Effective channel representations and composable security for passive-user Loop-Back quantum key distribution

This work develops effective channel representations and a composable-security framework for distributed passive-user Loop-Back quantum key distribution (QKD). Alice provides trusted preparation-and-measurement infrastructure at local service points, while two source-free and detector-free users, B₁ and B₂ , apply independently selected polarization rotations R(± π /8) to the same optical carrier. Two complementary derivations describe the honest operation. Direct averaging over the composed rotations gives the observable Loop-Back map, while an operationally equivalent representation on the Bennett–Brassard 1984 (BB84) preparation-and-measurement data is a balanced mixture of conjugate-basis dephasings, or an anisotropic Pauli channel with identity, X , and Z components and no Y component. Both recover the ideal conclusive-event probability Pconc=1/4 . Security is established without restricting the adversarial action to this effective channel. Eve may implement an arbitrary coherent attack with quantum memory on the single untrusted interface between the two local rotations. In the single-photon sector, a Choi-operator identity yields the exact ideal relation e1, p=3e1, b/2 , together with a low-error robust angular bound e1, p≤ κ ᵣₒₜe1, b . Phase-randomized weak coherent pulses, decoy-state estimation and generalized entropy accumulation then give a finite-key composable bound. A common optical model compares asymptotic and finite-block rates and quantifies their convergence with block size. The analysis also identifies the trusted-interface condition required to exclude loss-assisted operation-discrimination attacks and specifies the hardware assumptions needed for optical implementation.