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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.

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    $\begingroup$ 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. $\endgroup$
    – Rammus
    Commented May 8, 2023 at 9:08
  • $\begingroup$ I want to make the upper bound $\delta$ does not depend on $\rho$ and tighter. $\endgroup$
    – Kochan
    Commented May 8, 2023 at 17:52
  • $\begingroup$ 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 $\rho$ $\endgroup$ Commented May 27 at 16:38

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