# Prove that the conditional min-entropy is $H_{\rm min}(A|B)=\max_\sigma\sup\{\lambda:\,\rho\le 2^{-\lambda}(I\otimes\sigma)\}$

I have seen various definitions of quantum conditional min-entropy, which I believe are equivalent.

1. $$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \parallel I_A \otimes \sigma_B \right)$$ where in general $$D_{\max}(\rho \parallel \sigma) = \inf \{ \lambda : \rho \leq 2^\lambda \sigma \}$$

2. $$H_{\min}(A|B) = \max\limits_{\sigma_B} \sup \{ \lambda: \rho_{AB} \leq 2 ^{-\lambda} I_A \otimes \sigma_B \}$$

Why are the two definitions equivalent?

Observe that $$H_{\min}(A|B) = - \inf\limits_{\sigma_B} D_{\max} \left( \rho_{AB} \| I_A \otimes \sigma_B \right) = \sup\limits_{\sigma_B} [-D_{\max} \left( \rho_{AB} \| I_A \otimes \sigma_B \right)],$$ and $$-D_{\rm max}(\rho\|I\otimes \sigma)=-\inf \{\lambda: \,\, \rho\le 2^\lambda (I\otimes \sigma)\} \\ = \sup\{-\lambda: \,\, \rho\le 2^\lambda (I\otimes\sigma)\} \\ = \sup\{\lambda: \,\, \rho\le 2^{-\lambda} (I\otimes\sigma)\}.$$
It might help to take a slightly more abstract perspective here. What we're trying to compute is $$-\inf_x f(x)$$ where $$f(x)\equiv \inf\{y\in\mathbb{R}:\,\, y\in g(x)\}$$ for some function $$f$$, and some set $$g(x)\subseteq\mathbb{R}$$ that depends on $$x$$. And then $$-\inf_x f(x) = \sup_x[-f(x)] = \sup_x [-\inf\{y\in\mathbb{R}:\,\, y\in g(x)\}] \\ = \sup_x \sup\{-y: \,\, y\in\mathbb{R},\,\, y\in g(x)\} \\ = \sup_x \sup\{z\in\mathbb{R}: \,\, -z\in g(x)\}.$$ In words, we first minimise over the possible values of a function $$f$$ (restricting the possible inputs $$x$$ in a specific subset, and taking the opposite of the result), where $$f(x)$$ is defined as the smallest element of a subset of real numbers that depends on $$x$$. You can then just replace $$\sup\to\max$$ and $$\inf\to\min$$ whenever you know that a min/max exists.
• Thanks! That makes sense. In light of your answer, I came up across a result that I admittedly forgot: $-\inf(A) = \sup(-A)$. For interested readers, a proof can be found here: math.stackexchange.com/questions/522259/…