# Proof that joint-measurability means commutativity

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?

• 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. Nov 20 '21 at 19:42