Numerical evidence for bound secrecy from two-way postprocessing in quantum key distribution

Bound secret information is classical information that contains secrecy but from which secrecy cannot be extracted. The existence of bound secrecy has been conjectured but is currently unproven, and in this work we provide analytical and numerical evidence for its existence. Specifically, we consider two-way postprocessing protocols in prepare-and-measure quantum key distribution based on the well-known six-state signal states. In terms of the quantum bit-error rate $Q$ of the classical data, such protocols currently exist for $Q<\frac{5-\sqrt{5}}{10}\approx 27.6\%$. On the other hand, for $Q\ge \frac{1}{3}$ no such protocol can exist as the observed data are compatible with an intercept-resend attack. This leaves the interesting question of whether successful protocols exist in the interval $\frac{5-\sqrt{5}}{10}\le Q<\frac{1}{3}$. Previous work has shown that a necessary condition for the existence of two-way postprocessing protocols for distilling secret keys is breaking the symmetric extendability of the underlying quantum state shared by Alice and Bob. Using this result, it has been proven that symmetric extendability can be broken up to the $27.6\%$ lower bound using the advantage distillation protocol. In this work, we first show that to break symmetric extendability it is sufficient to consider a generalized form of advantage distillation consisting of one round of postselection by Bob on a block of his data. We then provide evidence that such generalized protocols cannot break symmetric extendability beyond $27.6\%$. We thus have evidence to believe that $27.6\%$ is an upper bound on two-way postprocessing and that the interval $\frac{5-\sqrt{5}}{10}\le Q<\frac{1}{3}$ is a domain of bound secrecy.

Figure 1

Key distillability as a function of the QBER $Q$ characterizing the quantum state ${\rho }_Q^{AB}$ from which Alice and Bob obtain their classical data by measurement in the standard basis. We are interested in whether a secret key can be distilled in the gap, that is, the yellow region.

Figure 2

Repetition code thresholds $Q_{{\mathcal{R}}_n}^{*}$ up to $n=30$.

Figure 3

The highest thresholds for some of the $(n,m)$ classes from Table 1.