Continuous-Variable Quantum Key Distribution Protocols Over Noisy Channels

A continuous-variable quantum key distribution protocol based on squeezed states and heterodyne detection is introduced and shown to attain higher secret key rates over a noisy line than any other one-way Gaussian protocol. This increased resistance to channel noise can be understood as resulting from purposely adding noise to the signal that is converted into the secret key. This notion of noise-enhanced tolerance to noise also provides a better physical insight into the poorly understood discrepancies between the previously defined families of Gaussian protocols.

Figure 1

Proposed experimental implementation of the new protocol. The source (Alice) is based on a master laser beam. A fraction of it is extracted to make the local oscillator (LO), while the rest is converted into second harmonic in a nonlinear crystal (SHG). After spectral filtering ($F_1$), the second harmonic beam pumps an optical parametric amplifier (OPA) which generates a squeezed vacuum state. Following the filtering of the second harmonic ($F_2$), this squeezed state is displaced by $a$. This is done by mixing the state on a beam splitter of high transmittivity ($T_{\mathrm{HT}}\sim 99\%$ for $B{S}_{\mathrm{HT}}$) with a coherent state of intensity $a^2/(1-T_{\mathrm{HT}})$, extracted from the LO. The attenuation (A) thus depends on $a$, which is distributed according to a Gaussian distribution $G(a)$ of variance $V_a$. Before time multiplexing ($M$) the quantum signal with the LO, Alice applies a phase shift $\theta =r\pi /2$ to it depending on the value of the random bit $r$. Then, the two components of the time multiplexed signal travel to Bob through the same fiber, thereby avoiding a spurious dephasing between the signal and LO. At Bob’s station, the two components are demultiplexed ($M^{\prime }$), and the quantum signal is heterodyne measured. The latter measurement consists in splitting the quantum signal (and LO) in two with the balanced beam splitters (${\mathrm{BS}}_B$), and then homodyning each beam. The LO used in the second measurement is dephased by $\pi /2$ in order to measure the conjugate quadrature. Each homodyne detector is composed of a balanced beam splitter (${\mathrm{BS}}_B$) and a pair of highly efficient photodiodes; the difference of the photocurrents gives the quadratures $b_x$ and $b_p$.

Figure 2

(a) Tolerable excess noise $ϵ$ (in shot-noise units) as a function of the channel losses (in dB) for RR protocols: new (solid line), squeezed states and homodyning (dashed line) [2], coherent states and homodyning (dotted line) [3], coherent states and heterodyning protocol (dash-dotted line) [6]. The optimal protocol with Bob’s added noise ${\chi }_D$ is also shown (crosses). The curves are plotted for $V\to \infty$. (b) Secret key rates as a function of the channel losses (in dB) for RR protocols: new (solid line), squeezed states and homodyning protocol (dashed line) [2]. The curves are plotted for an excess noise $ϵ=0.5$ and $V=40$.

Figure 3

Entanglement-based description of the new protocol with general Gaussian added noise on Bob’s side. The source of squeezed states on Alice’s side is replaced by an entangled pair (EPR) of variance $V$, followed by a homodyne measurement of mode $A$. The other mode is sent to Bob through the quantum channel. Before Bob’s homodyne detection, the state received by Bob is mixed with a thermal state (half of an EPR pair) of variance $N$ on a beam splitter of transmittivity $T_B$ [${\chi }_D=(1-T_B)N/T_B$].

Figure 4

(a) Optimal secret key rates as a function of the channel losses (dB) for a fixed excess noise $ϵ=0.5$ (solid line) compared to the protocol based on squeezed states and homodyning [2] (dotted line) and the new protocol proposed here (dashed line). (b) Optimal choice of ${\chi }_D$ (in shot-noise units) that maximizes the secret key rate. The curves are plotted for $V=40$.