The conditional min-entropy is defined as (wiki):
$$H_{\min}(A|B)_{\rho} \equiv -\inf_{\sigma_B}\inf_{\lambda}\{\lambda \in \mathbb{R}:\rho_{AB} \leq 2^{\lambda} \mathbb{I} \otimes \sigma_B\}$$
And the smooth min-entropy is defined as:
$$H_{\min}^{\epsilon}(A|B)_{\rho} \equiv \sup_{\rho'} H_{\min}(A|B)_{\rho'}$$
Which just means that $\rho'_{AB}$ is some $\epsilon$-bounded distance away from $\rho_{AB}$. I know how to write a semi-definite program of the conditional min-entropy, which is:
$$\text{min } \text{tr}(X) \\\text{such that:} \\\mathbb{I} \otimes X \ge \rho_{AB}\\X \in \text{Herm}(\mathcal{H_B})$$I can implement this program in cvx(matlab). But the trouble is, in order to calculate the smooth min-entropy, I have to take a maximization over all $\rho'_{AB}$ who are $\epsilon$-distance away from $\rho_{AB}$. This means I need to write a double objective function in the semidefinite program. Something like:
$$\text{max } \rho'_{AB} \\\text{min } \text{tr}(X) \\\text{such that:} \\\mathbb{I} \otimes X \ge \rho'_{AB}\\X \in \text{Herm}(\mathcal{H_B}) \\\rho'_{AB} \in \mathcal{B}^\epsilon(\rho_{AB})$$
What is the correct form of this double objective function? Is there any hope of writing it in cvx (matlab)?
