= Decoherence functional
{title2=$D(\alpha,\beta)=\operatorname{tr}(C_\alpha\rho C_\beta^\dagger)$}
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 $D(\alpha+\beta,\alpha+\beta)=D(\alpha,\alpha)+D(\beta,\beta)+2\operatorname{Re}D(\alpha,\beta)$. 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 $C_\alpha\rho C_\alpha^\dagger$ is positive.
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