Quantum Advantage for Two-Party Differential Privacy

We introduce information-theoretically private quantum protocols for two-party Hamming distance when both parties must output the same estimate. Classically, for input length $n$, information-theoretic protocols require $\Omega(\sqrt{n})$ error under pure differential privacy and $\Omega(\sqrt{n}/\log n)$ error under strong approximate differential privacy, whereas computational security permits $O(1)$ error. In Klauck's honest, nonpreemptive, message-preserving model, we give an $O(n)$-communication quantum protocol with pure $\varepsilon$ quantum differential privacy (QDP) and expected error at most $\frac{2}{\sinh \varepsilon}+\gamma$, for every $\gamma>0$. For approximate $(\varepsilon, \delta)$ QDP, an exact hockey-stick divergence calculation yields strictly smaller error, while preserving the $O(1)$-versus-$\Omega(\sqrt{n}/\log n)$ separation for $\delta=o(1/n)$. Thus, quantum communication achieves $O(1)$ information-theoretic error, matching the accuracy available classically only under computational assumptions. The main construction uses a guarded coherent round trip and an equal-Gram rigidity principle that prevents an honest player from retaining input-dependent complementary information. We separate this model from weaker prescribed-channel privacy, which already admits an exact classical realization, and from fully retention-robust security, against which measurement-and-abort attacks remain possible. Therefore, we identify preservation of non-orthogonal quantum messages as a resource for privacy.