  In this paper, we contribute to the mathematical foundations of the recently
established theory of Spectrum Broadcast Structures (SBS). These are
multipartite quantum states, encoding an operational notion of objectivity and
exhibiting a more advanced form of decoherence. We study SBS in the case of a
central system interacting with N environments via the von Neumann-type
measurement interactions, ubiquitous in the theory of open quantum systems. We
state and prove a novel sufficient condition for SBS to arise dynamically for
finite-dimensional systems. The condition is based on the Gram-Schmidt
orthogonalization rather than on the Knill-Barnum error estimation used before.
