Direct and Reverse Secret-Key Capacities of a Quantum Channel

We define the direct and reverse secret-key capacities of a memoryless quantum channel as the optimal rates that entanglement-based quantum-key-distribution protocols can reach by using a single forward classical communication (direct reconciliation) or a single feedback classical communication (reverse reconciliation). In particular, the reverse secret-key capacity can be positive for antidegradable channels, where no forward strategy is known to be secure. This property is explicitly shown in the continuous variable framework by considering arbitrary one-mode Gaussian channels.

Figure 1

Quantum channel $\mathcal{N}$ and its dilation.

Figure 2

Key-distribution protocol in direct reconciliation.

Figure 3

Key-distribution protocol in reverse reconciliation.

Figure 4

Canonical decomposition. In the center of the figure (Eve), we show the decomposition of a one-mode Gaussian channel $\mathcal{G}$ into a canonical form $\mathcal{C}(\tau ,r,\overline{n})$ up to a pair of unitaries $\widehat{U}$ and ${\widehat{U}}^{\prime }$. Noisy reverse protocol. The reverse protocol with the rate of Eq. (4) is achieved by applying two inverse unitaries ${\widehat{U}}^{-1}$ and ${\widehat{U}}^{\prime -1}$ (not shown in the figure), restricting the quantum distribution to a two-mode squeezed vacuum state $\Phi =|\mu ⟩⟨\mu |$, and providing Alice and Bob with homodyne detectors (see boxes in the figure). In particular, Bob’s homodyne detector is placed after one of the two output ports of a balanced beam splitter mixing the signal with the vacuum (the output of the other port is discarded).

Figure 5

Scaled thermal noise $\epsilon$ versus transmission $\tau \ne 1$. The thin solid curve ${\epsilon }_Q={\epsilon }_Q(\tau )$ refers to $Q^{(1,g)}(\mathcal{G})$, while the thick solid curve ${\epsilon }_R={\epsilon }_R(\tau )$ refers to $E_R(\mathcal{G})$. Above (below) these curves the corresponding capacities are zero (positive). Notice that $E_R(\mathcal{G})$ is positive in the region $0<\tau \le 1/2$ and $\epsilon <{\epsilon }_R(\tau )$, where the channel is antidegradable. The dashed curve ${\epsilon }_{◂}={\epsilon }_{◂}(\tau )$ corresponds to $R_{◂}=0$, where $R_{◂}$ is the rate given in Eq. (4).