 It is important to emphasize here that the test space \(\mathcal{A}_5\) has a perfectly unproblematic quantum-mechanical interpretation, as it can be realized as a set of orthonormal bases in a 3-dimensional Hilbert space \(\mathbf{H}\). However, the state \(\omega\) figuring in the Foulis-Randall example cannot arise quantum-mechanically (much less classically). (Indeed, this follows from the example itself: the canonical mapping \(\mathbf{H} \times \mathbf{H} \rightarrow \mathbf{H} \otimes \mathbf{H}\) provides a mapping satisfying the conditions (a) and (b) above. Since \(\mathbf{L}(\mathbf{H} \otimes \mathbf{H})\) is orthocoherent, the set S corresponds to a pairwise orthogonal family of projections, over which a quantum-mechanical state would have to sum to no more than 1.) 