1
$\begingroup$

The sandwiched Renyi divergence is defined as

$$\begin{equation} \tilde{D}_{\alpha}(\rho \| \sigma):=\frac{1}{\alpha-1} \log \operatorname{tr}\left[\left(\sigma^{\frac{1-\alpha}{2 \alpha}} \rho \sigma^{\frac{1-\alpha}{2 \alpha}}\right)^{\alpha}\right]. \end{equation} $$

One can define the limiting cases as $\alpha\rightarrow 1$ and $\alpha\rightarrow\infty$ to obtain the relative entropy and max-relative entropy respectively. I am not sure how to take these limits - does one apply L'Hôpital's rule and if yes, how should one deal with the terms inside the trace? The goal is to show

$$\lim_{\alpha\rightarrow 1}\tilde{D}_{\alpha}(\rho \| \sigma) = \text{tr}(\rho\log\rho - \rho\log\sigma)$$ and

$$\lim_{\alpha\rightarrow \infty}\tilde{D}_{\alpha}(\rho \| \sigma) = \inf \{\lambda: 2^\lambda\sigma\geq \rho \}$$

$\endgroup$
2
$\begingroup$

Both limits are dealt with in a fair amount of detail in the work that originally defined the sandwiched entropies: On quantum Renyi entropies: a new generalization and some properties. In particular, you'll find the relevant results in section IV.C.

$\endgroup$

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.