  Gleason-type theorems for quantum theory allow one to recover the quantum
state space by assuming that (i) states consistently assign probabilities to
measurement outcomes and that (ii) there is a unique state for every such
assignment. We identify the class of general probabilistic theories which also
admit Gleason-type theorems. It contains theories satisfying the no-restriction
hypothesis as well as others which can simulate such an unrestricted theory
arbitrarily well when allowing for post-selection on measurement outcomes. Our
result also implies that the standard no-restriction hypothesis applied to
effects is not equivalent to the dual no-restriction hypothesis applied to
states which is found to be less restrictive.
