Practical quantum teleportation with finite-energy codebooks

TL;DR

The paper advances practical continuous-variable quantum teleportation in the microwave regime by incorporating finite-energy codebooks, feedforward losses, and thermal noise into a full covariance-matrix framework. It demonstrates how a displacement-matching gain can mitigate feedforward imperfections and outlines how no-cloning thresholds evolve for truncated codebooks, moving beyond ideal Gaussian tails. By analyzing security against a public-channel eavesdropper using mutual information and the Holevo bound, it identifies a secure fidelity threshold that depends on squeezing and gain, finding $S\ge 2.39$ dB and $G\ge 20$ dB as a regime for genuine security. These results indicate that high-fidelity, unconditionally secure microwave quantum communication is feasible over short networks, underpinning the design of practical microwave quantum links and nodes.

Abstract

Quantum communication exploits non-classical correlations to achieve efficient and unconditionally secure exchange of information. In particular, the quantum teleportation protocol allows for a deterministic and secure transfer of unknown quantum states by using pre-shared quantum entanglement and classical feedforward communication. Quantum teleportation in the microwave regime provides an important tool for high-fidelity remote quantum operations, enabling distributed quantum computing with superconducting circuits and potentially facilitating short-range, open-air microwave quantum communication. In this context, we consider practical application scenarios for the microwave analog quantum teleportation protocol based on continuous-variable states. We theoretically analyze the effect of feedforward losses and noise on teleportation fidelities of coherent states and show that these imperfections can be fully corrected by an appropriate feedforward gain. Furthermore, we consider quantum teleportation with finite-size codebooks and derive modified no-cloning thresholds as a function of the codebook configuration. Finally, we analyze the security of quantum teleportation under public channel attacks and demonstrate that the corresponding secure fidelity thresholds may drastically differ from the conventional no-cloning values. Our results contribute to the general development of quantum communication protocols and, in particular, illustrate the feasibility of using quantum teleportation in realistic microwave networks for robust and unconditionally secure communication.

Figures (8)

• Figure 1: Schematic of a practical quantum teleportation protocol. A two-mode squeezed state is shared between Alice and Bob through a low-loss quantum channel. Alice performs a Bell-type measurement operation on an input code state superimposed with her part of the entangled state. Then, the measurement result is sent to Bob through a lossy and noisy feedforward channel. Finally, Bob performs a local unitary operation on his part of the entangled state according to the received feedforward signal in order to reconstruct the input state at his location. The input states are drawn from an ensemble of code states. The latter forms a certain codebook, which is represented by a truncated Gaussian distribution in this particular scheme.
• Figure 2: Microwave teleportation fidelities through thermal communication channels at the carrier frequency of $\omega/2\pi = 5G Hz$ and for a displacement photon number of $|\alpha|^2 = 10$, squeezing level of $S = 10dB$, and feedforward coupling of $\eta = -20dB$. Horizontal lines indicate characteristic mixing chamber, liquid helium, and liquid nitrogen temperatures of $50mK$, $4.2K$, and $77K$, respectively. Black solid and dashed lines indicate the asymptotic classical limit $F_\mathrm{cl} = 1/2$ and asymptotic no-cloning limit $F_\mathrm{nc} = 2/3$, respectively. (a) Teleportation fidelity $F$ as a function of the feedforward channel losses $\varepsilon_\mathrm{ff}$ and noise temperature $T_\mathrm{ff}$, where $\varepsilon_\mathrm{ent} = 0$. (b) Teleportation fidelity $F$ as a function of the entanglement distribution channel losses $\varepsilon_\mathrm{ent}$ and noise temperature $T_\mathrm{ent}$, where $\varepsilon_\mathrm{ff} = 0$. (c) Teleportation fidelity $F$ as a function of the entanglement distribution channel losses $\varepsilon_\mathrm{ent}$ and resource squeezing level $S$, where $\varepsilon_\mathrm{ff} = 0$.
• Figure 3: Teleportation fidelity as a function of various realistic experimental parameters. All experimental losses and noise sources are considered, based on the parameters in Ref. Yam2025. Horizontal lines indicate characteristic mixing chamber, liquid helium, and liquid nitrogen temperatures of $50mK$, $4.2K$, and $77K$, respectively. Black solid and dashed lines indicate the asymptotic classical limit $F_\mathrm{cl}=1/2$ and asymptotic no-cloning limit $F_\mathrm{nc}=2/3$, respectively. (a) Teleportation fidelity $F$ as a function of feedforward channel loss $\varepsilon_\mathrm{ff}$ and noise temperature $T_\mathrm{ff}$. The fidelity maintains a quantum advantage up to $\varepsilon_\mathrm{ff} \simeq 7dB$ and liquid helium temperatures. (b) Teleportation fidelity $F$ as a function of feedforward coupling strength $\eta$ and measurement gain $G$. Fidelity maxima occur near the displacement-matching condition $G\eta(1-\varepsilon_\mathrm{ff}) = 4$. (c) Teleportation fidelity $F$ as a function of feedforward coupling strength $\eta$ and noise temperature $T_\mathrm{ff}$, where the displacement-matching condition is fulfilled. The fidelity maintains quantum advantage up to room temperature at $\eta \simeq -24dB$.
• Figure 4: Probability distribution cross-sections of Gaussian, truncated uniform, and truncated Gaussian codebooks as a function of the complex displacement amplitude $\alpha$. The truncated uniform codebook is described by a cutoff photon number $N$ and is normalized to unit probability. The truncated Gaussian codebook is described by a Gaussian codebook with variance $\sigma^2$ and a cutoff photon number $N$, rescaled for normalization. Probability distribution functions in the phase space spanned by field quadratures ${p,q}$ for (b) Gaussian, (c) truncated uniform, and (d) truncated Gaussian codebooks.
• Figure 5: (a) No-cloning threshold for the truncated Gaussian codebook as a function of the cutoff photon number $N$ and codebook variance $\sigma^2$. Cross-sections of the no-cloning fidelity threshold for the truncated Gaussian codebook at various (b) $N$ and (c) $\sigma^2$. No-cloning thresholds for the Gaussian and truncated uniform codebooks are given by the plots where $N \to \infty$ and $\sigma^2 \to \infty$, respectively. Red dashed lines correspond to the asymptotic no-cloning threshold $F_\mathrm{nc} = 2/3$.