Decoherence functional ($D(\alpha ,\beta )=\mathrm{tr}(C_{\alpha }\rho C_{\beta }^{†})$)

For alternatives represented by projection operators at successive times, set $C_{\alpha }=P_{{\alpha }_n}(t_n)\cdots P_{{\alpha }_1}(t_1)$ in the Heisenberg picture. The displayed functional uses the initial density operator $\rho$. Combining two histories gives $D(\alpha +\beta ,\alpha +\beta )=D(\alpha ,\alpha )+D(\beta ,\beta )+2\mathrm{Re}D(\alpha ,\beta )$. Thus vanishing real cross-terms gives the probability sum rule; the stronger commonly used condition $D(\alpha ,\beta )=0$ for distinct histories is sufficient as well. Its diagonal is nonnegative because each $C_{\alpha }\rho C_{\alpha }^{†}$ is positive.