Prove that the trace distance is upper-bounded by the Hilbert-Schmidt distance

Question

In (Haah et al. 2015), in the third page, second column, the authors use the following result: given a pair of states $\rho,\sigma$, we have $$ \|\rho-\sigma\|_1 \le 2\sqrt{\min(\operatorname{rank}(\rho),\operatorname{rank}(\sigma))} \|\rho-\sigma\|_2, $$ where $\|A\|_1\equiv \operatorname{Tr}|A|$ is the trace norm, and $\|A\|_2\equiv \sqrt{\operatorname{Tr}(A^\dagger A)}$ is the Hilbert-Schmidt norm.

I haven't encountered this fact before, and I can't find a reference pointing to its source in the paper. What's a good way to prove this result (or equivalently, what's a reference discussing such result)?

Answer 1

Let $P - Q = \rho - \sigma$ be a Jordan-Hahn decomposition, meaning that $P$ and $Q$ are the unique positive semidefinite operators that satisfy that equation and have orthogonal images. We'll start by collecting some basic facts about this decomposition:

• It is the case that $\|\rho - \sigma\|_1 = \operatorname{Tr}(P) + \operatorname{Tr}(Q)$. Moreover, because $\rho$ and $\sigma$ are density operators, we have that $\rho - \sigma$ is traceless, and therefore $\operatorname{Tr}(P) = \operatorname{Tr}(Q)$. This implies $$ \|\rho - \sigma\|_1 = 2\operatorname{Tr}(P) = 2 \operatorname{Tr}(Q). $$

• By the Pythagorean theorem, we have $ \|\rho - \sigma\|_2^2 = \|P\|_2^2 + \|Q\|_2^2$, and so $$ \|P\|_2\leq\|\rho - \sigma\|_2 \quad \text{and}\quad \|Q\|_2\leq\|\rho - \sigma\|_2. $$

• Because $\rho$ and $\sigma$ are positive semidefinite, it is the case that $$ \operatorname{rank}(P) \leq \operatorname{rank}(\rho) \quad\text{and}\quad \operatorname{rank}(Q) \leq \operatorname{rank}(\sigma) $$ (in essence, by counting positive and negative eigenvalues).

Now we have what we need to get the inequality from the one that narip and JSdJ referred to in their comments (which follows from Cauchy-Schwarz for operators): $$ \begin{gathered} \|\rho - \sigma\|_1 = 2\operatorname{Tr}(P) \leq 2 \sqrt{\operatorname{rank}(P)} \,\|P\|_2 \leq 2 \sqrt{\operatorname{rank}(\rho)}\, \|\rho - \sigma\|_2\\ \|\rho - \sigma\|_1 = 2\operatorname{Tr}(Q) \leq 2 \sqrt{\operatorname{rank}(Q)} \,\|Q\|_2 \leq 2 \sqrt{\operatorname{rank}(\sigma)}\, \|\rho - \sigma\|_2 \end{gathered} $$ (and then get the minimum by chosing whichever is better).