Hiding and Masking Quantum Information in Complex and Real Quantum Mechanics

Classical information can be completely hidden in the correlations of bipartite quantum systems. However, it is impossible to hide or mask all quantum information according to the no-hiding and no-masking theorems derived recently. Here, by virtue of Hurwitz-Radon matrices (representations of the Clifford algebra), we show that information about real density matrices can be completely hidden in the correlations, although the minimum dimension of the composite Hilbert space required increases exponentially with the original Hilbert space. Moreover, the set of real density matrices is a maximal maskable set within quantum theory and has a surprising connection with maximally entangled states. Our study reveals a fundamental distinction between real quantum mechanics and complex quantum mechanics in hiding and masking quantum information, which is of intrinsic interest to a number of active research areas.