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For $i=1,2$, two measurements $m_{i}:\mathcal{X}_{i}\to\mathcal{L}(\mathcal{H})$, from alphabet $\mathcal{X}_{i}$ to set of bounded linear operators on Hilbert space $\mathcal{H}$, are compatible or jointly measurable if they are marginals of a measurement $m:\mathcal{X}_{1}\times\mathcal{X}_{2}\to\mathcal{L}(\mathcal{H})$. The marginals are aquired by summing out $x\in\mathcal{X_{i}}$ for $m_{j}$ $j\neq i$. This is the sense in which joint measurability or compatibility is defined in the literature 1 ,2. How does one prove commutativity of two jointly measurable observables?

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    $\begingroup$ Are you sure that the two measurements are on different Hilbert spaces? Because an operator on $H_1$ commutes with an operator on $H_2$ by definition. On a quick skim, I did not see anything about product spaces in the linked papers either. $\endgroup$
    – Noiralef
    Nov 20 '21 at 19:42

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