Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states

The trace distance between two quantum states, $\rho$ and $\sigma$, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of $\rho - \sigma$. In contrast, the Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt distance with a bound that is particularly strong when either $\rho$ or $\sigma$ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace norms. We also consider bounds that are useful not only for low-rank states but also for low-entropy states. Our results have relevance to quantum information theory, quantum algorithms design, and quantum complexity theory.