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.
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.
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?
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.