Duality of extremal quantum states in verification and data hiding (2509.01281v1)

Abstract: Quantum state verification (QSV) and quantum data hiding (QDH) have so far been studied separately. QSV decides whether a given quantum state is close to the ideal one, with significantly lower sample complexity compared with direct application of quantum tomography. On the other hand, QDH is a cryptographic protocol that encodes secret classical information in multipartite quantum states, providing stronger security guarantees than conventional classical secret-sharing schemes. Here, we consider two fundamental quantities of a pure quantum state, determining the sample complexity needed for QSV or the security level in QDH. We demonstrate that a pure state is most difficult to verify in QSV if and only if it is most secure in QDH with respect to those quantities. Furthermore, for such extremal states, the two fundamental quantities coincide. We also generalize this relationship to one between the security of QDH using mixed states and the sample complexity for quantum subspace verification, which is a generalization of QSV. As an application, we show the existence of efficient QSV protocols that are generalizations of the existing ones and examine the fundamental limitations on QDH by synthesizing research from both domains.