 Arbitrary quantum effects, and arbitrary effect-valued observables, arise quite naturally as models of actual experimental outcomes. Consider an isolated quantum system \(A\) with Hilbert space \(\mathbf{H}_A\), and an ancillary system \(B\), with Hilbert space \(\mathbf{H}_{B}\), maintained in a reference state represented by a density operator \(W^{B}_o\).  If \(A\) is in the state represented by a density operator \(W^{A}\) on \(\mathbf{H}_A\), thet state of the joint system is represented by \(W^{A} \otimes W^{B}_o\). If we make a yes-no measurement on \(AB\) represented by a projection operator \(P_{AB}\) on \(\mathbf{H}_{AB} = \mathbf{H}_{A} \otimes \mathbf{H}_{B}\) then the probability of obtaining a positive result is \(\textrm{Tr}(P_{AB}(W^{A} \otimes W^{B}_{o}))\). This defines a bounded convex-linear function of \(W^{A}\), and hence, there is a unique effect \(a\) with \(\textrm{Tr}((W^{A} \otimes W^{B}_{o})P_{AB}) = \textrm{Tr}(W^{A} a)\).  This effect \(a\) is called the compression of \(P_{AB}\) onto \(\mathbf{H}_{A}.\) In other words, we can understand \(a\) as representing the result of measuring \(P_{AB}\) on the combined system \(AB\), holding \(B\) in state \(W^{B}_o\), and then “forgetting about” the ancillary system \(B\).  It is not difficult to show that every every effect on \(A\) arises in this way from a projection on \(\mathbf{H}_{A} \otimes \mathbf{H}_{B}\) for a suitable Hilbert space \(\mathbf{H}_{B}\). More generally, a classic result in operator theory known as Naimark’s Theorem  asserts that any effect-valued observable \(a_1,...,a_n\) on \(A\) arises by compression of an ordinary projection-valued observable \(P_1,...,P_n\) on \(AB\) for a suitable quantum system \(B\).  Thus, all effects, and indeed all effect-valued observables, on \(A\) are physically realizable. In view of this, it is difficult to see why effect algebras should have any less claim to the status of a “quantum logic” than do, say, orthomodular posets. 