Finite-size secret-key rates of discrete modulation continuous-variable quantum key distribution under Gaussian attacks

TL;DR

This work considers continuous-variable communication protocols, where the sender Alice encodes information using a discrete modulation of phase-shifted coherent states, and the receiver Bob decodes by homodyne or heterodyne detection, and compute the Petz-R\'enyi and sandwiched R\'enyi conditional entropies associated with these setups.

Abstract

Quantum conditional entropies play a fundamental role in quantum information theory. In quantum key distribution, they are exploited to obtain reliable lower bounds on the secret-key rates in the finite-size regime, against collective attacks and coherent attacks under suitable assumptions. Here we consider continuous-variable communication protocols, where the sender Alice encodes information using a discrete modulation of phase-shifted coherent states, and the receiver Bob decodes by homodyne or heterodyne detection. We compute the Petz-Rényi and sandwiched Rényi conditional entropies associated with these setups, assuming either a passive eavesdropper or one that injects thermal photons into the channel, who gathers the quantum information leaked through a lossy communication line of known or bounded transmittance. Whereas our results do not directly provide reliable key-rate estimates, they do represent useful ball-park figures. We obtain analytical or semi-analytical expressions that do not require intensive numerical calculations. These expressions serve as bounds on the key rates that may be tight in certain scenarios. We compare different estimates, including known bounds that have already appeared in the literature and new bounds. The latter are found to be tighter for very short block sizes.

Figures (7)

• Figure 1: Phase-space representation of the encoding and decoding routines in BPSK (Fig. \ref{['bpskspace']}) and QPSK (Fig. \ref{['QpskScheme']}) protocols.
• Figure 2: Comparison between different kinds of entropies, computed on the CQ states at $\alpha=1$ and $a=1.2$ on $\rho_{YE}$ in Eqs. (\ref{['aaa']}) and (\ref{['CQ_qpsk']}) respectively for BPSK in \ref{['BPSKcomp']} and in \ref{['QPSKcomp']} for QPSK protocol. In particular: Petz-Rényi entropy $H^\downarrow_a(Y|E)$ (\ref{['HagiuMAIN']}); optimized Petz-Rényi entropy $H^\uparrow_a(Y|E)$ (\ref{['HasuMAIN']}); sandwiched Rényi entropy $\tilde{H}^\downarrow_a(Y|E)$ (\ref{['SandgiuMAIN']}); optimized sandwiched Rényi entropy $\tilde{H}^\uparrow_a(Y|E)$ (\ref{['HsupMAIN']}), von Neumann entropy $H(Y|E)$ (\ref{['VonNeumann']}); continuity bound $B_a(Y|E)$ (\ref{['VKBound']}). All the quantities converge to $1$ as $\eta\to0,1$ for BPSK and to $2$ for QPSK, while the minima are reached around $\eta_{\text{min}}\sim0.6$ in both cases.
• Figure 3: Secret-key rate estimates versus the number of signals $n$, for a fixed value of the transmittance at $\eta=0.9$, $\epsilon= \epsilon'=10^{-8}$. The plot compares the bounds $r_{\epsilon"}^{S}$, $r_{\epsilon"}^{B}$, $r_{\epsilon"}^{AEP}$ respectively in (\ref{['ratesandBPSK']}), (\ref{['rateboundBPSK']}) and \ref{['rateAEPBPSK']}) with $\epsilon" = \epsilon+\epsilon'$, for BPSK in \ref{['rateBPSK']} and for QPSK in \ref{['rateQ']}. The estimators $r_{\epsilon"}^{S}$ and $r_{\epsilon"}^{B}$ are optimized over the parameters $a$ and $\alpha$, while $r_{\epsilon"}^{AEP}$ is optimized over $\alpha$ alone. Asymptotically, the quantities converge to the same value, while in the finite-size region $r_{\epsilon"}^{S}$ is the tightest bound, being the only one still non-vanishing when $n<10^3$, and dropping to zero as $n < 5\cdot10^2$.
• Figure 4: Optimal value of $a>1$ for the estimator $r_{\epsilon"}^S$ based on a direct calculation of the sandwiched Rényi entropy. See other numerical details in the Mathematica code available on this github folder gioscaNPSK.
• Figure 5: Secret-key rates computed on the CQ states at $\alpha$ and $a$ on $\rho_{YE}$ in Eqs. (\ref{['rhoYEnoiseB']}) and (\ref{['rhoYEnoiseQ']}) respectively for BPSK in \ref{['fig:comparison_distance_BPSK_d']} and in \ref{['fig:comparison_distance_QPSK_d']} for QPSK protocol, in term of the distance. For QPSK an additional curve is added that reproduce the results in FlorianPRXQ2023.

Theorems & Definitions (2)

• Lemma 1
• proof