5
$\begingroup$

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)?

$\endgroup$

1 Answer 1

6
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Hi, I have a quick follow-up question to that, as I am dealing with a similar problem. I do not understand why there appear $\rho_{ABC}$ with and without tilde. According to the book, they denote with tilde the density matrices that achieve the extremum. But what about the rho without tilde? How do I know the $\rho$ that attains the extremum? Isn't that what I am actually looking for? Or are both $\sigma_{AB}$ and $\tilde{\rho}_{ABC}$ optimization variables in that problem? $\endgroup$
    – pcalc
    Nov 14, 2023 at 9:00
  • $\begingroup$ This is the definition of the smooth min-entropy. You want to compute the smooth min-entropy of some state $\rho_{AB}$ and you do so by optimizing over states close to $\rho_{AB}$ hence $\widetilde{\rho}_{AB}$ is an optimization variables, the other optimization variable $\sigma_B$ is coming from the definition of the min-entropy. $\endgroup$
    – Rammus
    Nov 16, 2023 at 9:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.