 Solution

For every orthogonal projector $P$, positivity givesThese inequalities do not require $P$ to commute with $Q$ or $R$: for example, $\mathrm{Tr}(PR)=\mathrm{Tr}(R^{1/2}P{R}^{1/2})\ge 0$, and apply the same observation to $I-P$ and $Q$.

Let $P_{+}$ project onto the positive spectral subspace of $X$. Then $P_{+}Q=Q$ and $P_{+}R=0$, so $\mathrm{Tr}(P_{+}X)=\mathrm{Tr}Q=D(\rho ,\sigma )$. Hence the variational characterization of trace distance over projectors isOne may include or exclude the zero eigenspace without changing the optimum.