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Like many ideas in quantum information theory, I think this is best understood using a $2$-party communication scenario. Suppose Alice has a classical random variable, $X$ which can take values $1,2, \cdots, k$ with probabilities $p_{X}(1), p_{X}(2), \cdots, p_{X}(k)$. Alice then encodes this information by encoding the classical index $j$ in the state $\rho^... 2 As @rnva points out these are not the same quantities. To give some clarity as to why they are both referred to as$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 $$\widetilde{D}_{\alpha}(\rho\|\sigma) = \frac{1}{\... 2 P is a projection operator in the limiting case where P represents a state that is completely known, i.e. a pure state, so the entropy is zero. As a limiting case, the valid and well-defined mathematical description is$$\lim_{P \rightarrow 0^+} P \log(P)=0.$$This is still a bit sloppy. Since$P$is a matrix, we're actually taking the trace and$P \...