Composable security of two-way continuous-variable quantum key distribution without active symmetrization

We present a general framework encompassing a number of continuous-variable quantum key distribution protocols, including standard one-way protocols, measurement-device-independent protocols, as well as some two-way protocols, or any other continuous-variable protocol involving only a Gaussian modulation of coherent states and heterodyne detection. The main interest of this framework is that the corresponding protocols are all covariant with respect to the action of the unitary group $\text{U}\left(n\right)$, implying that their security can be established thanks to a Gaussian de Finetti reduction. In particular, we give a composable security proof of two-way continuous-variable quantum key distribution against general attacks. We also prove that no active symmetrization procedure is required for these protocols, which would otherwise make them prohibitively costly to implement.

Figure 1

Optical scheme of the EB version of the two-way protocol: “Sq” is a two-mode squeezer; “BS” is a beamsplitter required to implement the random displacement by Bob.

Figure 2

Optical scheme of the Gaussian FL protocol.

Figure 3

Propagation of the unitaries through the circuit of the two-way protocol. “Sq” and “BS” stand respectively for two-mode squeezer and beamsplitter. One starts by applying unitaries $U$ or $\overline{U}$ to the input of the protocol and propagates these operators through the setup. Since the input of the setup is the vacuum and therefore invariant under the action of $U$ or $\overline{U}$, we infer that the protocol is invariant if Alice and Bob process their modes $A_1,A_2,B_1,B_2$ by $U\otimes \overline{U}\otimes U\otimes \overline{U}$. (a) Unitaries applied to vacuum, (b) Covariance with the initial squeezers, (c) Covariance with the forward channel, (d) Covariance with Bob's beamsplitter, (e) Covariance with the backward channel and (f) Commutation with Alice's second squeezer.

Figure 4

Propagation of the unitaries through the circuit of the FL protocol. “Sq” and “BS” stand respectively for two-mode squeezer and beamsplitter. One starts by applying unitaries $U$ or $\overline{U}$ to the input of the protocol and propagates these operators through the setup. Since the input of the setup is the vacuum and therefore invariant under the action of $U$ or $\overline{U}$, we infer that the protocol is invariant if Alice and Bob process their modes $A_1,A_2,A_3,B_1,B_2,B_3$ by $U\otimes \overline{U}\otimes \overline{U}\otimes U\otimes \overline{U}\otimes U$. (a) Unitaries applied to vacuum, (b) Covariance with the initial squeezers, (c) Covariance with Alice's beamsplitter, (d) Covariance with the forward channel, (e) Covariance with Bob's beamsplitter, (f) Covariance with Bob's second squeezer, and (g) Covariance with the backward channel.

Figure 5

Description of the EB version of the two-way protocol. Two pairs of two-mode squeezed vacuum states are initially prepared by Alice and Bob in modes $(A_1^{\prime },A_1^{″})$ and $(B_1,B_1^{\prime })$, respectively. Mode $A_1^{\prime }$ is sent through the quantum channel. The output mode $C_1$ is combined with $B_1^{\prime }$ in a beamsplitter: this effectively implements a Gaussian displacement of mode $C_1$. One output mode of the beamsplitter, $C_2$, is then sent back through the quantum channel and finally combined with $A_1^{″}$ with a two-mode squeezer. One of the two output modes, $A_2$, is then measured with heterodyne detection and the measurement result, $X_2$, serves as the raw key. In the prepare and measure version of the protocol, Alice and Bob would simply prepare $A_1^{\prime }$ and $B_1^{\prime }$ in a coherent state with a Gaussian modulation, Bob would apply the displacement operation (via the beamsplitter) and send back the output mode $C_2$ to Alice who would measure it with heterodyne detection. None of the three two-mode squeezers are needed in the PM version as they can always be simulated classically.

Figure 6

Secret key rate of the two-way CVQKD protocol (full line) and of the one-way no-switching protocol (dashed line), assuming a pure-loss channel ($\xi =0$) and perfect reconciliation efficiency ($\beta =1$).

Figure 7

Secret key rate of the two-way CVQKD protocol (full line) and of the one-way no-switching protocol (dashed line), assuming a noisy channel ($\xi =0.1$) and imperfect reconciliation efficiency ($\beta =0.95$).

Figure 8

Maximum tolerable excess noise $\xi$ for the two-way CVQKD protocol (full line) and the one-way no-switching protocol (dashed line), assuming perfect reconciliation efficiency ($\beta =1$).