Proposition: Explicit first-order product-formula error

For $t\ge 0$,

Proof. For two terms, a double Duhamel expansion compares their joint exponential with the ordered product; the error is bounded by ${\delta }^2\Vert [H_1,H_2]\Vert /2$ on an interval of length $\delta$. Iterating over the finite sum gives the one-step bound ${\delta }^2{C}_H/2$, since $\Vert [H_j,{\sum }_{k<j}{H}_k]\Vert \le {\sum }_{k<j}\Vert [H_j,H_k]\Vert$. Telescoping $n$ unitary steps yields the first estimate. The second follows from ${\Vert {\mathrm{Ad}}_U-{\mathrm{Ad}}_V\Vert }_{\diamond }\le 2\Vert U-V\Vert$.