Semi-definite program for smooth min-entropy

Question

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)?

Answer 1

You do not need a double objective function to solve this. Given $\rho_{AB}$ let $\rho_{ABC}$ be any purification of $\rho_{AB}$. Then we can write the smooth min-entropy as the following SDP \begin{align*} 2^{-H_{\min}^{\epsilon}(A|B)} = \min& \quad\mathrm{Tr}(\sigma_B) \\ \text{s.t.}& \quad I_A \otimes \sigma_B \geq \mathrm{Tr}_C(\widetilde{\rho}_{ABC}) \\ & \quad \mathrm{Tr}(\widetilde{\rho}_{ABC}) \leq 1 \\ & \quad \mathrm{Tr}(\widetilde{\rho}_{ABC} \rho_{ABC}) \geq 1 - \epsilon^2 \\ & \quad \widetilde{\rho}_{ABC} \geq 0 \\ & \quad \sigma_B \geq 0 \end{align*} For more details on this I would suggest reading Quantum Information Processing with Finite Resources. The SDP above is equation 6.37 in the linked book. From this you should be able to write the smooth min-entropy in CVX.