Rotational Bell-State Loop-Back Key Establishment and Agreement Certification for Passive Users

Abstract

Server-aided Bell-state networks with sequential local encoding are an established approach to mediated key distribution. Against this background, we study a Loop-Back architecture in which two source-free, detector-free users transform the same traveling qubit. Alice privately samples $r\in \mathbb{Z}\ast 2^2$, prepares $|{\beta }_r⟩$ over the complete Bell basis, retains one qubit, and uses $r$ as the reference for the returned pair. A deterministic Pauli-composition mode, $U\ast \mathrm{eff}=U_2{U}_1$, serves as a reference to prior serial-unitary protocols. The main extension instead uses the two same-axis settings $R(\pm \alpha )$ and a reference-versus-complement Bell test. Opposite rotations cancel, whereas equal rotations add, so $P(C\mid \mathrm{disagreement})=0$ and $P(C\mid \mathrm{agreement})={\sin }^2(2\alpha )$. At $\alpha =\pi /8$, agreement is certified with probability $1/2$, and the overall conclusive-key probability is $1/4$ for unbiased choices. The same event supplies one shared raw-key bit. For the Pauli mode, a one-qubit Pauli twirl hides either user's factor from an endpoint mediator that may prepare an arbitrary qubit--ancilla state and choose the final measurement, provided the other Pauli is uniform and the traveler is inaccessible between users. The rotational guarantee is narrower: common-sign privacy holds only for the prescribed Bell source and coarse-grained instrument. A polarization implementation can normalize the private reference to the singlet and use a beam splitter with verified two-photon detection; loss produces a third, discarded outcome. The proposal is theoretical and does not claim composable security or experimental validation.