Full characterization of informative subsets in Quantum Encrypted Cloning

Abstract

Quantum encrypted cloning, introduced by Yamaguchi and Kempf, is a Pauli-based protocol that distributes an unknown input qubit into multiple encrypted signal-noise pairs in such a way that redundancy is created without violating the no-cloning theorem, since at most one clone can later be perfectly recovered through an appropriate decoding procedure. In previous work we showed that unauthorized subsets of the storage register are not, in general, completely uninformative, and we identified a parity-dependent leakage pattern. In the present work we extend the analysis to subsets that also include the transformed source qubit A. Exploiting the purity of the global encoded state and the complementarity between storage-only subsets and subsets containing A, we derive a full classification of the informativeness of all sets of the form $H=\{A\}\cup C$. We show that these subsets are fully informative in the generic case. Two exceptions arise. First, if all pairs are incomplete and |C|<n, then the reduced state is completely uninformative. Second, if |C|=n, n is odd, and the number q of signal qubits in C is even, then the reduced state is partially informative. In this latter case, the residual dependence on the input state is confined to the y-component of the Bloch vector. These results provide a complete parity-based characterization of leakage for subsets containing the transformed input qubit.