Masking quantum information encoded in pure and mixed states

Masking of quantum information means that information is hidden from a subsystem and spread over a composite system. Modi et al. proved in [Phys. Rev. Lett. 120, 230501 (2018)] that this is true for some restricted sets of nonorthogonal quantum states and it is not possible for arbitrary quantum states. In this paper, we discuss the problem of masking quantum information encoded in pure and mixed states, respectively. Based on an established necessary and sufficient condition for a set of pure states to be masked by an operator, we find that there exists a set of four states that can not be masked, which implies that to mask unknown pure states is impossible. We construct a masker $S^\sharp$ and obtain its maximal maskable set, leading to an affirmative answer to a conjecture proposed in Modi's paper mentioned above. We also prove that an orthogonal (resp. linearly independent) subset of pure states can be masked by an isometry (resp. injection). Generalizing the case of pure states, we introduce the maskability of a set of mixed states and prove that a commuting subset of mixed states can be masked by an isometry $S^{\diamond}$ while it is impossible to mask all of mixed states by any operator. We also find the maximal maskable sets of mixed states of the isometries ${S^{\sharp}}$ and ${S^{\diamond}}$, respectively.