Lower Bounding the Secret Key Capacity of Bosonic Gaussian Channels via Optimal Gaussian Measurements

TL;DR

The paper tackles the problem of lower-bounding the secret key capacity of bosonic Gaussian channels using fully Gaussian protocols and optimal single-mode Gaussian measurements. By recasting private communication in an entanglement-based framework and analyzing Gaussian measurements, the authors derive a simple, single-parameter optimization for the achievable secret-key rate, $\mathcal{L}^G$. They show that for phase-insensitive channels (thermal loss, thermal gain, and added noise) this bound either matches or surpasses existing bounds, with proven optimality within the Gaussian single-mode-measurement constraint for loss and amplification and an improved bound for added noise. The results tighten the gap between known bounds and provide a practical, analytically tractable method to evaluate private communication performance in CV quantum channels.

Abstract

We find the maximum rate achievable in the private communication over a bosonic quantum channel with a fully Gaussian protocol based on optimal single-mode Gaussian measurements. This rate establishes a lower bound on the secret rate capacity of the channel. We focus on the class of phase-insensitive Gaussian channels. For the thermal-loss and thermal amplification channels, our results demonstrate the optimality, within the constraints of our analysis, of previously proposed protocols, while also providing a significantly simplified formula for their performance evaluation. For the added noise channel, our rate provides a better lower bound than any previously known.

Figures (5)

• Figure 1: Communication scheme in the entanglement-based representation. Alice sends one mode, $B$, of $\Phi_{AB}$ to Bob through a channel $\mathcal{E}$, while retaining mode $A$. Information is exchanged through measurements of the state, and guessing the outcome of either by Alice (direct reconciliation) or Bob (reverse reconciliation). The action of the channel can be interpreted as an eavesdropper, Eve, acting on an extended space to recover shared information.
• Figure 2: Thermal-Loss channel. A. We plot the security threshold $\omega_{th}$, defined in the main text, as a function of the transmissivity $\eta$, for the bound $\mathcal{L}_{\omega_\eta}$ and the channel's reverse coherent information $I^{RC}_{\eta,\omega}$. B. $\mathcal{L}_{\omega,\eta}$ is compared with the lower bounds $\mathcal{B}_{\omega,\eta}$ and $\mathcal{Q}_{\omega,\eta}$ (see the main text) at a fixed value $\omega=3$. We also plot $I^{RC}_{\eta,\omega}$ as well as the upper bound on the capacity $\mathcal{U}_{\omega,\eta}$.
• Figure 3: Thermal Amplification. A. Security threshold $\omega_{th}$, for the thermal amplification channel (see main text) as a function of the gain $g$, for the bound $\mathcal{L}_{\omega,g}$ and the reverse coherent information of the channel $I^{RC}_{\omega,g}$. B. Comparison of $\mathcal{L}_{\omega,g}$ with the lower bounds $\mathcal{B}_{\omega,g}$ and $\mathcal{Q}_{\omega,g}$ (see the main text) at $\omega=3$ with $I^{RC}_{\eta,\omega}$ and the upper bound $\mathcal{U}_{\omega,\eta}$ as reference.
• Figure 4: Added Noise.A. Comparison of the bound $\mathcal{L}_\zeta^G$ with $I^{C}_{\zeta}$ and $\mathcal{Q}_\zeta$. The upper bound $\mathcal{U}_{\zeta}$ is also reported for reference. B. $\delta(\gamma)$ (see main text) is plotted for parameters $\bar{\zeta}=0.36,0.38,0.40$. The value of $-I^{C}_{\zeta}$ is reported for reference in dashed lines.
• Figure 5: Dilation of the channels.A. Thermal loss and thermal amplification are dilated by a beam splitter/thermal amplifier with parameter $\eta/g$. The dilation consists of a two-mode interaction between $B$ and $E$, while $e$ is stored for later measurement. B. The dilation of an added noise channel is a universal cloner, a three-mode interaction involving $e,E$ and $B$, described by the unitary in Eq.(\ref{['eq:UC']}).