Quantum Computing Asked on January 18, 2021
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.
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 $$ mathrm{Tr}[tilderho_{ABC} rho_{ABC}] geq 1- epsilon^2~ mathrm{Tr}[tilderho_{ABC}] leq 1 ~ tilde rho_{ABC} geq 0 $$ 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 $$ begin{aligned} 2^{H^epsilon_{max}(A|B)_rho} &= min_{tilde{rho}_{AB} in B^epsilon(rho_{AB})} 2^{H_{max}(A|B)_{tilderho}} &= min_{tilde{rho}_{AB} in B^epsilon(rho_{AB})}min lambda &qquadmathrm{s.t.} quad Z_{AB} otimes I_C geq tilderho_{ABC} &qquad qquad lambda I_B geq mathrm{Tr}_A[Z_{AB}] &qquadqquad Z_{AB} geq 0, quad lambda geq 0 &= ,,,min quadlambda &qquadmathrm{s.t.} quad Z_{AB} otimes I_C geq tilderho_{ABC} &qquad qquad lambda I_B geq mathrm{Tr}_A[Z_{AB}] &qquad qquad mathrm{Tr}[tilderho_{ABC} rho_{ABC}] geq 1- epsilon^2~ &qquad qquad mathrm{Tr}[tilderho_{ABC}] leq 1 ~ &qquadqquad rho_{ABC} geq 0, quad Z_{AB} geq 0, quad lambda geq 0 end{aligned} $$
Correct answer by Rammus on January 18, 2021
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