Encrypted Redundancy as a Diagnostic Resource: Relational Diagnosis in Quantum Encrypted Cloning

Abstract

Quantum encrypted cloning encodes an unknown state into several encrypted components, each offering an alternative way to recover it later. We show that this redundancy can also serve for one-shot fault diagnosis: instead of inspecting the encrypted state, we measure relational Pauli observables testing consistency conditions imposed by the encoding. The canonical Yamaguchi--Kempf scheme encodes the input into \(n\) signal--key pairs, all carrying the same coherent Bell label, plus a transformed copy of the input qubit. Comparing the labels of two pairs yields a deterministic check that localizes an anomalous pair and identifies its Pauli-error class without revealing the label. Such checks cannot tell whether the signal or the key of that pair is faulty, and no measurement on the pairs can (we prove that they already generate the whole group of deterministic state-blind observables supported there, of rank \(2(n-1)\)). This ambiguity matters: a faulty signal costs one redemption path, a faulty key threatens them all. However, retaining also the transformed input qubit contributes exactly two further independent checks, at any multiplicity, and these suffice to identify any single fault drawn from the single-qubit Pauli set. For three clones this gives six checks in all, and we prove that no smaller set of state-blind observables achieves the same resolution. Across clone multiplicities, parity governs how many components must be measured jointly to reach this resolution, not the resolution itself. These findings are then abstracted into a general framework --- state-blind observables, deterministic healthy references, syndrome-induced fault partitions, redemption-oriented sufficiency --- yielding \textit{relational diagnosability}: a syndrome need not identify every fault, only enough of it to select a safe redemption path.