Tight winning probability for BB84-state unclonable encryption

Determine the exact optimal winning probability for unclonable encryption from BB84 states, and in particular establish whether the tight upper bound is \(\frac12+\frac12\cos^{2n}(\pi/8)\).

Background

The paper proves unclonable indistinguishability for the BB84-state encryption scheme by combining its general common-mask Goldreich–Levin reduction with the BB84 monogamy-of-entanglement bound. This yields an upper bound of 12+12cos⁡n/2(π/8)\frac12+\frac12\cos^{n/2}(\pi/8) on the optimal winning probability.

The authors explicitly note that this bound is probably not tight because the general reduction incurs a quartic loss. They report numerical evidence suggesting that the actual optimal bound is substantially stronger, namely 12+12cos⁡2n(π/8)\frac12+\frac12\cos^{2n}(\pi/8). Establishing this sharper value remains a separate quantitative problem beyond the result proved in the paper.

References

We remark that the upper bounds in Corollaries~\ref{cor:intro-bb84-ue} and~\ref{cor:intro-coset-ue} are most likely not tight, as they are obtained by combining our general search-to-decision reduction, which pays a quartic loss, with the corresponding search bounds. In particular, numerical evidence strongly suggests that the tight upper bound for unclonable encryption from BB84 states is actually $\frac12+\frac12\cos{2n}!\left(\frac{\pi}{8}\right)$.

— Unclonable encryption from BB84 states: a simultaneous Goldreich-Levin reduction  (2608.17629 - Coladangelo et al., 18 Aug 2026) in Section 1, immediately following Corollary 1 (paragraph beginning “We remark that the upper bounds…”).