All joint measurability structures are quantum realizable

The general impossibility of jointly implementing arbitrary sets of measurements is a fundamental feature of quantum theory. Indeed, it is a key ingredient that enables a demonstration of the nonclassicality of quantum theory in proofs of Bell’s theorem [1] and the Kochen-Specker theorem [2]. A finite set of measurements is called jointly measurable or compatible if there exists a single measurement whose various coarsegrainings recover the original measurements. The problem of characterizing the joint measurability of observables has been studied in the literature [3, 4], and at least the joint measurability of binary qubit observables has been completely characterized [5, 6]. The connection between Bell inequality violations and the joint measurability of observables has also been quantitatively studied [7, 8].