Quantum Relative entropy- the math and intuition

Question

I am new to quantum information theory and have been reading Mark Wilde's notes on quantum relative entropy. http://www.markwilde.com/teaching/2015-fall-qit/lectures/lecture-19.pdf

I have three basic questions here.

1. Why does $\sigma$ belong to the space of bounded linear operators when $\rho$ belongs to the space of density operators? All density operators are anyway bounded operators with trace 1. So why is this distinction between the spaces that $\rho$ and $\sigma$ belong to? If I look at Watrous's notes (https://cs.uwaterloo.ca/~watrous/TQI/TQI.5.pdf) he refers to both $\rho$ and $\sigma$ as belonging to the space of positive semidefinite operators and doesn't draw the distinction that Wilde does.

2. How does the condition $supp(\rho)\subseteq supp(\sigma)$ figure into the relative entropy being defined in this way? in other words where are we using this condition to form our definition?

3. What is the intuition behind quantum relative entropy? Why is it exactly defined this way? specifically the second term $Tr(\rho(log\sigma))$ of the relative entropy expression.

Answer 1

To answer your questions briefly:

1. This is purely a matter of normalization. It doesn't matter really as you could define it for just positive semidefinite operators like Watrous and they will be the same up to some additive normalization factor. For the second part it is important to have $\sigma$ not normalized in the definition as you define conditional entropy as $H(A|B) := - D(\rho_{AB} \|I \otimes \rho_B)$ and notice $\sigma = I \otimes \rho_B$ is not trace one.
2. As mentioned in the comments this is related to when the quantity is well defined. Things are all very nice when everything is full rank but how do you define $|\psi\rangle \langle \psi| \log |\phi \rangle \langle \phi|$? Roughly if $\mathrm{supp}(\rho) \subseteq \mathrm{supp}(\sigma)$ then we can work inside the subspace $\mathrm{supp}(\sigma)$ where $\sigma$ is then full rank.
3. For intuition think classical information theory and look at the Kullback-Liebler divergence which should be recovered when we insert classical (diagonal) states.