Quantum Teleportation Fidelity Assessed in Expanding Friedmann-Robertson-Walker Universes with Scalar Fields

Scientists are increasingly exploring the fundamental limits of quantum information transfer within cosmological contexts. Babak Vakili from the Department of Physics, CT.C., Islamic Azad University, Tehran, Iran, alongside Babak Vakili and et al., present new research detailing quantum teleportation protocols in an expanding Friedmann-Robertson-Walker universe. Their work, utilising a field-theoretical approach and Bogoliubov transformations, demonstrates how spacetime curvature and cosmic expansion impact the fidelity of quantum teleportation. This is significant because it sheds light on whether information transfer is possible across vast cosmological distances and offers insights into the interplay between quantum mechanics and general relativity, potentially influencing our understanding of the early universe and the nature of information itself.

They also compare their results with the flat Minkowski case to highlight the role of cosmic expansion in degrading or preserving quantum information . Quantum teleportation is a remarkable achievement in quantum information science, allowing the transmission of an arbitrary unknown quantum state without the physical transfer of the particle itself . This protocol exploits prior shared Entanglement between sender (Alice) and receiver (Bob), and the transmission of two classical bits of information . It relies on the nonlocal correlations present in entangled states, enabling state reconstruction at the receiver’s side following a joint Bell-state measurement and classical communication .

Quantum teleportation has been realized in various physical systems, including Photonic qubits, trapped ions, and Superconducting circuits, and is a cornerstone protocol in quantum communication, distributed quantum computing, and quantum networks . Its theoretical significance lies in the insight it provides into the nonlocality and foundational aspects of quantum mechanics . While most theoretical and experimental treatments assume a flat (Minkowski) spacetime, there is increasing interest in understanding how relativistic effects and spacetime curvature influence the protocol . This is especially relevant for long-distance quantum communication in space-based platforms or where gravitational fields cannot be neglected .
Quantum field theory in curved spacetime predicts phenomena such as particle creation, entanglement degradation and observer-dependent vacuum structure, all of which may influence quantum information protocols . For instance, uniform acceleration (through the Unruh effect) or proximity to a black hole horizon can reduce entanglement and decrease teleportation fidelity . Following these foundational works, several authors extended the analysis to relativistic or curved backgrounds, revealing how motion and spacetime curvature affect quantum information transfer . Despite these advances, much existing work has focused on idealized or static backgrounds .

Realistic cosmological settings involve time-dependent geometries, such as those described by the Friedmann, Robertson, Walker (FRW) metric . The FRW universe, the standard model of modern cosmology, describes a spatially homogeneous and isotropic universe undergoing expansion . The dynamical nature of this geometry modifies the vacuum structure and mode decomposition of quantum fields through time-dependent Bogoliubov transformations, giving rise to particle creation and decoherence, which can influence quantum information processes . While the general structure of the quantum teleportation protocol, entanglement distribution, local gate operations, Bell measurement, and classical communication, remains valid in curved spacetimes, the physical realization of each step becomes more involved .

In particular, the definition and implementation of quantum gates and measurements in expanding or curved backgrounds require careful treatment due to ambiguities in mode localization, time evolution, and causal structure . Two main approaches have been discussed in the literature: the operational approach, which assumes localized observers able to perform ideal quantum operations in their local frames, and a more fundamental field-theoretic formulation, where quantum gates and protocols are modeled directly in quantum field theory on curved backgrounds . While entanglement degradation in expanding universes has been studied from a field-theoretic perspective, teleportation fidelity provides a task-oriented and operationally meaningful measure of how spacetime dynamics influence the practical use of entanglement as a communication resource . This paper studies the quantum teleportation of a single qubit between two inertial observers in a spatially flat FRW universe .

By explicitly analyzing the field mode transformations in this time-dependent background, they evaluate the resulting fidelity of the teleportation protocol and its sensitivity to the scale factor and particle creation rate . For instance, Ball et al analysed the entanglement between modes of a quantum field in expanding universes, while Bruschi et al examined particle creation and correlation dynamics in FRW spacetimes . More recent analyses have explored the dependence of entanglement entropy on cosmological parameters and the possibility of preserving quantum correlations in dynamically evolving backgrounds . This work contributes to this direction by systematically comparing different expansion regimes, including de Sitter and power-law models .

They consider a spatially flat FRW spacetime with the line element given by ds2 = −dt2 + a2(t) dx2 + dy2 + dz2, where a(t) is the scale factor encoding the expansion of the universe . This background metric serves as the curved spacetime arena for the quantum teleportation process . They introduce the conformal time η, defined by dη = dt a(t), so that the metric becomes conformally flat ds2 = a2(η) −dη2 + dx2 + dy2 + dz2 . This transformation is useful when quantizing fields, as it allows the use of plane-wave modes in the conformal coordinates . They employ the conformal representation to analyse quantum teleportation between two comoving observers in expanding backgrounds, focusing on the power law expansion characterized by a(t) ∝tα, with α 0, and the exponential expansion de Sitter universe with the scale function a(t) ∝eHt, leading to a(η) = −1/(Hη) with H the Hubble parameter .

They model the quantum system as a real, minimally coupled, massless scalar field φ(η, x) propagating in the FRW background, with the action given by S = −1 2 Z d4x √−g gμν∂μφ ∂νφ, leading to the equation of motion □φ = 1 √−g∂μ √−g gμν∂νφ = 0 . This leads to φ′′ + 2a′ a φ′ −∇2φ = 0, where primes denote derivatives with respect to conformal time η . Given the conformal flatness of the metric, the equation simplifies to ∂2 η −∇2 −a′′(η) a(η) (a(η)φ) = 0 . They define the rescaled field χk(η) ≡a(η)uk(η), transforming the equation into χ′′ k + k2 −a′′ a χk = 0, which resembles a harmonic oscillator with a time-dependent effective mass term −a′′/a .

This equation will serve as the foundation for their analysis of mode evolution and the corresponding vacuum structure in various cosmological scenarios . They focus on a specific form of the scale factor a(η), depending on the cosmological model under consideration, such as de Sitter, radiation-dominated or matter- dominated universes . The two observers, traditionally labeled Alice and Bob, are assumed to be comoving with the cosmic fluid . Alice prepares an entangled state by locally coupling to two modes of the scalar field, then performs a teleportation protocol by making a Bell measurement and sending classical information to Bob, who applies a local operation to reconstruct the quantum state . However, due to the nonstationary nature of spacetime, the field modes associated with the observers are no longer globally orthogonal, leading to effective decoherence and information loss . They will compute the explicit form of the Bogoliubov coefficients for a given scale factor and analyse how the resulting mode-mix .

Teleportation fidelity in expanding Friedmann-Robertson-Walker spacetimes

The study pioneered a method for quantifying entanglement degradation in Gaussian states by modelling cosmological particle creation as a thermal-like noise channel. Experiments constructed the covariance matrix of field modes from equal-time two-point functions, defined as σij = 1/2⟨{Ri, Rj}⟩, where R = (qk, pk) represents the phase-space vector for mode k, with qk = φk and pk = πk = a2(η)φ′ k. Explicitly, the components were calculated as σqq = ⟨qkqk⟩= |φk(η)|2, σpp = ⟨pkpk⟩= |a2(η)φ′ k(η)|2, and σqp = R φk(η)a2(η)φ′∗ k (η). These quantities fully characterise the Gaussian state, forming the basis for fidelity calculations.

Scientists harnessed the standard formulae from continuous-variable quantum information theory to calculate teleportation fidelity, expressed as F = 2 p det (2σ + I), where σ is the 4 × 4 covariance matrix and I is the identity matrix. For symmetric two-mode states and vacuum or coherent inputs, the formula simplified to F = det A + 1 2I2 −1/2, utilising the local 2 × 2 covariance block A for one mode. This formalism enabled a systematic cross-check of analytical fidelity results and clarified the geometric contribution of spacetime expansion to quantum correlation loss. The normalized positive-frequency mode function was defined as χk(η) = √−η H(2) ν (−kη), where H(2) ν is the Hankel function of the second kind and ν = 1−3α 2(1−α).

To understand fidelity behaviour, the team considered asymptotic limits. In the subhorizon limit, k|η| ≫1, the Hankel function approximated to H(2) ν (x) ≈ r 2 πxe−i(x−πν 2 −π 4), resulting in a high fidelity, Fsub → 1. Conversely, in the superhorizon limit, k|η| ≪1, H(2) ν (x) ≈i πΓ(ν) x 2 −ν, causing covariance matrix elements to grow and fidelity to fall, Fsuper ≪1, demonstrating suppressed teleportation due to coherence loss.

Cosmic expansion impacts quantum teleportation fidelity

The research compared these findings to the flat Minkowski case, highlighting the role of cosmic expansion in either degrading or preserving quantum information. Researchers recorded that the operational viewpoint, focusing on teleportation fidelity, provides a task-oriented measure of how spacetime dynamics influence entanglement as a communication resource. Further analysis established that the fidelity is sensitive to the initial entanglement shared between the observers, with higher initial entanglement partially mitigating the effects of cosmic expansion. Measurements confirm that the mode frequency also plays a critical role, with lower frequencies exhibiting greater susceptibility to degradation due to spacetime curvature.

Cosmic expansion impacts quantum teleportation fidelity

Researchers observed that cosmic expansion can degrade or preserve information, contrasting with the flat Minkowski spacetime case. The study introduces an effective squeezing parameter to model entanglement loss resulting from particle creation during expansion, providing a quantifiable measure of degradation. The authors acknowledge that their model simplifies the teleportation process by using a continuous-variable TMSV state as the shared quantum resource and encoding the teleported state as an effective single qubit.