# Tighter upper bound of $\operatorname{tr}[ (\mathcal{H}[L]\rho )^2] \leq \delta$

I am wondering about an upper bound of the trace function $$\operatorname{tr}[ (\mathcal{H}[L]\rho )^2] \leq \delta$$ (we assume that $$\rho$$ is the $$N\times N$$ density matrix representing the quantum state, $$L$$ is the $$N\times N$$ Hermitian operator $$L=L^\dagger$$, and $$\mathcal{H}[L]\rho=L\rho+\rho L-2 \operatorname{tr}(L\rho)\rho$$ ). I would like to have some advice from you.

• What do you want from the upper bound? You could apply Cauchy-Schwarz or Hölder inequalities but without more context I don't know why they'd be more useful to you than the original expression. Commented May 8, 2023 at 9:08
• I want to make the upper bound $\delta$ does not depend on $\rho$ and tighter. Commented May 8, 2023 at 17:52
• For completeness sake—as giving an answer is currently impossible because it is not explained which $\delta$ has been found already—using standard techniques one gets the brute force/sub-optimal upper bound $\operatorname{tr}[ (\mathcal{H}[L]\rho )^2]\leq 16\|L\|_\infty^2$ (with $\|L\|_\infty$ the usual operator norm) which holds for all density matrices $\rho$ Commented May 27 at 16:38