{V}$. Such a pair of random variables is usually referred to as binning
variables. Then, for each $m_{B}\in M_{B}$ and $m_{C}\in M_{C}$, one generates
codewords $u^{n}(m_{B})$ and $v^{n}(m_{C})$ according to the reduced
probability distributions $p_{U}(u)$ and $p_{V}(v)$. Next, the codewords are
mixed together into a single codeword $x^{n}(m_{B},m_{C})$ according to a
deterministic function $x=f(u,v)$. With this approach, it follows that a rate
pair $(R_{B},R_{C})$ is achievable if it satisfies [3]

$\displaystyle 0\leq R_{B}$ $\displaystyle\leq I(U;B)_{\sigma},$ (68)
$\displaystyle 0\leq R_{C}$ $\displaystyle\leq I(V;C)_{\sigma},$ (69)
$\displaystyle R_{B}+R_{C}$ $\displaystyle\leq
I(U;B)_{\sigma}+I(V;C)_{\sigma}-I(U,V)_{\sigma},$ (70)

where

$I(X;Y)_{\rho}\equiv S(X)+S(Y)-S({XY})$ (71)

is the mutual information of a state $\rho^{XY}$, with

$S(\alpha)=-{\rm tr}\left(\rho^{\alpha}\log\rho^{\alpha}\right),$

$\alpha=X,Y$, being the von Neumann entropy of $\rho^{\alpha}$, $\alpha=X,Y$.
Here, $\rho^{X}={\rm tr}_{Y}\rho^{XY}$ and $\rho^{Y}={\rm tr}_{X}\rho^{XY}$.
The states $\sigma$ in Eqs. (68)-(70) are obtained by suitably (partially)
tracing out the degrees of freedom of the density matrix

$\sigma^{UVBC}\equiv\sum_{u,v}p(u,v)|u\rangle\langle
u|^{U}\otimes|v\rangle\langle
v|^{V}\otimes\mathcal{E}\left(\rho^{A}_{f(u,v)}\right),$ (72)

with $p(u,v)$ being the joint probability distribution of the random variables
$U$ and $V$.

We begin our analysis by deriving bounds for the achievable rates through the
Marton coding scheme applied to our relativistic quantum broadcast channel. To
evaluate Eq. (68), we take partial traces relative to $V$ and $C$ in Eq. (72),
obtaining

$\sigma^{UB}\equiv\sum_{u}p_{U}(u)|u\rangle\langle
u|^{U}\otimes\omega^{B}_{u},$ (73)

where we have written $p(u,v)=p_{V|U}(v|u)p_{U}(u)$, whereas

$\omega^{B}_{u}\equiv\sum_{v}p_{V|U}(v|u)\mathcal{E}_{B}\left(\rho^{A}_{f(u,v)}\right).$
(74)

A state like $\sigma^{UB}$ in Eq. (73) is called a classical-quantum state.
For this class of states, a straightforward calculation shows that [2]

$I(U;B)_{\sigma}=S\left[\sum_{u}p_{U}(u)\omega^{B}_{u}\right]-\sum_{u}p_{U}(u)S\left[\omega^{B}_{u}\right].$
(75)

In order to compute $\omega^{B}_{u}$ and its von Neumann entropy, let us
decompose the initial state of Alice’s qubit in terms of Bloch vectors, i.e.,

$\rho^{A}_{f(u,v)}=\frac{1}{2}\left(I_{A}+\mathbf{r}_{f(u,v)}\cdot\boldsymbol{\sigma}_{A}\right),$
(76)

where $\mathbf{r}_{f(u,v)}\equiv({x}_{f(u,v)},{y}_{f(u,v)},{z}_{f(u,v)})$,
$I_{A}$ is the identity in $\mathcal{H}_{A}$,
$\boldsymbol{\sigma}_{A}\equiv(\sigma^{\mathrm{x}}_{A},\sigma^{\mathrm{y}}_{A},\sigma^{\mathrm{z}}_{A})$,
and
$\|\mathbf{r}_{f(u,v)}\|^{2}=x_{f(u,v)}^{2}+y_{f(u,v)}^{2}+z_{f(u,v)}^{2}\leq
1$. From Eqs. (III), (74), and (76) we get

$\displaystyle\omega^{B}_{u}$
$\displaystyle=\frac{1}{2}I_{B}+\frac{\nu_{B}}{2}\cos{[2\Delta(f_{A},f_{B})]}\sigma^{\mathrm{y}}_{B}$
$\displaystyle-\overline{z}_{u}\frac{\nu_{B}}{2}\sin{[2\Delta(f_{A},f_{B})]}\sigma^{\mathrm{x}}_{B},$
(77)

where $\overline{z}_{u}\equiv\sum_{v}p_{V|U}(v|u){z}_{f(u,v)}$, and thus we
can further write

$\displaystyle\omega^{B}\equiv\sum_{u}p_{U}(u)\omega^{B}_{u}$
$\displaystyle=\frac{1}{2}I_{B}+\frac{\nu_{B}}{2}\cos{[2\Delta(f_{A},f_{B})]}\sigma^{\mathrm{y}}_{B}$
$\displaystyle-\overline{z}\frac{\nu_{B}}{2}\sin{[2\Delta(f_{A},f_{B})]}\sigma^{\mathrm{x}}_{B},$
(78)

where $\overline{z}\equiv\sum_{u,v}p(u,v)z_{f(u,v)}$.

Now, by using standard diagonalization, we find that $\omega^{B}_{u}$ has
eigenvalues $p^{B}_{u}$ and $1-p^{B}_{u}$, where

$p^{B}_{u}\equiv\frac{1}{2}+\frac{\nu_{B}}{2}\sqrt{\overline{z}_{u}^{2}\sin^{2}{[2\Delta(f_{A},f_{B})]}+\cos^{2}{[2\Delta(f_{A},f_{B})]}},$
(79)

whereas $\omega^{B}$ has eigenvalues $p^{B}$ and $1-p^{B}$, with

$p^{B}\equiv\frac{1}{2}+\frac{\nu_{B}}{2}\sqrt{\overline{z}^{2}\sin^{2}{[2\Delta(f_{A},f_{B})]}+\cos^{2}{[2\Delta(f_{A},f_{B})]}}.$
(80)

Therefore, we can now write Eq. (75) as

$I(U;B)_{\sigma}=H\left(p^{B}\right)-\sum_{u}p_{U}(u)H\left(p^{B}_{u}\right),$
(81)

where $H(x)\equiv-x\log_{2}{x}-(1-x)\log_{2}{(1-x)}$, $x\in[0,1]$. Following
similar steps, we can show that

$I(V;C)_{\sigma}=H\left(p^{C}\right)-\sum_{v}p_{V}(v)H\left(p^{C}_{v}\right),$
(82)

where

$\displaystyle p^{C}_{v}\equiv\frac{1}{2}$
$\displaystyle+\frac{\nu_{C}}{2}|\cos{[2\Delta(f_{B},f_{C})]}|$ (83)
$\displaystyle\times\sqrt{\overline{z}_{v}^{2}\sin^{2}{[2\Delta(f_{A},f_{C})]}+\cos^{2}{[2\Delta(f_{A},f_{C})]}}$

with $\overline{z}_{v}\equiv\sum_{u}p_{U|V}(u|v){z}_{f(u,v)}$, and

$\displaystyle p^{C}\equiv\frac{1}{2}$
$\displaystyle+\frac{\nu_{C}}{2}|\cos{[2\Delta(f_{B},f_{C})]}|$ (84)
$\displaystyle\times\sqrt{\overline{z}^{2}\sin^{2}{[2\Delta(f_{A},f_{C})]}+\cos^{2}{[2\Delta(f_{A},f_{C})]}}.$

Now, let us note that $H(x)$ is a monotonically decreasing function when
$x\geq 1/2$. From Eqs. (79) and (80), we have

$p^{B}_{u}\leq\frac{1}{2}+\frac{\nu_{B}}{2}$ (85)

and

$p^{B}\geq\frac{1}{2}+\frac{\nu_{B}}{2}|\cos{[2\Delta(f_{A},f_{B})]}|,$ (86)

and thus it follows that

$H(p^{B}_{u})\geq H\left(\frac{1}{2}+\frac{\nu_{B}}{2}\right)$ (87)

and

$H(p^{B})\leq
H\left(\frac{1}{2}+\frac{\nu_{B}}{2}\left|\cos{[2\Delta(f_{A},f_{B})]}\right|\right).$
(88)

As a result, from Eq. (81), we conclude that

$I(U;B)_{\sigma}\leq\mathcal{C}(\mathcal{E}_{B}),$ (89)

where

$\mathcal{C}(\mathcal{E}_{B})\equiv
H\left(\frac{1}{2}+\frac{\nu_{B}}{2}\left|\cos{[2\Delta(f_{A},f_{B})]}\right|\right)-H\left(\frac{1}{2}+\frac{\nu_{B}}{2}\right)$
(90)

is the classical capacity of the reduced channel $\mathcal{E}_{B}$, given in
Eq. (III), as shown in [28]. We note that the upper bound in Eq. (89) can be
attained if we choose random variables $U,V=\\{0,1\\}$ with $p(u,v)=1/4$ for
all $u,v$, associated with Bloch vectors
$\mathbf{r}_{f(0,0)}=\mathbf{r}_{f(0,1)}=(0,0,+1)$ and
$\mathbf{r}_{f(1,0)}=\mathbf{r}_{f(1,1)}=(0,0,-1)$. By using such choices
together with Eq. (68), we conclude that Alice can reliably convey classical
information to Bob at rates arbitrarily close to
$\mathcal{C}(\mathcal{E}_{B})$.

Similarly, we can show from Eqs. (82)-(84) that

$I(V;C)_{\sigma}\leq\mathcal{C}(\mathcal{E}_{C}),$ (91)

where

$\displaystyle\mathcal{C}(\mathcal{E}_{C})$ $\displaystyle\equiv
H\left(\frac{1}{2}+\frac{\nu_{C}}{2}\left|\cos{[2\Delta(f_{B},f_{C})]}\cos{[2\Delta(f_{A},f_{C})]}\right|\right)$
$\displaystyle-H\left(\frac{1}{2}+\frac{\nu_{C}}{2}|\cos{[2\Delta(f_{B},f_{C})]}|\right),$
(92)

is the classical capacity of the reduced channel $\mathcal{E}_{C}$ given in
Eq. (III). The upper bound can be attained, e.g., if we choose random
variables $U,V=\\{0,1\\}$ with $p(u,v)=1/4$ for all $u,v$, associated with
Bloch vectors $\mathbf{r}_{f(0,0)}=\mathbf{r}_{f(1,0)}=(0,0,+1)$ and
$\mathbf{r}_{f(0,1)}=\mathbf{r}_{f(1,1)}=(0,0,-1)$. Hence, from Eq. (69), we
conclude that Alice can reliably convey classical information to Charlie as
well at rates arbitrarily close to $\mathcal{C}(\mathcal{E}_{C})$.

It is important to highlight that causality is explicitly manifest on the
bounds of the achievable rates. First, we note that the achievable rates
$R_{B}$ between Alice and Bob are bounded by $\mathcal{C}(\mathcal{E}_{B})$,
which does not depends on the interaction between Charlie’s qubit and the
quantum field. This should indeed be the case as, from hypothesis (QB2) in
Sec. III, Charlie cannot influence the communication between Alice and Bob
since he does not perform any actions before Bob finishes his measurement
process. Furthermore, the presence of the commutator $\Delta(f_{B},f_{C})$ in
Eq. (IV.1) indicates that when Bob and Charlie let their qubits interact with
the quantum field in causally connected regions of the spacetime, noise form
Bob’s actions can influence on the rate $R_{C}$ of communication between Alice
and Charlie. Additionally, we note that whenever $\Delta(f_{A},f_{j})=0$, we
have

$\mathcal{C}(\mathcal{E}_{j})=0$ (93)

for $j=B,C$. Hence, when Alice and Bob (Charlie) interact with the field in
causally disconnected regions of the spacetime, the achievable rate in Eq.
(68) (or Eq. (69)) will reduce to $R_{B}=0$ (or $R_{C}=0$).

To this day, no one has been able to prove that the Marton rate region given
by Eqs. (68)-(70) is optimal for general broadcast channels, not even in the
classical case. However, it is generally conjectured that the Marton rate
region indeed represents the full capacity region of general broadcast
channels. If this is the case, then our analysis shows that causality will not
be violated when transmitting classical information, no matter which
communication protocol is chosen.

###  IV.2 A father protocol for quantum broadcast channels

Let us now turn our attention to the communication of quantum information.
Following [5], we present a father protocol for entanglement-assisted quantum
communication through quantum broadcast channels that can be used to
investigate at which rates Alice can send classical or quantum information to
Bob and Charlie when they share an unlimited supply of entanglement. Then, we
show how this protocol can be adapted to investigate communication rates for
classical information transmission using entanglement as well as for
unassisted quantum communication.

Let us suppose that Alice has access to two quantum systems $T_{A}$ and
$T_{A^{\prime}}$ while Bob and Charlie possess similar quantum systems $T_{B}$
and $T_{C}$, respectively. All systems possesses the same dimension
$d_{T_{A}}\equiv\mathrm{dim}\mathcal{H}_{T_{A}}$. Suppose further that Alice
shares maximally entangled states with both Bob and Charlie:

$\left|\Phi^{T_{A}T_{k}}\right\rangle=\frac{1}{\sqrt{d_{T_{A}}}}\sum_{i=0}^{d_{T_{A}}-1}|i\rangle_{T_{A}}\otimes|i\rangle_{T_{k}},$
(94)

where the above state is defined on
$\mathcal{H}_{T_{A}}\otimes\mathcal{H}_{T_{k}}$, with $k=B,C$ and
$\\{|i\rangle_{T_{\alpha}}$ is an orthonormal set of vectors on
$\mathcal{H}_{T_{\alpha}}$, $\alpha=A,B,C$.

In order to study the transmission of quantum information, we first note that
whenever Alice is able to transmit the entanglement she shares with some
reference system to each receiver, she will be able to send arbitrary quantum
states to each of them. Hence, suppose that Alice possesses two quantum
systems $A_{1}$ and $A_{2}$ respectively entangled with reference systems
$R_{1}$ and $R_{2}$ and that these systems are in states
$|\Phi^{A_{j}R_{j}}\rangle$ defined on
$\mathcal{H}_{A_{j}}\otimes\mathcal{H}_{R_{j}}$ for $j=1,2$555As a result, the
quantum state being transmitted by Alice to the receiver $j$, $j=B,C$, is
$\rho^{A_{j}}\equiv{\rm
tr}_{R_{j}}|\Phi^{A_{j}R_{j}}\rangle\langle\Phi^{A_{j}R_{j}}|.$. Her goal is
to send her share of $|\Phi^{A_{1}R_{1}}\rangle$ and
$|\Phi^{A_{2}R_{2}}\rangle$ to Bob and Charlie, respectively.

The initial global state of the system is

$\left|\varphi\right\rangle\equiv\left|\Phi^{A_{1}R_{1}}\right\rangle\left|\Phi^{A_{2}R_{2}}\right\rangle\left|\Phi^{T_{A}T_{B}}\right\rangle\left|\Phi^{T_{A^{\prime}}T_{C}}\right\rangle$
(95)

and We will denote
$\rho_{\varphi}\equiv\left|\varphi\right\rangle\left\langle\varphi\right|$. In
order to use the quantum channel $\mathcal{E}$ to share her entanglement with
$R_{1}$ and $R_{2}$ to Bob and Charlie (and hence, convey quantum
information), Alice uses a CPTP map
$\mathcal{C}:{\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}\otimes\mathcal{H}_{T_{A}}\otimes\mathcal{H}_{T_{A^{\prime}}}}\rightarrow\mathcal{H}_{A}^{\otimes
n}$ in order to encode her shares of the quantum
systems–$T_{A},T_{A^{\prime}},A_{1},$ and $A_{2}$, into a state of $n$ qubits.
The global state will then reads

$\tilde{\rho}^{A_{n}R_{1}R_{2}T_{B}T_{C}}\equiv\left(\mathcal{C}\otimes
I^{R_{1}R_{2}T_{B}T_{C}}\right)\left(\rho_{\varphi}\right),$ (96)

where $I^{R_{1}R_{2}T_{B}T_{C}}$ is the identity operator of the joint system
${R_{1}R_{2}T_{B}T_{C}}$. Next, by making $n$ independent uses of the channel
$\mathcal{E}$, Alice sends her total encoded state to Bob and Charlie, which
results in the global state

$\omega^{B_{n}C_{n}R_{1}R_{2}T_{B}T_{C}}\equiv\left(\mathcal{E}^{\otimes
n}\otimes
I^{R_{1}R_{2}T_{B}T_{C}}\right)\left(\tilde{\rho}^{A_{n}R_{1}R_{2}T_{B}T_{C}}\right).$
(97)

Bob and Charlie decode their share of the global state by using the CPTP maps
$\mathcal{D}_{B}:{\mathcal{H}_{B}^{\otimes
n}\otimes\mathcal{H}_{T_{B}}}\rightarrow\mathcal{H}_{B^{\prime}}$ and
$\mathcal{D}_{C}:{\mathcal{H}_{C}^{\otimes
n}\otimes\mathcal{H}_{T_{C}}}\rightarrow\mathcal{H}_{C^{\prime}}$,
respectively. Hence, the final global state is

$\zeta^{B^{\prime}C^{\prime}R_{1}R_{2}}\equiv\left(\mathcal{D}_{C}\otimes\mathcal{D}_{B}\otimes
I^{R_{1}R_{2}}\right)\left(\omega^{B_{n}C_{n}R_{1}R_{2}T_{B}T_{C}}\right).$
(98)

We define the entanglement-assisted quantum communication rates as

$\widetilde{Q}_{B}\equiv\frac{1}{n}\log_{2}d_{A_{1}}\;,\;\;\widetilde{Q}_{C}\equiv\frac{1}{n}\log_{2}d_{A_{2}},$
(99)

where $d_{A_{j}}\equiv\mathrm{dim}\mathcal{H}_{A_{j}}$ and $j=1,2$. These
rates of quantum communication measure how many qubits are being sent per
channel use.

The communication process will be good if given a small $\epsilon>0$ we have

$\left\|\zeta^{B^{\prime}C^{\prime}R_{1}R_{2}}-\rho_{\varphi}^{B^{\prime}C^{\prime}R_{1}R_{2}}\right\|_{1}\leq\epsilon,$
(100)

where

$\left\|\mathcal{O}\right\|_{1}\equiv\mathrm{tr}\left(\sqrt{\mathcal{O}^{\dagger}\mathcal{O}}\right)$
(101)

is the trace norm of an operator $\mathcal{O}$. Here,
$\rho_{\varphi}^{B^{\prime}C^{\prime}R_{1}R_{2}}$ is the analogous of the
initial state in the composite system $B^{\prime}C^{\prime}R_{1}R_{2}$, i.e.,
given the initial state in Alice’s laboratory

$\rho_{\varphi}^{A_{1}A_{2}R_{1}R_{2}}\equiv\left|\Phi^{A_{1}R_{1}}\right\rangle\left\langle\Phi^{A_{1}R_{1}}\right|\otimes\left|\Phi^{A_{2}R_{2}}\right\rangle\left\langle\Phi^{A_{2}R_{2}}\right|,$
(102)

we define

$\rho_{\varphi}^{B^{\prime}C^{\prime}R_{1}R_{2}}\equiv\left(\mathcal{I}^{A_{1}\rightarrow
B^{\prime}}\otimes\mathcal{I}^{A_{2}\rightarrow
C^{\prime}}\right)\left(\rho_{\varphi}^{A_{1}A_{2}R_{1}R_{2}}\right),$ (103)

where $\mathcal{I}^{A_{1}\rightarrow B^{\prime}}$ (or
$\mathcal{I}^{A_{2}\rightarrow C^{\prime}}$) is the identity map between the
quantum systems $A_{1}$ (or $A_{2})$ and $B^{\prime}$ (or $C^{\prime}$).

The communication protocol described here is named as a
$(n,\widetilde{Q}_{B},\widetilde{Q}_{C},\epsilon)$ code if it satisfies Eq.
(100) for every input state $\rho_{\varphi}^{A_{1}A_{2}R_{1}R_{2}}$. Again, we
say that a rate pair $(\widetilde{Q}_{B},\widetilde{Q}_{C})$ is achievable if
given any $\epsilon,\delta>0$ there exists a
$(n,\widetilde{Q}_{B}-\delta,\widetilde{Q}_{C}-\delta,\epsilon)$ code for
sufficiently large $n$.

Now, given a general broadcast channel $\mathcal{E}:A\rightarrow BC$ and an
arbitrary mixed state $\rho^{AA_{1}A_{2}}$ defined on
$\mathcal{H}_{A}\otimes\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}$, it can
be shown [5] that a entanglement-assisted quantum rate pair
$(\widetilde{Q}_{B},\widetilde{Q}_{C})$ is achievable if

$\displaystyle 0\leq\widetilde{Q}_{B}$
$\displaystyle\leq\frac{1}{2}I(A_{1};B)_{\sigma},$ (104) $\displaystyle
0\leq\widetilde{Q}_{C}$ $\displaystyle\leq\frac{1}{2}I(A_{2};C)_{\sigma},$
(105) $\displaystyle\widetilde{Q}_{B}+\widetilde{Q}_{C}$
$\displaystyle\leq\frac{1}{2}[I(A_{1};B)_{\sigma}+I(A_{2};C)_{\sigma}-I(A_{1};A_{2})_{\sigma}],$
(106)

where the mutual information quantities are evaluated relative to the state

$\sigma^{A_{1}A_{2}BC}\equiv\left(\mathcal{E}\otimes
I^{A_{1}A_{2}}\right)\left(\rho^{AA_{1}A_{2}}\right).$ (107)

In addition to the entanglement-assisted quantum communication, the father
protocol presented here can be adapted to obtain achievable rates for
entanglement-assisted classical communication and for unassisted quantum
communication, as we shall see in the next two sections.

###  IV.3 Unassisted quantum communication

We note that the father protocol presented above can be modified to describe
quantum communication unassisted by entanglement simply by ignoring the
existence of the quantum systems $T_{A},T_{A^{\prime}},T_{B}$, and $T_{C}$ and
following the exact same procedure. As shown in [5], given an arbitrary mixed
state $\rho^{AA_{1}A_{2}}$ defined on
$\mathcal{H}_{A}\otimes\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}$, it
follows that the following unassisted quantum rate region is achievable:

$\displaystyle 0\leq Q_{B}$ $\displaystyle\leq I(A_{1}\rangle B)_{\sigma},$
(108) $\displaystyle 0\leq Q_{C}$ $\displaystyle\leq I(A_{2}\rangle
C)_{\sigma},$ (109)

where $\sigma$ is given by Eq. (107) and

$I(A\rangle B)\equiv S(B)-S(AB)$

is the quantum coherent information between systems $A$ and $B$.

To analyze if Alice can send entanglement (and, as a result, an arbitrary
state $\rho^{A}$) to Bob through the broadcast channel, let us note that we
may purify the mixed state $\rho^{AA_{1}A_{2}}$ by adding an environment
system $E$ such that

$\rho^{AA_{1}A_{2}}=\mathrm{tr}_{E}\left(|\psi^{AA_{1}A_{2}E}\rangle\langle\psi^{AA_{1}A_{2}E}|\right),$
(110)

where
$|\psi^{AA_{1}A_{2}E}\rangle\in{\mathcal{H}_{A}\otimes\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}}\otimes\mathcal{H}_{E}}$
is a pure state. Let us decompose it as

$|\psi^{AA_{1}A_{2}E}\rangle=\sum_{a=0}^{1}\sum_{a_{1}=0}^{1}\sum_{a_{2}=0}^{1}\sum_{e=0}^{d-1}c_{aa_{1}a_{2}e}|a\rangle_{A}|a_{1}\rangle_{A_{1}}|a_{2}\rangle_{A_{2}}|e\rangle_{E},$
(111)

where $|a\rangle_{A}$,$|a_{1}\rangle_{A_{1}}$, and $|a_{2}\rangle_{A_{2}}$ are
eigenstates of $\sigma^{\mathrm{z}}_{A}$, $\sigma^{\mathrm{z}}_{A_{1}}$, and
$\sigma^{\mathrm{z}}_{A_{2}}$, respectively. Furthermore,
$\\{|e\rangle_{E}\\}$ is some orthonormal basis for $\mathcal{H}_{E}$, with
$d\equiv{\rm dim}\mathcal{H}_{E}$ being as large as needed, and

$\sum_{a=0}^{1}\sum_{a_{1}=0}^{1}\sum_{a_{2}=0}^{1}\sum_{e=0}^{d-1}|c_{aa_{1}a_{2}e}|^{2}=1.$
(112)

By defining

$\left|\zeta_{a}\right\rangle_{A_{1}A_{2}E}\equiv\sum_{a_{1}=0}^{1}\sum_{a_{2}=0}^{1}\sum_{e=0}^{d-1}c_{aa_{1}a_{2}e}|a_{1}\rangle_{A_{1}}|a_{2}\rangle_{A_{2}}|e\rangle_{E}$
(113)

and

$\zeta_{aa^{\prime}}^{A_{1}A_{2}}\equiv\mathrm{tr}_{E}\left(\left|\zeta_{a}\right\rangle_{A_{1}A_{2}E}{}_{A_{1}A_{2}E}\left\langle\zeta_{a^{\prime}}\right||\right),$
(114)

we can write Eq. (110) as

$\rho^{AA_{1}A_{2}}=\sum_{a=0}^{1}\sum_{a^{\prime}=0}^{1}\zeta_{aa^{\prime}}^{A_{1}A_{2}}\otimes|a\rangle_{A}{}_{A}\langle
a^{\prime}|.$ (115)

By using Eq. (115) in Eq. (107) and taking the partial trace over $C$ and
$A_{2}$ we obtain

$\sigma^{A_{1}B}=\sum_{a=0}^{1}\zeta_{aa}^{A_{1}}\otimes\mathcal{E}_{B}\left(|a\rangle_{A}{}_{A}\langle
a|\right),$ (116)

where $\zeta_{aa}^{A_{1}}\equiv\mathrm{tr}_{A_{2}}(\zeta_{aa}^{A_{1}A_{2}})$
and we have used the fact that

$\mathcal{E}_{B}\left(|a\rangle_{A}{}_{A}\langle
a^{\prime}|\right)=\delta_{aa^{\prime}}\mathcal{E}_{B}\left(|a\rangle_{A}{}_{A}\langle
a|\right),$ (117)

which can be proven by a direct calculation using Eq. (III). We define now the
density matrices

$\mathfrak{S}^{B}_{a}\equiv\mathcal{E}_{B}\left(|a\rangle_{A}{}_{A}\langle
a|\right),$ (118)

and

$\tau^{A_{1}}_{a}\equiv\left\|\zeta_{a}\right\|^{-2}\zeta_{aa}^{A_{1}},$ (119)

with $\left\|\zeta_{a}\right\|^{2}\equiv\mathrm{tr}(\zeta_{aa}^{A_{1}})$. This
allows us to rewrite Eq. (116) as

$\sigma^{A_{1}B}=\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}\tau^{A_{1}}_{a}\otimes\mathfrak{S}^{B}_{a},$
(120)

and we note that
$\mathrm{tr}(\mathfrak{S}^{B}_{a})=\mathrm{tr}(\tau^{A_{1}}_{a})=1$ and

$\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}=\sum_{a=0}^{1}\sum_{a_{1}=0}^{1}\sum_{a_{2}=0}^{1}\sum_{e=0}^{d-1}|c_{aa_{1}a_{2}e}|^{2}=1.$
(121)

Hence, we have shown that $\sigma^{A_{1}B}$ is a separable state, which
implies that the reduced channel from Alice to Bob lies in the class of the
entanglement-breaking channels. As shown in [33], the coherent information is
non-positive for separable states like $\sigma^{A_{1}B}$, i.e.,

$I(A_{1}\rangle B)_{\sigma}\leq 0.$ (122)

Following similar steps, one can also show that $I(A_{2}\rangle
C)_{\sigma}\leq 0$. As a result, the achievable rate region given by Eqs.
(108) and (109) reduces to

$Q_{B}=Q_{C}=0.$ (123)

It should be noted that it is not known, to this day, if the region defined by
Eqs. (108)-(109) characterizes the full capacity region for general quantum
broadcast channels. If this is the case, our analysis implies that Alice
cannot send qubits to the receivers without prior shared entanglement. Since
the reduced channels $\mathcal{E}_{B}$ and $\mathcal{E}_{C}$ are entanglement-
breaking, Alice cannot transmit the needed entanglement to establish quantum
communication by using only the quantum broadcast channel $\mathcal{E}$.

###  IV.4 Entanglement-assisted quantum communication

We have seen in Sec. IV.1 that Alice can reliably transmit classical messages
to Bob and Charlie through the quantum broadcast channel $\mathcal{E}$
provided that their interactions with the quantum field are causally
connected. On the other hand, we have seen in Sec. IV.3 that (probably) Alice
can never convey qubits to Bob or Charlie if they do not share prior
entanglement. Now, we investigate if this limitation changes if the three
observers perform an entanglement-assisted quantum communication protocol as
described in Sec. IV.2. In this scenario, we recall that Eqs. (104)-(106) give
an achievable (entanglement-assisted) quantum rate region that we shall
investigate now.

We begin by deriving upper bounds for this region. Recall that the information
bounds are evaluated with respect to the final global state
$\sigma^{A_{1}A_{2}BC}$ given by Eq. (107). Following the procedure described
in Sec. IV.3, we can take partial traces over Charlie’s qubit space $C$ and
system $A_{2}$ and write the reduced state
$\sigma^{A_{1}B}\equiv\mathrm{tr}_{A_{2},C}(\sigma^{A_{1}A_{2}BC})$ in a
separable form given by Eq. (120). Then, by using the concavity of the von
Neumann entropy, together with its addictive property for product states [2],
we find that

$\displaystyle S(\sigma^{A_{1}B})$
$\displaystyle\geq\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}S(\tau^{A_{1}}_{a}\otimes\mathfrak{S}^{B}_{a})$
$\displaystyle=\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}S(\tau^{A_{1}}_{a})+\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}S(\mathfrak{S}^{B}_{a})$
$\displaystyle\geq\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}S(\mathfrak{S}^{B}_{a}).$
(124)

Hence, by the definition of quantum mutual information, we have

$\displaystyle I(A_{1};B)_{\sigma}$
$\displaystyle=S(\sigma^{A_{1}})+S(\sigma^{B})-S(\sigma^{A_{1}B})$
$\displaystyle\leq
S(\sigma^{A_{1}})+S(\sigma^{B})-\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}S(\mathfrak{S}^{B}_{a})$
$\displaystyle\leq
S(\sigma^{B})-\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}S(\mathfrak{S}^{B}_{a}).$
(125)

A direct calculation using Eqs. (118) and (III) yields

$\displaystyle\mathfrak{S}^{B}_{0}$
$\displaystyle=\frac{1}{2}I_{B}+\frac{\nu_{B}}{2}\cos{[2\Delta(f_{A},f_{B})]}\sigma^{\mathrm{y}}_{B}$
$\displaystyle-\frac{\nu_{B}}{2}\sin{[2\Delta(f_{A},f_{B})]}\left\|\zeta_{0}\right\|^{2}\sigma^{\mathrm{x}}_{B},$
(126) $\displaystyle\mathfrak{S}^{B}_{1}$
$\displaystyle=\frac{1}{2}I_{B}+\frac{\nu_{B}}{2}\cos{[2\Delta(f_{A},f_{B})]}\sigma^{\mathrm{y}}_{B}$
$\displaystyle+\frac{\nu_{B}}{2}\sin{[2\Delta(f_{A},f_{B})]}\left\|\zeta_{1}\right\|^{2}\sigma^{\mathrm{x}}_{B},$
(127)

and hence, as
$\sigma^{B}=\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}\mathfrak{S}^{B}_{a}$,
we get

$\displaystyle\sigma^{B}$
$\displaystyle=\frac{1}{2}I_{B}+\frac{\nu_{B}}{2}\cos{[2\Delta(f_{A},f_{B})]}\sigma^{\mathrm{y}}_{B}$
$\displaystyle-\frac{\nu_{B}}{2}\sin{[2\Delta(f_{A},f_{B})]}\left(\left\|\zeta_{0}\right\|^{2}-\left\|\zeta_{1}\right\|^{2}\right)\sigma^{\mathrm{x}}_{B}.$
(128)

By standard diagonalization, we can show that the eigenvalues of
$\mathfrak{S}^{B}_{a}$ are $p^{B}_{a}$ and $1-p^{B}_{a}$, where

$p^{B}_{a}\equiv\frac{1}{2}+\frac{\nu_{B}}{2}\sqrt{\left\|\zeta_{a}\right\|^{4}\sin^{2}{[2\Delta(f_{A},f_{B})]}+\cos^{2}{[2\Delta(f_{A},f_{B})]}}.$
(129)

Similarly, the eigenvalues of $\sigma^{B}$ are $p^{B}$ and $1-p^{B}$, where

$p^{B}\equiv\frac{1}{2}+\frac{\nu_{B}}{2}\sqrt{\zeta_{01}^{4}\sin^{2}{[2\Delta(f_{A},f_{B})]}+\cos^{2}{[2\Delta(f_{A},f_{B})]}},$
(130)

with
$\zeta_{01}^{2}\equiv\left\|\zeta_{0}\right\|^{2}-\left\|\zeta_{1}\right\|^{2}$.
This implies that the RHS of Eq. (125) can be written as

$S(\sigma^{B})-\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}S(\mathfrak{S}^{B}_{a})=H(p^{B})-\sum_{a=0}^{1}\left\|\zeta_{a}\right\|^{2}H(p_{a}^{B}),$
(131)

with $H(x)$ defined below Eq. (81). Following similar steps as the ones in
Sec. IV.1, we note that $H(x)$ is a monotonically decreasing function for
$x\geq 1/2$. Hence, as

$\frac{1}{2}\leq p^{B}_{a}\leq\frac{1}{2}+\frac{\nu_{B}}{2}$ (132)

and

$p^{B}\geq\frac{1}{2}+\frac{\nu_{B}}{2}|\cos{[2\Delta(f_{A},f_{B})]}|,$ (133)

we get by Eqs. (125) and (131) that

$I(A_{1};B)_{\sigma}\leq\mathcal{C}(\mathcal{E}_{B}),$ (134)

where $\mathcal{C}(\mathcal{E}_{B})$ is given by Eq. (90). By an analogous
reasoning, we can show that

$\displaystyle I(V;C)_{\sigma}$
$\displaystyle\leq\mathcal{C}(\mathcal{E}_{C}),$ (135)

where $\mathcal{C}(\mathcal{E}_{C})$ is given by Eq. (IV.1).

Hence, we can conclude that any entanglement-assisted rate pair satisfying
Eqs. (104)-(106) must satisfy the upper bounds

$\displaystyle\widetilde{Q}_{B}$
$\displaystyle\leq\frac{1}{2}\,\mathcal{C}(\mathcal{E}_{B}),$ (136)
$\displaystyle\widetilde{Q}_{C}$
$\displaystyle\leq\frac{1}{2}\,\mathcal{C}(\mathcal{E}_{C}),$ (137)

i.e., the individual rates are bounded by half of the classical capacities of
the reduced channels $\mathcal{E}_{B}$ and $\mathcal{E}_{C}$, respectively. In
fact, we can show that both bounds are (not simultaneously) attainable by
making different choices of the input state $\rho^{AA_{1}A_{2}}$. To see this,
let us choose

$\rho^{AA_{1}A_{2}}\equiv|\Phi^{AA_{1}}\rangle\langle\Phi^{AA_{1}}|\otimes\rho^{A_{2}},$
(138)

where $\rho^{A_{2}}$ is arbitrary and $|\Phi^{AA_{1}}\rangle$ is the maximally
entangled state

$|\Phi^{AA_{1}}\rangle\equiv\frac{1}{\sqrt{2}}\sum_{a=0}^{1}|a\rangle_{A}|a\rangle_{A_{1}}.$
(139)

For this particular state, Eq. (120) can be written as

$\sigma^{A_{1}B}=\frac{1}{2}\sum_{a=0}^{1}|a\rangle_{A_{1}}{}_{A_{1}}\langle
a|\otimes\mathfrak{S}^{B}_{a},$ (140)

which implies that

$\displaystyle I(A_{1};B)_{\sigma}=\mathcal{C}(\mathcal{E}_{B}),$ (141)
$\displaystyle I(A_{2};C)_{\sigma}=0,$ (142) $\displaystyle
I(A_{1};A_{2})_{\sigma}=0.$ (143)

In view of Eqs. (104)-(106), this implies that Alice will be able to convey
quantum information to Bob at a rate arbitrarily close to
$\widetilde{Q}_{B}=\mathcal{C}(\mathcal{E}_{B})/2$ when they initially share
unlimited amounts of entanglement. Note that this is in contrast with the
unassisted case discussed in Sec. IV.3. Similarly, by switching $A_{1}$ by
$A_{2}$ in Eq. (138), we show that Alice will be able to transmit quantum
states to Charlie at a rate arbitrarily close to
$\widetilde{Q}_{C}=\mathcal{C}(\mathcal{E}_{C})/2$ when they communicate
assisted by shared entanglement.

Furthermore, initial tripartite entangled states $\rho^{AA_{1}A_{2}}$ will, in
general, lead to simultaneously nonvanishing rate pairs provided that sender
and receivers interact with the field in causally connected regions of
spacetime. In contrast, in view of the upper bounds in Eqs. (136)-(137), we
can see that whenever Alice, Bob, and Charlie try to communicate being in
causally disconnected regions, we have that $\Delta(f_{A},f_{j})=0$, with
$j=B,C$, and the entanglement-assisted quantum region reduces to

$\widetilde{Q}_{B}=\widetilde{Q}_{C}=0.$ (144)

Although an expression for the full capacity region is not known, this result
suggests that whenever Alice and Bob/Charlie interact with the field in
causally disconnected regions of spacetime, no quantum information can be sent
from her to them, not even with unlimited prior shared entanglement.

##  V Conclusions

In this paper, we have built a quantum broadcast channel by using a bosonic
quantum field in a general globally hyperbolic spacetime. In this context, we
have explored relativistic effects on the communication of classical and
quantum information in a covariant manner, where the parts conveying the
information are moving in arbitrary states of motion with the field being
assumed to be in an arbitrary (quasifree) state.

To construct the quantum broadcast channel, we have considered that Alice (the
sender) prepares some input state $\rho^{A}_{-\infty}$ for her qubit and
switches on its interaction with the field for a finite time. After that, Bob
(the first receiver) lets his qubit interact with the field for a finite time
interval, thus obtaining a final state possibly containing information encoded
by Alice. Similarly, after Bob finishes his measurement, Charlie performs an
interaction between his qubit and the quantum field to try to recover
information imprinted by Alice in the field state. We were able to trace the
field degrees of freedom non-perturbatively and showed that suitable initial
states for Bob’s and Charlie’s qubits can be chosen in order to maximize the
signaling between Alice and the receivers. This procedure defines a fully
relativistic quantum broadcast channel $\mathcal{E}$.

With this channel, we were able to investigate at which rates Alice can
reliably convey classical and quantum information to Bob and Charlie. By
considering first a scenario where the three observers do not share prior
entanglement, we found that Alice can reliably convey classical information to
both Bob and Charlie and at which rates she can perform this task. However, we
have shown that the broadcast channel presented here breaks entanglement and
thus, Alice cannot convey quantum information to Bob and Charlie following an
unassisted strategy. Nevertheless, we have shown that this situation changes
when they perform entanglement-assisted quantum communication. In this
scenario, we were able to find achievable rates that Alice can achieve when
sending qubits to the receivers provided that they initially share entangled
states.

We were also able to show that all rates that were analyzed here vanish when
the interactions between qubits and field occur in causally disconnected
regions, an effect that is manifest in all expressions bounding the classical
and quantum rates of communication even with the use of quantum resources like
entanglement. Thus, our investigation provides good evidence that causality is
preserved throughout the communication process, reinforcing the fundamental
principles of relativistic physics.

Our study shows that quantum network information theory in general spacetimes
is a rich and promising area of research, shedding light on several aspects of
the interplay between quantum information theory and relativity. We believe
that this work may provide tools to investigate open problems concerning
quantum gravity, in particular, the fate of the information that has fallen in
(evaporating) black holes. The preservation of causality observed in our
analysis reaffirms the robustness of fundamental physical principles, even in
the realm of quantum information theory in curved spacetimes. We hope that
following the path we presented here could lead us to unveil fundamental
aspects of physics that should be present in a full quantum theory of gravity.

###### 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.

##  Appendix A Full expression for the quantum broadcast channel map

As discussed in Sec. III, each $\Gamma_{\alpha\beta\gamma\delta\epsilon\zeta}$
coefficient defined in Eq. (III) can be evaluated by using Eqs. (40) and (41)
together with the product relation given by Eq. (46). Then, we substitute
these coefficients in Eq. (III), obtaining

$\displaystyle\rho^{BC}$
$\displaystyle=\frac{1}{4}(1+\nu_{B}\cos{[2\Delta(f_{A},f_{B})]}+\nu_{C}\cos{[2\Delta(f_{A},f_{C})]}\cos{[2\Delta(f_{B},f_{C})]})\rho^{BC}_{-\infty}$
$\displaystyle+\frac{1}{4}(1-\nu_{B}\cos{[2\Delta(f_{A},f_{B})]}+\nu_{C}\cos{[2\Delta(f_{A},f_{C})]}\cos{[2\Delta(f_{B},f_{C})]})\sigma_{B}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}$
$\displaystyle+\frac{1}{4}(1+\nu_{B}\cos{[2\Delta(f_{A},f_{B})]}-\nu_{C}\cos{[2\Delta(f_{A},f_{C})]}\cos{[2\Delta(f_{B},f_{C})]})\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{C}^{\mathrm{z}}$
$\displaystyle+\frac{1}{4}(1-\nu_{B}\cos{[2\Delta(f_{A},f_{B})]}-\nu_{C}\cos{[2\Delta(f_{A},f_{C})]}\cos{[2\Delta(f_{B},f_{C})]})\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}$
$\displaystyle+\frac{i\nu_{B}}{4}\sin{[2\Delta(f_{A},f_{B})]}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}[\rho^{BC}_{-\infty}+\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{C}^{\mathrm{z}},\sigma_{B}^{\mathrm{z}}]$
$\displaystyle+\frac{i\nu_{C}}{4}\sin{[2\Delta(f_{A},f_{C})]}\cos{[2\Delta(f_{B},f_{C})]}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}[\rho^{BC}_{-\infty}+\sigma_{B}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}},\sigma_{C}^{\mathrm{z}}]$
(145)
$\displaystyle+\frac{\Lambda^{+}_{c}}{8}\left(\rho^{BC}_{-\infty}-\sigma_{B}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}-\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{C}^{\mathrm{z}}+\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}\right)$
$\displaystyle+\frac{\Lambda^{-}_{c}}{8}\left(\\{\rho^{BC}_{-\infty},\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}\\}-\sigma_{B}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{C}^{\mathrm{z}}-\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}\right)$
$\displaystyle+\frac{i\Lambda^{+}_{s}}{8}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}[\rho^{BC}_{-\infty}-\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{C}^{\mathrm{z}},\sigma_{B}^{\mathrm{z}}]+\frac{i\Lambda^{-}_{s}}{8}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}[\rho^{BC}_{-\infty}-\sigma_{B}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}},\sigma_{C}^{\mathrm{z}}]$
$\displaystyle+\frac{i\nu_{C}}{4}\cos{[2\Delta(f_{A},f_{C})]}\sin{[2\Delta(f_{B},f_{C})]}\left([\rho^{BC}_{-\infty},\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{z}}]+\sigma_{B}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{C}^{\mathrm{z}}-\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{B}^{\mathrm{z}}\right)$
$\displaystyle-\frac{\nu_{C}}{4}\sin{[2\Delta(f_{A},f_{C})]}\sin{[2\Delta(f_{B},f_{C})]}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\\{\rho^{BC}_{-\infty}-\sigma_{C}^{\mathrm{z}}\rho^{BC}_{-\infty}\sigma_{C}^{\mathrm{z}},\sigma_{B}^{\mathrm{z}}\\},$

where we have defined the following coefficients:

$\displaystyle\Lambda^{\pm}_{c}$
$\displaystyle\equiv\nu_{BC}^{+}\cos{[2\Delta(f_{A},f_{B}+f_{C})]}\pm\nu_{BC}^{-}\cos{[2\Delta(f_{A},f_{B}-f_{C})]},$
(146) $\displaystyle\Lambda^{\pm}_{s}$
$\displaystyle\equiv\nu_{BC}^{+}\sin{[2\Delta(f_{A},f_{B}+f_{C})]}\pm\nu_{BC}^{-}\sin{[2\Delta(f_{A},f_{B}-f_{C})]},$
(147) $\displaystyle\nu_{j}$
$\displaystyle\equiv\omega_{\mu}\left(W[E(2f_{j})]\right),$ (148)
$\displaystyle\nu_{BC}^{\pm}$
$\displaystyle\equiv\omega_{\mu}\left(W[E(2f_{B}\pm 2f_{C})]\right).$ (149)

As discussed in Sec. III, we are motivated to fix the initial states for Bob’s
and Charlie’s qubit as given in Eqs. (52) and (58). We write these states in
terms of their Bloch vectors, i.e.,

$\rho^{j}_{-\infty}=\frac{I_{j}+\sigma^{\mathrm{y}}_{j}}{2},$ (150)

where $j=B,C$. By substituting Eq. (150) in Eq. (A), and by using the standard
commutation relations of the Pauli matrices, we obtain the following
expression describing the quantum broadcast channel map:

$\displaystyle\mathcal{E}(\rho^{A}_{-\infty})$
$\displaystyle=\frac{1}{16}(1+\nu_{B}\cos{[2\Delta(f_{A},f_{B})]}+\nu_{C}\cos{[2\Delta(f_{A},f_{C})]}\cos{[2\Delta(f_{B},f_{C})]})\left(I_{B}+\sigma^{\mathrm{y}}_{B}\right)\otimes\left(I_{C}+\sigma^{\mathrm{y}}_{C}\right)$
$\displaystyle+\frac{1}{16}(1-\nu_{B}\cos{[2\Delta(f_{A},f_{B})]}+\nu_{C}\cos{[2\Delta(f_{A},f_{C})]}\cos{[2\Delta(f_{B},f_{C})]})\left(I_{B}-\sigma^{\mathrm{y}}_{B}\right)\otimes\left(I_{C}+\sigma^{\mathrm{y}}_{C}\right)$
$\displaystyle+\frac{1}{16}(1+\nu_{B}\cos{[2\Delta(f_{A},f_{B})]}-\nu_{C}\cos{[2\Delta(f_{A},f_{C})]}\cos{[2\Delta(f_{B},f_{C})]})\left(I_{B}+\sigma^{\mathrm{y}}_{B}\right)\otimes\left(I_{C}-\sigma^{\mathrm{y}}_{C}\right)$
$\displaystyle+\frac{1}{16}(1-\nu_{B}\cos{[2\Delta(f_{A},f_{B})]}-\nu_{C}\cos{[2\Delta(f_{A},f_{C})]}\cos{[2\Delta(f_{B},f_{C})]})\left(I_{B}-\sigma^{\mathrm{y}}_{B}\right)\otimes\left(I_{C}-\sigma^{\mathrm{y}}_{C}\right)$
(151)
$\displaystyle-\frac{\nu_{B}}{4}\sin{[2\Delta(f_{A},f_{B})]}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\left(\sigma_{B}^{\mathrm{x}}\otimes
I_{C}\right)-\frac{\nu_{C}}{4}\sin{[2\Delta(f_{A},f_{C})]}\cos{[2\Delta(f_{B},f_{C})]}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\left(I_{B}\otimes\sigma_{C}^{\mathrm{x}}\right)$
$\displaystyle+\frac{\Lambda^{+}_{c}}{8}\sigma_{B}^{\mathrm{y}}\otimes\sigma_{B}^{\mathrm{y}}-\frac{\Lambda^{-}_{c}}{8}\left(\sigma_{B}^{\mathrm{x}}\otimes\sigma_{C}^{\mathrm{y}}\right)-\frac{\Lambda^{+}_{s}}{8}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\left(\sigma_{B}^{\mathrm{x}}\otimes\sigma_{C}^{\mathrm{y}}\right)-\frac{\Lambda^{-}_{s}}{8}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\left(\sigma_{B}^{\mathrm{y}}\otimes\sigma_{C}^{\mathrm{x}}\right)$
$\displaystyle-\frac{\nu_{C}}{4}\cos{[2\Delta(f_{A},f_{C})]}\sin{[2\Delta(f_{B},f_{C})]}\left(\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{x}}\right)-\frac{\nu_{C}}{4}\sin{[2\Delta(f_{A},f_{C})]}\sin{[2\Delta(f_{B},f_{C})]}\langle\sigma_{A}^{\mathrm{z}}\rangle_{\rho^{A}_{-\infty}}\left(\sigma_{B}^{\mathrm{z}}\otimes\sigma_{C}^{\mathrm{y}}\right).$

By taking partial traces relative to each qubit, one recovers Eqs. (III) and
(III).

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