 Gleason’s theorem can now be invoked to identify the states on \(\mathcal{A}_{\mathbf{H}}\) with the density operators on \(\mathbf{H}\): to each state \(\omega\) in \(\Omega(\mathcal{A}_{\mathbf{H}})\) there corresponds a unique density operator \(W\) such that, for every unit vector \(x\) of \(\mathbf{H}, \omega(x) = \langle Wx,x\rangle = Tr(WP_x), P_x\) being the one-dimensional projection associated with \(x\). Conversely, of course, every such density operator defines a unique state by the formula above. We can also represent simple real-valued random variables operator-theoretically. Each bounded simple random variable \(f\) gives rise to a bounded self-adjoint operator \(A = \sum_{x\in E} f(x)P_x\). The spectral theorem tells us that every self-adjoint operator on \(\mathbf{H}\) can be obtained by taking suitable limits of operators of this form.