Proposition: Certified positivity repair

Let $\hat{S}={\hat{S}}^{†}$, $\mathrm{Tr}\hat{S}=1$ exactly, and suppose a certified number $\delta \ge 0$ gives $\hat{S}\ge -\delta I_N$. Define

Then $S_{+}$ is a density matrix and

If ${\Vert \hat{S}-S\Vert }_1\le E$, then

Proof. Positivity and trace normalization follow directly. The negative spectral mass of $\hat{S}$ is at most $N\delta$, so ${\Vert \hat{S}\Vert }_1\le 1+2N\delta$. Also

Taking norms gives

The final assertion is the triangle inequality.