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The trace distance between two quantum states, $\rho$ and $\sigma$, is an operationally meaningful quantity in quantum information theory.
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For example, the following bound would be natural conjecture: D ( ρ , σ ) 2 ⩽ D HS ( ρ , σ ) / Tr ( ρ 2 ) D(\rho,\sigma)^{2}\leqslant D_{\text{HS}}(\rho,\sigma)/{\rm Tr}(\rho^{2}) . However, we numerically found this bound to be violated
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Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T Sornborger, and Patrick J Coles, “Quantum-assisted quantum compiling,” Quantum 3
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