Gaussian one-way thermal quantum cryptography with finite-size effects

We study the impact of finite-size effects on the security of thermal one-way quantum cryptography. Our approach considers coherent and/or squeezed states at the preparation stage, on the top of which the sender adds trusted thermal noise. We compute the key rate incorporating finite-size effects, and we obtain the security threshold at different frequencies. As expected, finite-size effects deteriorate the performance of thermal quantum cryptography. Our analysis is useful to quantify the impact of this degradation on relevant parameters like tolerable attenuation and transmission frequencies at which one can achieve security.

Figure 1

Initial mode $A$ is in a thermal state with variance $V_{\text{th}}+V_s$ and modulated with variance $V_M$. After mode $A$ is sent through the channel, Bob receives mode $B$ and applies a homodyne detection on either $q$ or $p$ quadrature or a heterodyne detection measuring $q$ and $p$ quadrature at the same time. The thermal-loss channel is modeled by a beam splitter with transmissivity $\tau$. The Gaussian eavesdropping takes the form of an entangling cloner attack, where Eve's system is described by the modes $e$ and $E$ in a TMSV state with variance $\omega$. According to this description, an optical fiber is simulated with transmissivity $\tau ={10}^{-\frac{0.2d}{10}}$, where $d$ (in km) is the length of the fiber, and excess noise variance $V_{\varepsilon }=\tau \varepsilon$, where $\varepsilon =\frac{(1-\tau )(\omega -1)}{\tau }$.

Figure 2

Key rate in the optical regime. Left: key rate vs channel attenuation given in dB. The red solid curve describes the ideal key rate, using just coherent states. The blue-dashed curve describes the ideal key rate assuming a preparation noise with variance $V_{\mathrm{th}}=9$ SNU. Then, we keep the same $V_{\mathrm{th}}$ and plot the finite-size rate for block size with $\overline{N}={10}^9$ (black dashed line) and $\overline{N}={10}^7$ (gray dashed) with $\xi =0.98$ and $\omega =1$. The key rate is optimized over the Gaussian modulation $V_M$. Right: key rate as a function of block size ($\overline{N}$). We fix channel attenuation to 1 dB, and we assume pure loss attack $\omega =1$ while $\xi =0.98$. The plot shows the convergence of the key rates toward the asymptotic values (dashed curves) for different values of the preparation noise $V_{\mathrm{th}}=$ 0, 1, 10, 100, and 150 SNU, from top to bottom.

Figure 3

The case for signal point block size of $\overline{N}={10}^9$, reconciliation efficiency $\beta =98\%$, trusted thermal noise of $V_{\mathrm{th}}=9$ SNU, and pure loss attack $\omega =1$. We compare the finite-size key rate obtained when Alice starts from coherent states ($V_s^{\left(q\right)}=V_s^{\left(p\right)}=1$) with the case when $V_s^{\left(q\right)}={10}^{-1}$, i.e., Alice's encoding is performed by adding thermal noise on moderate squeezed states. We notice that in such a case the achievable distance is only incrementally improved. Moreover the performance of the protocol saturates for stronger initial squeezing, i.e., for $V_s^{\left(q\right)}={10}^{-2}$ (blue-dashed) and ${10}^{-5}$ (green line).

Figure 4

The red line shows the security threshold (frequencies vs channel transmissivity) for a shot-noise level attack $\omega =V_{th}+1$ without finite-size effects, and assuming infinite Gaussian modulation. Then we have the case for block size with $\overline{N}={10}^6$ signal points (black) and $\overline{N}={10}^9$ (blue).