Theorem: Exact residual fluctuations

The fluctuation family has the exact residual representation

Its commutators, on the conjugated common core or as the corresponding form identities, are

The unnormalized observables instead satisfy

Concentration does not determine a unique fluctuation limit. A sufficient additional condition for convergence of bounded fluctuation expectations is trace-norm convergence of the residual states in the fixed representation.

Proof. The representation follows directly from (eq:displacement); the commutators follow from (eq:CCR). For every bounded operator $B$,

Apply this to fluctuation Weyl products or bounded spectral functions.

For nonuniqueness, let ${ϵ}_k=1/k$, and alternate ${\rho }_{{ϵ}_k}=|0⟩⟨0|$ and $|1⟩⟨1|$ in one mode. Then $n_{{ϵ}_k}\le 1$, so concentration holds. For $v=e$,

The two subsequences have distinct characteristic functions.