6

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}_{...


4

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{...


2

I think I have an answer. The following should be the CVX code for one of the formulations found in this link. cvx_begin sdp variable X(2, 2) hermitian minimize(trace(id' * X)) % id is eye(2) subject to kron(id, X) >= rho_ab % the tensor product of two density matrices a, b X >= 0 cvx_end The optimal value found in this program is $$\text{optval} = ...


1

I found a way to do it for the $2\to1$ QRAC. I simply guessed that we could leave the measurement bases as they are, $Z$ for the first bit, and $X$ for the second bit, and added $Y$ as the basis with which to extract the XOR of the two bits. From the guess obtaining the optimal encoding states is then easy, we just need to diagonalize the relevant operators. ...


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