 Solution

Set $X=\rho -\sigma$. The trace distance is $D(\rho ,\sigma )=\frac{1}{2}\Vert X{\Vert }_1$, where $\Vert X{\Vert }_1=\mathrm{Tr}\sqrt{X^{†}X}$. Since $X$ is a Hermitian operator, its spectral decomposition is $X={\sum }_j{x}_j|j⟩⟨j|$. Define its positive part of a Hermitian operator and negative part of a Hermitian operator byThey are positive semidefinite operators, satisfy $X=Q-R$ and $QR=0$, and obey $|X|=Q+R$. ThereforeThe states have equal trace, so $\mathrm{Tr}X=0$ and $\mathrm{Tr}Q=\mathrm{Tr}R=D(\rho ,\sigma )$. This is the spectral-parts formula for trace distance.