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In this note, we analyze joint probability distributions that arise fromoutcomes of sequences of quantum measurements performed on sets of quantumstates. First, we identify some properties of these distributions that need tobe fulfilled to get a classical behavior. Secondly, we prove that a jointdistribution exists iff measurement operators "on-state" permute (permutabilityis the commutativity of more than two operators). By "on-state" we meanproperties of operators that hold only on a subset of states in the Hilbertspace. Then, we disprove a conjecture proposed by  Carstens, Ebrahimi, Tabia, and Unruh (eprint 2018), which states that theproperty of partial on-state permutation implies full on-state permutation. Wedisprove this conjecture with a counterexample where pairwise "on-state"commutativity does not imply on-state permutability, unlike in the case ofcommutativity for all states in the Hilbert space.  Finally, we explore the new concept of on-state commutativity by showing asimple proof that if two projections almost on-state commute, then there is acommuting pair of operators that are on-state close to the originals. Thisresult was originally proven by Hastings (Communications in MathematicalPhysics, 2019) for general operators.

On-State Commutativity of Measurements and Joint Distributions of Their Outcomes