  Suppose N parties describe the state of a quantum system by N possibly
different density operators. These N state assignments represent the beliefs of
the parties about the system. We examine conditions for determining whether the
N state assignments are compatible. We distinguish two kinds of procedures for
assessing compatibility, the first based on the compatibility of the prior
beliefs on which the N state assignments are based and the second based on the
compatibility of predictive measurement probabilities they define. The first
procedure leads to a compatibility criterion proposed by Brun, Finkelstein, and
Mermin [BFM, Phys. Rev. A 65, 032315 (2002)]. The second procedure leads to a
hierarchy of measurement-based compatibility criteria which is fundamentally
different from the corresponding classical situation. Quantum mechanically none
of the measurement-based compatibility criteria is equivalent to the BFM
criterion.
