# Definition of quantum min-relative entropy

In John Watrous' lectures, he defines the quantum min-relative entropy as

$$D_{\min}(\rho\|\sigma) = -\log(F(\rho, \sigma)^2),$$

where $$F(\rho,\sigma) = tr(\sqrt{\rho\sigma})$$. Here, I use this question and answer to make the definition simpler although one should note that the linked question uses a different definition of fidelity (squared vs not squared).

On the other hand, one of the early papers introducing this quantity (see Definition 2 of this paper) defines it as

$$D_{\min}(\rho\|\sigma) = -\log(tr(\Pi_\rho\sigma)),$$

where $$\Pi_\rho$$ is the projector onto the support of $$\rho$$. It's not clear if these definitions are equivalent since I can change $$\rho$$ without altering its support.

How are the two definitions related to each other, if at all?

• They are not the same. Later in Watrous' lecture, the second definition is covered and it goes under the name of hypothesis testing relative entropy with $\varepsilon = 1$. The confusion is due to different definitions by different authors. – rnva Jul 17 at 17:15

As @rnva points out these are not the same quantities. To give some clarity as to why they are both referred to as $$D_{\min}$$ it is best to look at the as limiting cases of $$\alpha$$-R'enyi divergences.
First, we have the sandwiched divergences which for $$\alpha \in (0, 1) \cup (1, \infty)$$ are defined as $$\widetilde{D}_{\alpha}(\rho\|\sigma) = \frac{1}{\alpha - 1} \log \mathrm{Tr}\left[ (\sigma^{\frac{1-\alpha}{2\alpha}} \rho \sigma^{\frac{1-\alpha}{2\alpha}} )^\alpha \right].$$ These divergences are monotonically increasing in $$\alpha$$ and satisfy the data processing inequality (DPI) for all $$\alpha \geq 1/2$$. Thus the smallest divergence in this family satisfying the DPI is $$\widetilde{D}_{\min}(\rho \| \sigma) = \widetilde{D}_{1/2}(\rho \|\sigma) = - \log \mathrm{Tr}[\sqrt{\rho} \sqrt{\sigma}]^2.$$
Another well studied family of divergences are the so-called Petz divergences defined for $$\alpha \in (0,1) \cup (1, \infty)$$ to be $$\overline{D}_{\alpha}(\rho \| \sigma) = \frac{1}{\alpha - 1} \log \mathrm{Tr}[\rho^{\alpha} \sigma^{1-\alpha}].$$ This family satisfies the DPI for $$\alpha \in (0,1) \cup(1,2]$$ and they are also monotonically increasing in $$\alpha$$. Thus, the smallest divergence satisfying the DPI in this family is $$\overline{D}_{\min}(\rho \| \sigma) = \lim_{\alpha \to 0^+} \overline{D}_{\alpha}(\rho \|\sigma) = -\log \mathrm{Tr}[\Pi_\rho \sigma ].$$