Theorem: Gaussian continuity

For positive semidefinite $K,L\in {\mathbb{C}}^{J\times J}$,

If $K,L⪰mI$, $m>0$, then also

Proof. Let $Z$ be a standard circular complex Gaussian vector. The coherent-state mixtures with amplitudes $\sqrt{K}Z$ and $\sqrt{L}Z$ have the characteristic functions of ${\rho }_K$ and ${\rho }_L$; degenerate Gaussian measures cover singular matrices. Since

convexity and Cauchy–Schwarz give the Frobenius bound.

Set $\delta ={\Vert K-L\Vert }_{\infty }$. Operator monotonicity of the square root gives

and the reverse inequality follows by interchanging $K,L$. Thus ${\Vert \sqrt{K}-\sqrt{L}\Vert }_{\infty }\le \sqrt{\delta }$.

For strictly positive matrices, $X=\sqrt{K}-\sqrt{L}$ solves $\sqrt{K}X+X\sqrt{L}=K-L$, and

The integral's Frobenius norm is bounded by ${\Vert K-L\Vert }_{\mathrm{F}}/(2\sqrt{m})$.