Theorem: Exact identities and concentration criterion

For Definition def:states, write

and let ${\mu }_{ϵ}$ be the spectral law of ${\hat{O}}_{ϵ}$ in ${\tilde{\rho }}_{ϵ}$. Then

For a fixed named $v,o$, the following are equivalent:

1. the mean tends to $o$ and the variance tends to zero;

2. $ϵM_{ϵ}(v)\to 0$;

3. $W_2({\mu }_{ϵ},{\delta }_o)\to 0$.

A sufficient, not necessary, condition is

Under that condition,

Proof. The exact self-adjoint conjugation is

Unitary covariance of spectral measures gives the mean, variance, and second-moment identities. Their moments exist by (eq:state-number). Equation (eq:pointmass) proves the Wasserstein identity.

For every probability law with a finite second moment,

Both summands are nonnegative, proving the equivalences. Cauchy–Schwarz gives $b_{ϵ}(v{)}^2\le M_{ϵ}(v)$; (eq:state-number) then proves the sufficient condition and all three estimates.