Conditions for compatibility of quantum-state assignments

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)]. Them 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.