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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.