Unitary variational formula for the trace norm ($\Vert X{\Vert }_1={\max }_U|\mathrm{Tr}(XU)|$)

For a finite square matrix $X$, maximize over unitary operators. Write its singular value decomposition as $X=PD{Q}^{†}$. Then $|\mathrm{Tr}(XU)|=|\mathrm{Tr}(D{Q}^{†}UP)|\le {\sum }_j{D}_{jj}$, since every diagonal entry of a unitary has modulus at most one. Choosing $U=Q{P}^{†}$ attains equality, proving the formula for the trace norm.