Alice and Bob in an expanding spacetime

As quantum technologies progress and cutting-edge experiments incorporate relativistic regimes, studying the effects of gravity and motion on these technologies becomes increasingly more important. Consequently, relativistic aspects of quantum information theory in both flat and curved spacetimes have been extensively studied [1]. A good example is quantum metrology, which uses quantum properties of probes to measure precision observables such as time [2], field strengths [3], gravity waves [4], Unruh temperatures [5], effects of gravity on entanglement using Bose-Einstein condensates [6], Schwarzschild parameters of the Earth [7], etc. From the theoretical standpoint, the study of effective models such as quantum field theory on curved spacetime [8] has been revisited with an eye towards both estimating and understanding quantum metrology and quantum information protocols [9].

Quantum teleportation is a fundamental tool for the transmission of quantum information [10,11] where a sender (Alice) aims to transmit an unknown state to the receiver (Bob) without sending the system itself, which is thus never transported between the two sites during teleportation. However, a small amount of classical information must be telegraphed between the two sites, preventing quantum teleportation from being instantaneous which is in accordance with special relativity. A nice account of relativistic effects on quantum teleportation can be found in [12].

Besides the issues of creating and maintaining long-distance entanglement, it is a matter of paramount importance in quantum information protocols to understand the interplay between quantum nonlocality and relativistic locality, and space-like correlations and causality. In [13], the quantum fidelity of a teleported state between two moving cavities in relativistic motion (Alice at rest and Bob uniformly accelerated) was computed in order to study how fidelity is degraded by the Unruh effect. Moreover, the Unruh effect is used in [14] to investigate the teleportation of quantum states when one of the entangled qubits used in the process is under the influence of some external force, which leads to the sudden death of entanglement and teleportation fidelity loss. Scalar, fermionic and vector fields in accelerated cavities were discussed in [15].

In [16,17] it was suggested that information on the underlying spacetime can be recovered from the entanglement of scalar/fermionic field modes. It would be interesting to investigate such idea in the context of a quantum information task, for instance, in the analysis of the fidelity in quantum teleportation in an expanding spacetime. Moreover, entanglement plays a crucial role in quantum information protocols, and the study of its degradation/creation mechanisms in dynamical spacetimes is important [13,18,19].

In the present work we investigate the effect on quantum teleportation of entanglement frame dependence in a general relativistic setting, in the context of an expanding, asymptotically flat Robertson-Walker (RW) spacetime. For the sake of generality we explore both fermions and scalar field modes. For the latter we consider a non-minimal coupling to 4-dimensional spacetime since there may occur substantial differences between minimal and conformal coupling [9]. Assuming that the metric is flat in both the distant past and future, we consider for this purpose the following gedankenexperiment: at the distant Minkowskian past, two comoving observers Alice and Bob share a maximally entangled, EPR Bell state $|\beta\rangle$. It is a Bell state of qubits, encoded in terms of the modes of a real massive scalar field or of a fermionic field. Each observer uses an isolated, non-interacting ideal cavity capable of supporting such field modes to maintain the qubit in his/her possession. Furthermore, Alice holds an extra ideal cavity with a qubit superposition state that she wants to teleport to Bob within the standard protocol T₀ of Bennett et al. (also called the BBCJPW protocol) [10], using the Bell state $|\beta\rangle$ as a quantum channel. If the teleportation protocol is performed when Alice and Bob are in the distant past, then the corresponding fidelity of course is maximal. But instead, the experiment we describe consists of Alice waiting to perform the teleportation task only in the distant future, after spacetime expansion saturates (evidently, the roles of Alice and Bob may be exchanged, as they are both comoving observers).

In our gedankenexperiment, we rely on the single-mode approximation (SMA) —namely, on the fact that the local physics in the cavity is well described by a single mode of the extended field. Such SMA was shown not to hold in general in the case of the study of field mode entanglement degradation for observers moving in relative uniform acceleration [20]. Apart from some peculiarities with fermion modes, their results beyond SMA are argued to be qualitatively equivalent to SMA. For the sake of simplicity, we shall therefore rely on the SMA approximation, although a generalization beyond SMA for teleportation in RW spacetimes would be interesting.

We will show that in this case the fidelity of teleportation reduces from the point of view of Bob as the degree of entanglement of the Bell state $|\beta\rangle$ degrades due to spacetime expansion. We study how these reductions are influenced by the coupling of the scalar field to the scalar curvature of spacetime, and discuss how they are affected by the rapidity of the expansion. We will also extend the results to the case of the teleportation of fermions and explore how the fidelity is affected by the bosonic or fermionic character of the fields.

The period of spacetime expansion affects the degree of entanglement of the quantum channel $|\beta\rangle$ because it implies particle creation: under the above assumptions, Bob perceives Alice receding away from him in his local inertial frame as the experiment proceeds, and for him the vacuum state inside Alice's cavity becomes a particle-populated, thermal state in the distant future. We proceed to describe and quantify this creation process.

In conformal coordinates the metric reads

$$\begin{equation} \text{d}s^2 = a^2(\eta)\left(\text{d}\eta ^2 - \text{d}x_{1}^{2} - \text{d}x_{2}^{2} - \text{d}x_{3}^{2}\right), \end{equation} \tag{ 1 }$$

where a is the scale factor characterizing the expansion and η is the conformal time, related to Alice's proper time t by $\text{d}\eta = a^{-1}\text{d}t$. The dynamics over the spacetime described by eq. (1) of a real non-interacting scalar field ϕ with mass m coupled to gravity is given by the covariant form of the Klein-Gordon equation,

$$\begin{equation} (\square + m^2 + \xi R)\phi = 0,\end{equation} \tag{ 2 }$$

where ξ is the coupling of the field to the Ricci scalar curvature R and □ is the covariant D'Alambertian. The cases $\xi = 0$ and $\xi = \frac{1}{6}$ correspond to minimal and conformal coupling to gravity, respectively. Therefore, the field equation (2) in the conformal coordinates $(\eta,\mathbf{x})$ and in terms of an auxiliary field variable $\chi(\eta,\mathbf{x}) \equiv a(\eta)\phi(\eta,\mathbf{x})$ reduces to

$$\begin{equation} \eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\chi + a^2(\eta)\left[m^2 + \left(\xi - \frac{1}{6}\right)R(\eta)\right]\chi = 0. \end{equation} \tag{ 3 }$$

Let as assume a scale factor of the form $C(\eta) = a^2 (\eta ) = 1 + \epsilon(1 + \tanh(\rho\eta))$, where the rate of spacetime expansion is given by the parameter $\rho >0$ and the total amount of expansion by $\epsilon >0$. Then $C(\eta)$ conveniently represents a spacetime that undergoes a period of smooth expansion and becomes flat in the distant past $(\eta \rightarrow -\infty)$ and in the far future $(\eta \rightarrow +\infty)$, namely $C(\eta \rightarrow -\infty) \rightarrow 1$ and $C(\eta \rightarrow +\infty) \rightarrow 1 + 2\epsilon$. In these two asymptotic regions, the in region $(\eta \rightarrow -\infty)$ and the out region $(\eta \rightarrow +\infty)$, there is a time-like Killing vector field $\partial_{\eta}$ such that $\mathcal{L}_{\partial_{\eta}}u_{\mathbf{k}} = -i\omega_{k}u_{\mathbf{k}}$ for some $\omega_{k} > 0$, where $\mathcal{L}$ denotes the Lie derivative. Hence, we can define particle states in each of these regions in terms of positive frequency modes, as well as a vacuum state. Treating $\chi(\eta,\mathbf{x})$ as a field operator, we can express $\chi(\eta,\mathbf{x})$ in the in region (respectively, out region) as a combination of positive frequency $u_{\mathbf{k}}^{\mathrm{in}}(\eta,\mathbf{x})$ and negative frequency $u_{\mathbf{k}}^{\mathrm{in}\ast }(\eta,\mathbf{x})$ solutions (respectively, $u_{\mathbf{k}}^{\mathrm{out}}(\eta,\mathbf{x})$ and $u_{\mathbf{k}}^{\mathrm{out}\ast }(\eta,\mathbf{x})$) to eq. (3):

$$\begin{equation} \chi^{\mathrm{in,out}}(\eta,\mathbf{x}) = \int_{\mathbf{k}}\left[a_{\mathbf{k}}^{\mathrm{in,out}}u_{\mathbf{k}}^{\mathrm{in,out}} + a^{\mathrm{in,out}\dagger }_{\mathbf{k}}u_{\mathbf{k}}^{\mathrm{in,out}\ast }\right], \end{equation} \tag{ 4 }$$

where $\int _{\mathbf{k}}\equiv\int \text{d}^{3}\mathbf{k}$, and $a_{\mathbf{k}}^{\mathrm{in}},a^{\mathrm{in}\dagger }_{\mathbf{k}}$ (respectively, $a_{\mathbf{k}}^{\mathrm{out}},a^{\mathrm{out}\dagger }_{\mathbf{k}}$) are annihilation/creation operators characterizing the in-vacuum $|0\rangle_{\mathrm{in}}$ (respectively, out-vacuum $| 0\rangle_{\mathrm{out}}$) state of the field, subject to the usual commutation relations. From the values of C(η) at $\eta \to \pm\infty$, where the Ricci scalar is $R \rightarrow 0$, the mode function solutions to eq. (3) are

$$\begin{equation} u_{\mathbf{k}}^{\mathrm{in}} \longrightarrow \frac{e^{i(\mathbf{k}\cdot\mathbf{x} - \omega_{\mathrm{in}}\eta)}}{(2\pi)^{\frac{3}{2}}\sqrt{2\omega_{\mathrm{in}}}}, \quad u_{\mathbf{k}}^{\mathrm{out}} \longrightarrow \frac{e^{i(\mathbf{k}\cdot\mathbf{x} - \omega_{\mathrm{out}}\eta)}}{(2\pi)^{\frac{3}{2}}\sqrt{2\omega_{\mathrm{out}}}}, \end{equation} \tag{ 5 }$$

for $\omega_{\mathrm{in}}= \sqrt{k^2 + m^2}$, $\omega_{\mathrm{out}} = \sqrt{k^2 + (1 + 2\epsilon)m^2}$, and ${k=|\mathbf{k}|}$.

The relationship between in and out modes is given by a Bogoliubov transformation,

$$\begin{equation} a^{\mathrm{in}}_{{\mathbf{k}}} = \int \text{d}^3 \mathbf{k}\left[\alpha_{k}a^{\mathrm{out}}_{\mathbf{k}} - \beta_{k}a^{\mathrm{out}\dagger}_{-\mathbf{k}}\right], \end{equation} \tag{ 6 }$$

mixing only modes of the same momentum, where the (Bogoliubov) coefficients are determined by the scalar product of mode functions: $\alpha_{k} = \left(u^{\mathrm{out}}_{\mathbf{k}},u^{\mathrm{in}}_{\mathbf{k}}\right)$ and $\beta_{k} = \left(u^{\mathrm{out}}_{\mathbf{k}},u^{\mathrm{in}\ast}_{\mathbf{k}}\right)$. The corresponding relation between the in and out vacua is

$$\begin{equation} | 0\rangle_{\mathrm{in}} \equiv \mathcal{N}\exp\left(-\frac{1}{2}\sum_{k}\tfrac{\beta_{k}}{\alpha_{k}}a^{\mathrm{out}\dagger}_{\mathbf{k}}a^{\mathrm{out}\dagger}_{-\mathbf{k}}\right)| 0\rangle_{\mathrm{out}}, \end{equation} \tag{ 7 }$$

where $\mathcal{N}$ is a normalization factor. It shows that the in-vacuum is a two-mode squeezed out-vacuum state, which implies in particular that the out-region number operator $N_{\mathbf{k}}^{\mathrm{out}}=a^{\mathrm{out}\dagger}_{\mathbf{k}}a^{\mathrm{out}}_{\mathbf{k}}$ has a positive in-vacuum expectation value $_{\mathrm{in}}\langle 0| N_{\mathbf{k}}^{\mathrm{out}}| 0\rangle_{\mathrm{in}} >0$ in the Heisenberg picture, which describes the dynamical creation of particles from the vacuum in Alice's cavity from Bob's point of view.

We thus see that, for Bob, an initial vacuum state inside Alice's cavity $|0\rangle_{\mathrm{in}}$ becomes populated with particles in the asymptotic future $|{0 _\infty }\rangle$, which is just the right-hand side of (7) at $\eta \rightarrow \infty$. After some algebra in eq. (7), we get

$$\begin{equation} |{0 _\infty}\rangle = \sqrt{1 - \gamma}\sum_{n=0}^{\infty}\gamma^{n}|n_{\mathbf{k}}n_{-\mathbf{k}}\rangle, \end{equation} \tag{ 8 }$$

where $\gamma \equiv \left|\frac{\beta_{k}}{\alpha_{k}}\right|^2$. Similarly, the one-particle excitation in the in-vacuum $|1\rangle_{in}$ evolves as $\eta \to \infty$ into the state $|{1 _\infty }\rangle$ given in terms of out-region Fock states by

$$\begin{equation} |{1_\infty}\rangle = (1 - \gamma)\sum_{n=0}^{\infty}\gamma^{n}\sqrt{n + 1}| n + 1_{\mathbf{k}}n_{-\mathbf{k}}\rangle. \end{equation} \tag{ 9 }$$

In what follows, we will need therefore to have the explicit value of Bogoliubov coefficients. To obtain it, let us begin by noticing that spatial translation invariance allows us to express the mode functions for the field equation in the form $u_{\mathbf{k}}(\eta,\mathbf{x}) = (2\pi)^{-\frac{3}{2}}e^{i\mathbf{k}\cdot\mathbf{x}}f_{\mathbf{k}}(\eta)$, in which $f_{\mathbf{k}}(\eta)$ satisfies

$$\begin{equation} \left[\eta^{\mu\nu}\partial_{\mu}\partial_{\nu} + m^2\right]f_{\mathbf{k}}(\eta) + V(\eta)f_{\mathbf{k}}(\eta) = 0 \end{equation} \tag{ 10 }$$

for $V(\eta)$ given by

$$\begin{equation} V(\eta) \equiv [C(-\infty) - C(\eta)]m^2 - \left(\xi - \frac{1}{6}\right)C(\eta)R(\eta). \end{equation} \tag{ 11 }$$

Following [8,21], observing that $C(\eta) R(\eta) \to 0$ as $\eta \to \pm \infty$ if $\xi \ne 1/6$ and $C(\eta) \to C(\pm \infty) < \infty$ as $\eta \to \pm \infty$ if $m \ne 0$, we may treat $V(\eta)$ as small to solve eq. (10) by iteration to the lowest order in $V(\eta)$ in terms of the momentum space propagator. Using (5) as initial condition, the solution $f_{\mathbf{k}}(\eta)$ of (10) is given by the following convolution integral involving the retarded propagator:

$$\begin{equation} f_{\mathbf{k}}(\eta) = f^{in}_{\mathbf{k}}(\eta) - \int^{\infty}_{-\infty} G_{r}(\eta,\eta')V(\eta')f_{\mathbf{k}}(\eta')\text{d}\eta', \end{equation} \tag{ 12 }$$

which satisfies $\left[\eta^{\mu\nu}\partial_{\mu}\partial_{\nu} + m^2\right]G_{r}(\eta,\eta') = \delta(\eta - \eta')$ and in momentum space it reads

$$\begin{equation} G_{r}(\eta,\eta') = \frac{1}{2\pi}\int \dfrac{e^{-ik'_{0}(\eta' - \eta)}}{k'^{2}_{0} - \omega^{2}_{k} - i\varepsilon k'_{0}}\text{d}k'_{0}. \end{equation} \tag{ 13 }$$

The momentum integral can be performed by closing the integration contour in the upper-half complex momentum plane. However, for the calculation of the Bogoliubov coefficients, it suffices to notice that in the limit $\eta \longrightarrow + \infty$, $f_{\mathbf{k}}(\eta)$ can be written in terms of the in-mode functions as $f_{\mathbf{k}}(\eta) \longrightarrow (2\omega)^{-\frac{1}{2}}[\alpha_{k}e^{-i\omega\eta} + \beta_{k}e^{i\omega\eta}]$, where $\omega \equiv \omega_{\mathrm{in}}$. Taking this into (12) yields the following expressions for the Bogoliubov coefficients:

$$\begin{eqnarray} \begin{array}{rcl} \alpha_{k} &=& \displaystyle1 + \frac{i}{2\omega}\int^{\infty}_{-\infty}e^{i\omega\eta'} V(\eta') f_{\mathbf{k}}(\eta')\text{d}\eta', \\ \beta_{k}&=&-\displaystyle\frac{i}{2\omega}\int^{\infty}_{-\infty}e^{-i\omega\eta'}V(\eta') f_{\mathbf{k}}(\eta')\text{d}\eta'. \end{array} \end{eqnarray} \tag{ 14 }$$

Assuming the conditions that justify a perturbative solution, eq. (14) allows us to calculate $\alpha_{k}$ and $\beta_{k}$ to lowest order in $V(\eta)$. To leading order $f_{\mathbf{k}}(\eta) \cong f^{\mathrm{in}}_{\mathbf{k}}(\eta)$, which together with eq. (11) gives

$$\begin{eqnarray} \beta_{k}&=\frac{i}{2\omega}\int^{\infty}_{-\infty} e^{-2i\omega\eta'}\biggl[m^2\epsilon(1 + \tanh(\rho\eta'))\nonumber \\ &\quad+\,\left(\xi - \frac{1}{6}\right)C(\eta)R(\eta)\biggr] \text{d}\eta'. \end{eqnarray} \tag{ 15 }$$

The integrals in (15) can be easily evaluated in the complex plane by replacing the real variable η by the complex variable z and integrating around a closed counterclockwise rectangular contour Σ with vertices at $-\Lambda$, Λ, $\Lambda + \frac{i\pi}{\rho}$, $-\Lambda + \frac{i\pi}{\rho}$. In the upper horizontal path we use that $\tanh\rho\eta = \tanh\rho(\eta + \frac{i\pi}{\rho})$. The two vertical paths parallel to the imaginary axis vanish exponentially as $\Lambda \rightarrow \infty$. For the first term in eq. (15), we have

$$\begin{equation} I_{1} = \frac{im^2\epsilon}{2\omega}\frac{1}{1 - e^{2\pi\omega}}\oint_\Sigma \text{d}z e^{-2i\omega z}(1 + \tanh(\rho z)), \end{equation} \tag{ 16 }$$

whose integrand has a pole at $z = \frac{i\pi}{2\rho}$. Using Cauchy's integral formula we find $I_{1} =m^2\epsilon \pi [ 2\omega\rho \sinh(\pi\omega{\rho}^{-1})]^{-1}$.

The second term in eq. (15) can be evaluated in a similar way. The resulting expression for $\beta_{k}$ is

$$\begin{equation} \beta_{k} \approx \left[\frac{m^2\epsilon}{2\omega\rho} - (6\xi - 1)\frac{\omega\epsilon}{\rho}\right]\frac{\pi}{\sinh(\frac{\pi\omega}{\rho})}. \end{equation} \tag{ 17 }$$

The calculation of the Bogoliubov coefficient $\alpha_{k}$ is analogous and to this order $\alpha_{k}\approx 1 + O(\epsilon)$.

Notice that this calculation also shows how the particle number seen by Bob inside Alice's cavity relates to the coupling of the field to the scalar curvature. The spectral behaviour of particle number for different values of ξ is summarized in fig. 1. In particular, for $\xi = \tfrac{1}{4}$ the generation due to spacetime expansion of bosonic particles reproduces the behavior of fermions [16].

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Suppose that Alice's and Bob's cavities support the orthogonal modes $\mathbf{k}_1$, $\mathbf{k}_2$ of frequency $|\mathbf{k}_1 |=|\mathbf{k}_2 |\equiv k$ of ϕ, labeled, respectively, as A_i and B_i, $i = 1,2$. Assume that the Bell state $|\beta\rangle$ shared by Alice and Bob in the distant past is the maximally entangled, EPR Bell state

$$\begin{equation} |\beta \rangle = \frac{1}{\sqrt{2}}\left(|\textbf{0}_{A}\rangle_{\mathrm{in}}|\textbf{0}_{B}\rangle + |\textbf{1}_{A}\rangle _{\mathrm{in}} |\textbf{1}_{B}\rangle \right). \end{equation} \tag{ 18 }$$

The states $|\textbf{0}_{A}\rangle$ and $|\textbf{1}_{A}\rangle$ are defined in terms of the dual-rail basis as suggested in [13], $|\textbf{0}_{A}\rangle = |1_{A_{1}}\rangle|0_{A_{2}}\rangle$, $|\textbf{1}_{A}\rangle = | 0_{A_{1}}\rangle| 1_{A_{2}}\rangle$, with similar expression for Bob's cavity (in Bob's local frame, the in and out modes are the same and therefore the subscript "in" will be omitted). Then, it follows from the previous considerations that in the distant future, after spacetime expansion saturates, the state in Alice and Bob's possession will no longer be $|\beta\rangle$; instead, it will be the mixed state

$$\begin{equation} \hat{\tau }\equiv Tr_{\tiny{-B_{1},-B_{2}}}\left(|\beta _\infty \rangle \langle \beta _\infty |\right) \end{equation} \tag{ 19 }$$

which we obtain by tracing the negative momenta $-\mathbf{k}_1,-\mathbf{k}_2$ degrees of freedom out of Bob's party in $|{\beta _\infty }\rangle\langle{\beta _\infty}|$, where

$$\begin{equation} |\beta _\infty \rangle = \frac{1}{\sqrt{2}}\left(|\textbf{0}_{A}\rangle _{\infty }| \textbf{0}_{B}\rangle + |\textbf{1}_{A}\rangle _{\infty}|\textbf{1}_{B}\rangle \right). \end{equation} \tag{ 20 }$$

Here, we defined $|\textbf{0}_{A}\rangle_{\infty }=|1_{A_{1}}\rangle_{\infty }| 0_{A_{2}}\rangle_{\infty }$ and $|\textbf{1}_{A}\rangle_{\infty } = |0_{A_{1}}\rangle_{\infty }| 1_{A_{2}}\rangle_{\infty }$, following the notation introduced in eqs. (8), (9).

The mixed state $\hat{\tau}$ is still an entangled state and may be used to perform the teleportation of a qubit superposition state from Alice to Bob. However, it is no longer a maximally entangled state. In fact, a direct computation shows that the logarithmic negativity of $\hat{\tau }$ is given in terms of γ by

$$\begin{equation} \mathrm{E}_{\mathcal{N}}\left( \hat{\tau } \right) = \log _2 \left( 1+(1-\gamma )^3 \left[ \displaystyle\sum _{j=0}^{\infty } \gamma ^{2j}\sqrt{j+1} \right] ^2 \right), \end{equation} \tag{ 21 }$$

which attains the upper bound $\mathrm{E}_{\mathcal{N}}=1$ only in the limit $\gamma =0$ when there is no particle creation inside Alice's cavity due to spacetime expansion.

This reduction of the degree of entanglement of the quantum channel results in degradation of the fidelity of teleportation. In order to compute the fidelity, assume that Alice has an additional cavity containing the qubit superposition state $|\psi\rangle = a| \textbf{0}\rangle + b| \textbf{1}\rangle$. This state is unknown to Bob. The full input state is $|\psi_{0}\rangle_{\mathrm{in}} = |\psi\rangle _{\mathrm{in}} |\beta\rangle$. If Alice made a Bell measurement on her two qubits in the distant past with the result $|\textbf{i}\rangle\otimes|\textbf{j}\rangle$, $i,j\in \left\lbrace0,1\right\rbrace$, then after Alice's measurement the full state would be projected into

$$\begin{equation} |\psi\rangle_{\mathrm{in}} = |\textbf{i}\rangle\otimes|\textbf{j}\rangle\otimes|\phi_{i,j}\rangle_{\mathrm{in}}, \end{equation} \tag{ 22 }$$

where the final state received by Bob would be

$$\begin{equation*} |\phi_{i,j}\rangle_{\mathrm{in}} \equiv x_{ij}| \textbf{0}\rangle_{\mathrm{in}} + y_{ij}| \textbf{1}\rangle_{\mathrm{in}} \end{equation*}$$

with four possible conditional state amplitudes given by $(x_{00},y_{00}) = (a,b)$, $(x_{01},y_{01}) = (b,a)$, $(x_{10},y_{10}) = (a,-b)$, $(x_{11},y_{11}) = (-b,a)$. However, when Alice makes the Bell measurement in the distant future, the final state received by Bob would be obtained by tracing the $-\mathbf{k}_1,-\mathbf{k}_2$ degrees of freedom out of

$$\begin{equation*} |\phi_{i,j}\rangle_{\infty \infty} \langle \phi_{i,j}|, \end{equation*}$$

where $|\phi_{i,j}\rangle_{\infty }=x_{ij}|\textbf{0}\rangle_{\infty} + y_{ij}|\textbf{1}\rangle_{\infty }$. Explicitly, this state is

$$\begin{eqnarray} \rho^{\mathbf{k}_1,\mathbf{k}_2 }_{ij} &= \text{Tr}_{-\mathbf{k}_1,-\mathbf{k}_2}\left(|\phi_{i,j}\rangle_{\infty \infty }\langle\phi_{i,j}|\right) = \sum^{\infty}_{n=0}p_{n}\rho^{\mathbf{k}_1,\mathbf{k}_2 }_{ij,n}\nonumber\\ &= (1 - \gamma)^3\sum^{\infty}_{n=0}\sum^{n}_{m=0}\biggl[\gamma^{n - 1}\left[(n - m)|x_{ij}|^2 + m|y_{ij}|^2\right]\nonumber\\ &\times | m, n - m\rangle\langle m,n - m| + \Bigl(x_{ij}y^{*}_{ij}\gamma^{n}\sqrt{n - m - 1}\nonumber\\ &\times\sqrt{m + 1}| m, n - m + 1\rangle\langle m + 1,n - m| + \text{h.c.} \Bigr)\biggr],\nonumber\\ \end{eqnarray} \tag{ 23 }$$

where $p_{0} = 0$, $p_{1} = (1 - \gamma)^3$, and $p_{n} = (1 - \gamma)^3\gamma^{n - 1}$.

The teleportation will be complete only when Bob receives from Alice the classical information $\{i,j\}$ concerning the result of her measurement and applies a unitary transformation to verify the protocol in his local frame. Notice that this will be feasible as long as Alice and Bob remain causally connected as $\eta \to \infty$ (smooth expansion)¹ . From eq. (23), we obtain the fidelity corresponding to the teleportation,

$$\begin{equation} F \equiv \text{Tr}\left(|\psi\rangle\langle\psi| \rho^{\mathbf{k}_1,\mathbf{k}_2 }\right) = \left(1 - \gamma\right)^3. \end{equation} \tag{ 24 }$$

This result, together with the results found for the Bogoliubov coefficients, allows us to determine the influence of spacetime expansion on the protocol. It suggests that even a small disturbance in conformal symmetry results in degradation of the quality of the teleportation. In fact, it is only under conformal symmetry that there is no particle creation due to spacetime expansion and no response on the logarithmic negativity of the quantum teleportation channel and on the fidelity F. A numerical study of (24) shows that the fidelity of teleportation always degrades due to conformal symmetry breaking (non-zero mass or/and non-conformal coupling).

Moreover, for $m \ne 0$, as the coefficient ξ increases from minimal to conformal, the fidelity increases. Concerning the role played by the parameters of the expansion, we find that the rapidity ρ of the expansion leads to a reduction of the fidelity, but only up to a threshold. If the expansion is too fast (large ρ), the fidelity and logarithmic negativity are not affected. Our analysis is summarized in fig. 2. Because the creation of high-mass and high $|k|$ modes particles demands a larger amount of energy from spacetime expansion, only low $|k|$ modes significantly contribute to fidelity degradation.

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Let us extend our considerations to the teleportation of fermionic modes to assess the effect of the Grassman character of the field on the teleportation fidelity. For this purpose consider a Dirac spinless field of mass m. In this case, it can be shown [22] that the relation between the in- and out-region vacua is

$$\begin{equation} | 0\rangle_{\mathrm{in}} \equiv \overline{\mathcal{N}} \exp\left(\int \text{d}^3\mathbf{k}\tfrac{\beta^{(-)*}_{k}}{\alpha^{(-)*}_{k}}X(k)a^{\mathrm{out}\dagger}_{\mathbf{k}}b^{\mathrm{out}\dagger}_{-\mathbf{k}}\right)| 0\rangle_{\mathrm{out}}, \end{equation} \tag{ 25 }$$

where $\overline{\mathcal{N}}$ is a normalization constant, $a^{\dagger },b^{\dagger }$ denote creation operators for particles and antiparticles, respectively, and X(k) is the polarization tensor given by

$$\begin{equation} X(k) = \frac{\sqrt{1 + 2\epsilon}\, m}{k}\left(1 - \frac{\omega_{\mathrm{out}}}{\sqrt{1 + 2\epsilon}\,m}\right). \end{equation} \tag{ 26 }$$

The Bogoliubov coefficients can also be found in a perturbative approach. The result is identical to the scalar case, save that $V(\eta)$ is substituted with

$$\begin{equation} V_f(\eta) \equiv [C(-\infty) - C(\eta)]m^2 - im\frac{C'(\eta)}{\sqrt{C(\eta)}}. \end{equation} \tag{ 27 }$$

Following the notation used in [16,22] we redefine $\alpha_{k} \rightarrow \alpha^{(-)}_{k}$ and $\beta_{k} \rightarrow \beta^{(-)}_{k}$. The resulting expression for $\frac{\beta^{(-)}_{k}}{\alpha^{(-)}_{k}}$ is

$$\begin{equation} \frac{\beta^{(-)}_{k}}{\alpha^{(-)}_{k}} \approx \left(\frac{m^2\epsilon}{\omega\rho} - \frac{m\epsilon}{\rho}\right)\frac{\pi}{\sinh(\frac{\pi\omega}{\rho})}. \end{equation} \tag{ 28 }$$

This result coincides to leading order in in which $\omega_{+} = \omega +\tfrac{m^2 \epsilon }{\omega }$ and $\omega _{-} = \tfrac{m^2 \epsilon }{\omega }$, with $\gamma^{-}$ obtained in [16]. After some algebra in (25), we verify that $|0\rangle_{\mathrm{in}}$ evolves in the asymptotic future into the following state in terms of particle and antiparticle modes:

$$\begin{equation} | 0\rangle_{\infty } = \overline{\mathcal{N}}\prod_{\mathbf{k}}\left(| 0\rangle - \gamma^{-*}| 1_{\mathbf{k}}1_{-\mathbf{k}}\rangle\right). \end{equation} \tag{ 29 }$$

Notice that due to the Pauli exclusion principle the number of fermionic excitations is limited to {1, 0}. Similarly, the 1-particle excitation state in the in region evolves into

$$\begin{equation} |1\rangle_{\infty } = |1_{\mathbf{k}}0_{-\mathbf{k}}\rangle. \end{equation} \tag{ 30 }$$

The experiment can be considered along the same lines as for the scalar field case. A dual rail basis state can be interpreted in terms of particle excitations in one of two possible spatial modes in Alice's or Bob's cavity. Assuming they shared a Bell state of the same form as in eq. (18) in the distant past, the entangled state $\hat{\tau }$ available to perform the teleportation at $\eta \to \infty$ would result from tracing out the antiparticle degrees of freedom out of Alice's party. It would have the same form shown in eqs. (19), (20) but with the expressions (29) and (30) for the evolved Fock states. The resulting logarithmic negativity and fidelity of teleportation are found to be

$$\begin{eqnarray} \begin{array}{rcl} \mathrm{E}_{\mathcal{N}}&=&\displaystyle\log _2 \left(1+ \frac{1}{1+|\gamma^{-}X(k)|^2 }\right),\\ F&=&\displaystyle\frac{1}{1 + |\gamma^{-}X(k)|^2}. \end{array} \end{eqnarray} \tag{ 31 }$$

The numerical analysis of the expression shows that the reduction of the fidelity in the present case is qualitatively similar to the conformally coupled bosonic case. This is a consequence of the already observed fact that the generation of bosonic particles for $\xi = \tfrac{1}{4}$ is equivalent to the generation of fermionic particles, which can also be traced back to the fact that the Dirac equation in curved spacetime can be written as a Klein-Gordon–type equation with a non-minimal coupling to gravity and covariant derivatives.

HA and MS acknowledge financial support from CNPq. PM acknowledges support from STFC under the consolidated Grants ST/J000426/1 and ST/L000407/1. His research is also supported in part by the Marie Curie network GATIS (gatis.desy.eu) of the European Union's Seventh Framework Programme FP7/2007-2013/ under REA Grant Agreement No. 317089.