Semi-definite program for conditional smooth max-entropy

Question

I am aware of a SDP formulation for smooth min-entropy: question link. That program for smooth min-entropy was found in this book by Tomachiel: page 91. However, I am yet to come across a semi-definite formulation for smooth max-entropy. There is however, a formulation for the non-smoothed version of max-entropy, found in this link: page 4, lemma 8. Here is the detailed program for a bipartite density operator $\rho_{AB}$, $2^{H_{\text{max}}(A|B)_\rho}$ =
$$ \text{minimize }\lambda \\ \text{subject to} \\ Z_{AB} \otimes \mathbb{I} \ge \rho_{ABC} \\ \lambda \mathbb{I}_B \ge \text{tr}_A [Z_{AB}] \\ Z_{AB} \ge 0 \\ \lambda \ge 0 $$

Where $Z_{AB}$ runs over all positive semi-definite operators in $\mathcal{H}_{AB}$, $\lambda$ is a real number. The smooth max-entropy is then: $$ H^{\epsilon}_{\text{max}}(A|B)ρ := \underset{\rho'_{AB} \in \mathcal{B}^\epsilon (\rho_{AB})}{\min}H_{\text{max}}(A|B)_{\rho'} $$ i.e., just the minimum over all bipartite operators which are at most $\epsilon$ distance away from $\rho_{AB}$. But the primal or dual SDP formulation for the smooth version of max-entropy was not found anywhere. Is there one? How could I transform it into a smooth version? TIA.

Answer 1

Yes, you can formulate the smooth max-entropy as an SDP. The author of the book you linked notes this when they explain how to derive the SDP for the smooth min-entropy that you reference on page 91.

In particular they say that the smoothing constraint $\tilde{\rho}_{AB} \in B^\epsilon(\rho_{AB})$ can be reformulated as the triple of constraints $$ \mathrm{Tr}[\tilde\rho_{ABC} \rho_{ABC}] \geq 1- \epsilon^2~\\ \mathrm{Tr}[\tilde\rho_{ABC}] \leq 1 ~\\ \tilde \rho_{ABC} \geq 0 $$ where $\rho_{ABC}$ is any purification of $\rho_{AB}$.

Now we can incorporate these extra constraints with the SDP formulation of $H_{\max}(A|B)$. In particular $$ \begin{aligned} 2^{H^\epsilon_{\max}(A|B)_\rho} &= \min_{\tilde{\rho}_{AB} \in B^\epsilon(\rho_{AB})} 2^{H_{\max}(A|B)_{\tilde\rho}} \\ &= \min_{\tilde{\rho}_{AB} \in B^\epsilon(\rho_{AB})}\min \lambda \\ &\qquad\mathrm{s.t.} \quad Z_{AB} \otimes I_C \geq \tilde\rho_{ABC} \\ &\qquad \qquad \lambda I_B \geq \mathrm{Tr}_A[Z_{AB}] \\ &\qquad\qquad Z_{AB} \geq 0, \quad \lambda \geq 0 \\ &= \,\,\,\min \quad\lambda \\ &\qquad\mathrm{s.t.} \quad Z_{AB} \otimes I_C \geq \tilde\rho_{ABC} \\ &\qquad \qquad \lambda I_B \geq \mathrm{Tr}_A[Z_{AB}] \\ &\qquad \qquad \mathrm{Tr}[\tilde\rho_{ABC} \rho_{ABC}] \geq 1- \epsilon^2~\\ &\qquad \qquad \mathrm{Tr}[\tilde\rho_{ABC}] \leq 1 ~\\ &\qquad\qquad \rho_{ABC} \geq 0, \quad Z_{AB} \geq 0, \quad \lambda \geq 0 \\ \end{aligned} $$