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Expand the following function around $t=0$ $$t \mapsto\ln(A - tB)$$ where $A$ and $B$ are non-commuting finite-dimensional operators over a Hilbert space.


I would like to do something in the lines of

$$\ln(a-tb)=\ln(a)+\ln(1-bt/a)$$

and use the Taylor series for $\ln(1-x)$. This doesn't seem trivial for me in the matrix valued case but I thought it might be feasible by means of some Baker-Campbell-Hausdorff like expansion.

It might also help to provide further assumptions given the general context. I have a time-evolved state

$$\rho'=e^{-tH}\rho e^{+tH}$$

where $H$ is a self-adjoint operator. Since $t \ll 1$, $\rho' \approx \rho -i t [ H, \rho ]$. Then, I want to compute (general case)

$$\ln[e^{-tH} \rho e^{+tH}]$$

So, if either a formula for the general case would help or if assuming that $A$ is a state $\rho^\dagger = \rho$ with $\text{Tr}{\rho} = 1$ could give any improvement.

Thanks in advance.

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