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I am reading paper Channel Identification and its Impact on Quantum LDPC Code Performance where the authors discuss the scenario where the decoder of a Quantum LDPC code uses an estimation of the depolarization probability of the channel in order to correct the errors.

When doing so, they establish the quantum Cramer-Rao bound in terms of the Fisher information. The Fisher information is defined as

$J_f = \mathrm{Tr}(\rho_f L_f^2)$,

where $L_f$ is the symmetric logarithmic derivative $2\partial_f \rho_f=L_f\rho_f+\rho_f L _f$. They then state that when the probe for the estimation is an unnetangled pure quantum state, such Fisher information can be proved to be $J_f = [f(2-f)]^{-1}$, and if the probe is a maximally-entangled qubit pair, it equals $J_f=[f(\frac{4}{3}-f)]^{-1}$.

I have been trying to prove such equalities for those cases of the Fisher information, but I haven't been able to find something similar to it nor prove it.

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