  Multipartite quantum states that cannot be uniquely determined by their
reduced states of all proper subsets of the parties exhibit some inherit
`high-order' correlation. This paper elaborates this issue by giving necessary
and sufficient conditions for a pure multipartite state to be locally
undetermined, and moreover, characterizing precisely all the pure states
sharing the same set of reduced states with it. Interestingly, local
determinability of pure states is closely related to a generalized notion of
Schmidt decomposition. Furthermore, we find that locally undetermined states
have some applications to the well-known consensus problem in distributed
computation. To be specific, given some physically separated agents, when
communication between them, either classical or quantum, is unreliable and they
are not allowed to use local ancillary quantum systems, then there exists a
totally correct and completely fault-tolerant protocol for them to reach a
consensus if and only if they share a priori a locally undetermined quantum
state.

Quantum Physics