Proposition: An additional hypothesis for a fluctuation limit

Suppose, for fixed directions $v_1,\dots ,v_J$,

for a density operator ${\rho }_{\ast }$. Then all the ordered fluctuation expectations in (eq:fluctuation-Weyl) converge to those of ${\rho }_{\ast }$. Each marginal fluctuation spectral law converges weakly to the law of $\Phi (v_j)$ in ${\rho }_{\ast }$.

Trace-norm convergence alone does not imply convergence of unbounded second moments.

Proof. The operators $e^{\mathrm{i}\Phi (V(t))}$ have norm one, so the difference of their expectations is at most $\Vert {\rho }_{ϵ}-{\rho }_{\ast }{\Vert }_1$. The ordered phase is unchanged. Marginal characteristic functions therefore converge to the characteristic function of the stated probability law, which is continuous at zero. The usual characteristic-function criterion gives weak convergence. The final caution is demonstrated explicitly in Section sec:examples.