In this note, we analyze joint probability distributions that come from the outcomes of quantum measurements performed on sets of quantum states. First, we identify the properties of these distributions that need to be fulfilled to recover a classical behavior. Secondly, we connect the existence of a joint distribution with the “on-state” permutability (commutativity of more than two operators) of measurement operators. By “on-state” we mean properties of operators that can hold only on a subset of states in the Hilbert space. Then, we disprove a conjecture proposed by Carstens, Ebrahimi, Tabia, and Unruh (eprint 2018), which states that partial on-state permutation imply full on-state permutation. We disprove such a conjecture with a counterexample where pairwise “on-state” commutativity does not imply on-state permutability, unlike in the case where the definition is valid for all states in the Hilbert space. Finally, we explore the new concept of on-state commutativity by showing a simple proof that if two projections almost on-state commute, then there is a commuting pair of operators that are on-state close to the originals. This result was originally proven by Hasting (Communications in Mathematical Physics, 2019) for general operators.
In this note, we analyze joint probability distributions that come from the outcomes of quantum measurements performed on sets of quantum states. First, we identify the properties of these distributions that need to be fulfilled to recover a classical behavior. Secondly, we connect the existence of a joint distribution with the “on-state” permutability (commutativity of more than two operators) of measurement operators. By “on-state” we mean properties of operators that can hold only on a subset of states in the Hilbert space. Then, we disprove a conjecture proposed by Carstens, Ebrahimi, Tabia, and Unruh (eprint 2018), which states that partial on-state permutation imply full on-state permutation. We disprove such a conjecture with a counterexample where pairwise “on-state” commutativity does not imply on-state permutability, unlike in the case where the definition is valid for all states in the Hilbert space. Finally, we explore the new concept of on-state commutativity by showing a simple proof that if two projections almost on-state commute, then there is a commuting pair of operators that are on-state close to the originals. This result was originally proven by Hasting (Communications in Mathematical Physics, 2019) for general operators.