 We can regard \(\mathcal{A}\) as arising from \(\mathcal{A}^{\sim}\) by deletion of the record of which test was performed to secure a given outcome. Note that every state on \(\mathcal{A}\) defines a state \(\omega^{\sim}\) on \(\mathcal{A}^{\sim}\) by \(\omega^{\sim} (x,E) = \omega(x)\). The mapping \(\omega \rightarrow \omega^{\sim}\) is plainly injective; thus, we may identify the state-space of \(\mathcal{A}\) with a subset of the state-space of \(\mathcal{A}^{\sim}\). Notice that there will typically be many states on \(\mathcal{A}^{\sim}\) that do not descend to states on \(\mathcal{A}\). We might wish to think of these as “non-physical”, since they do not respect the (presumably, physically motivated) outcome-identifications whereby \(\mathcal{A}\) is defined.