Quantum multipartite maskers vs quantum error-correcting codes

TL;DR

This paper explores the relationship between quantum multipartite maskers and quantum error-correcting codes, establishing new theoretical links and proving limitations on universal masking in quantum systems.

Contribution

It introduces a formal connection between quantum maskers and error-correcting codes, proving non-existence of certain universal maskers and extending understanding of quantum information hiding.

Findings

No isometric universal masker from a22 to a22a22a22 exists.

States of a23 cannot be masked isometrically into a22a22a22.

Quantum states of a2d can be hidden in correlations between any two subsystems of a tripartite system.

Abstract

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)].