Unidimensional continuous-variable quantum key distribution using squeezed states

The possibility of using squeezed states in the recently suggested unidimensional continuous-variable quantum key distribution based on a single quadrature modulation is addressed. It is shown that squeezing of the signal states expands the physicality bounds of the effective entangled state shared between the trusted parties due to the antisqueezing noise in the unmodulated quadrature. Modulation of the antisqueezed quadrature, on the other hand, effectively shrinks the physicality bounds due to the squeezing in the unmodulated quadrature and also provides noise on the reference side of the protocol, thus limiting the possibility of eavesdropping in noisy channels. This strategy is practical for low-loss (i.e., short-distance) channels, especially if direct reconciliation scheme is applied.

Figure 1

Scheme of the squeezed-state UD CV QKD protocol. Alice prepares quadrature-squeezed states using, e.g., an optical parametric oscillator, and then modulates a state by displacing it along the modulated quadrature using the modulator M so that the modulation variance is $V_M$. The states travel through an untrusted, generally phase-sensitive channel (with transmittance values ${\eta }_x$ and ${\eta }_p$ and excess noise values ${\epsilon }_x$ and ${\epsilon }_p$ in the $x$- and $p$-quadratures, respectively) to a remote party, Bob, who performs homodyne measurement of the modulated quadrature. Inset: The equivalent entanglement-based scheme using a two-mode squeezed vacuum source; mode A is measured by Alice using a homodyne detector, and mode B is squeezed on the squeezer S and sent to a channel.

Figure 2

Modulation schemes for unidimensional protocols: (a) using coherent states [37], (b) using $x$-quadrature squeezed states and modulation in the squeezed quadrature, and (c) using $p$-quadrature squeezed states and modulation in the antisqueezed quadrature.

Figure 3

Physicality (solid lines) and security within the physicality (dotted lines, DR; dashed lines, RR) regions of the UD protocol. Modulation variance $V_M=10$, channel transmittance in $x{\eta }_x=0.9$, noise in $x{\epsilon }_x=3\%$ SNR. Plots are given for coherent-state ($V_S=1$; middle, black lines), squeezed-state ($V_S=0.9$;lower, red lines), and anti-squeezed-state ($V_S=1.1$; upper, blue lines) protocols.

Figure 4

Physicality (solid lines) and security within the physicality (dotted lines, DR; dashed lines, RR) regions of the UD protocol in channels with symmetric transmittance ${\eta }_x={\eta }_p=0.9$ with respect to excess noise ${\epsilon }_p$. Modulation variance $V_M=10$, noise in $x{\epsilon }_x=3\%$ SNR. Plots are given for coherent-state ($V_S=1$; middle, black lines), squeezed-state ($V_S=0.9$; lower, red lines), and anti-squeezed-state ($V_S=1.1$; upper, blue lines) protocols.

Figure 5

Secure key rate versus channel attenuation (on dB scale), secure against collective attacks for the DR (left) and RR (right) protocols, for the coherent-state (solid lines), squeezed-state ($V_S=0.5$; dashed lines), and anti-squeezed-state ($V_S=2$; dotted lines) protocols. Channel noise $\epsilon =3\%$ SNU, modulation variance $V_M=100$.

Figure 6

Maximal tolerable channel noise $\epsilon$ versus channel attenuation (on dB scale) for the protocols, secure against collective attacks for DR (left) and RR (right), based on coherent states (solid lines), squeezed states ($V_S=0.5$; dashed lines), and antisqueezed states ($V_S=2$; dotted lines). Modulation variance $V_M=100$.