Joint measurability through Naimark's dilation theorem

We use Naimark's dilation theorem in order to characterize the joint measurability of two POVMs. Then, we analyze the joint measurability of two commutative POVMs F1 and F2 which are the smearing of two self-adjoint operators A1 and A2, respectively. We prove that the compatibility of F1 and F2 is connected to the existence of two compatible self-adjoint dilations A1+ and A2+ of A1 and A2, respectively. As a corollary we prove that each couple of self-adjoint operators can be dilated to a couple of compatible self-adjoint operators. Next, we analyze the joint measurability of the unsharp position and momentum observables and show that it provides a master example of the scheme we propose. In other words, the scheme we have in the case of the position and momentum operators can be generalized to the case of an arbitrary couple of self-adjoint operators. Finally, we give a sufficient condition for the compatibility of two effects.