Quantum multipartite maskers vs quantum error-correcting codes

Since masking of quantum information was introduced by Modi et al. in [PRL 120, 230501 (2018)], many discussions on this topic have been published. In this paper, we consider relationship between quantum multipartite maskers (QMMs) and quantum error-correcting codes (QECCs). We say that a subset $Q$ of pure states of a system $K$ can be masked by an operator $S$ into a multipartite system $\H^{(n)}$ if all of the image states $S|\psi\>$ of states $|\psi\>$ in $Q$ have the same marginal states on each subsystem. We call such an $S$ a QMM of $Q$. By establishing an expression of a QMM, we obtain a relationship between QMMs and QECCs, which reads that an isometry is a QMM of all pure states of a system if and only if its range is a QECC of any one-erasure channel. As an application, we prove that there is no an isometric universal masker from $\C^2$ into $\C^2\otimes\C^2\otimes\C^2$ and then the states of $\C^3$ can not be masked isometrically into $\C^2\otimes\C^2\otimes\C^2$. This gives a consummation to a main result and leads to a negative answer to an open question in [PRA 98, 062306 (2018)]. Another application is that arbitrary quantum states of $\C^d$ can be completely hidden in correlations between any two subsystems of the tripartite system $\C^{d+1}\otimes\C^{d+1}\otimes\C^{d+1}$, while arbitrary quantum states cannot be completely hidden in the correlations between subsystems of a bipartite system [PRL 98, 080502 (2007)].
Comment: This is a revision about arXiv:2004.14540. In the present version, $k$ and $j$ old Eq. (2.2) have been exchanged and the followed three equations have been corrected