Random approximate quantum information masking

Abstract: Masking information into quantum correlations is a cornerstone of many quantum information applications. While there exist the no-hiding and no-masking theorems, approximate quantum information masking (AQIM) offers a promising means of circumventing the constraints. Despite its potential, AQIM still remains underexplored, and constructing explicit approximate maskers remains a challenge. In this work, we investigate AQIM from multiple perspectives and propose using random isometries to construct approximate maskers. First, different notions of AQIM are introduced and we find there are profound intrinsic connections among them. These relationships are characterized by a set of figures of merit, which are introduced to quantify the deviation of AQIM from exact QIM. We then explore the possibility of realizing AQIM via random isometries in bipartite and multipartite systems. In bipartite systems, we identify a fundamental lower bound for a key figure of merit, implying that almost all random isometries fail to realize AQIM. This surprising result generalizes the original no-masking theorem to the no-random-AQIM theorem for bipartite systems. In contrast, in multipartite systems, we show almost all random isometries can realize AQIM. Remarkably, the number of physical qubits required to randomly mask a single logical qubit scales only linearly. We further explore the implications of these findings. In particular, we show that, under certain conditions, approximate quantum error correction is equivalent to AQIM. Consequently, AQIM naturally gives rise to approximate quantum error correction codes with constant code rates and exponentially small correction inaccuracies. Overall, our results establish quantum information masking as a central concept in quantum information theory, bridging diverse notions across multiple domains.