# Fast algorithm for maximizing the quantum fidelity

Consider the following optimization problem: Given a quantum state $\sigma$, a constant $b$ and a Hermitian operator $A$, find

$\underset{\rho} \max F(\rho,\sigma)$

subject to $\text{Tr}(\rho A)>b$,

where $F(\rho,\sigma)=||\sqrt{\rho} \sqrt{\sigma}||_1$ is the fidelity function.

Does anyone know of a fast algorithm for this problem? Even better, an available package that implements such an algorithm?

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$F(\lambda \rho_1+(1-\lambda)\rho_2,\lambda \sigma_1+(1-\lambda)\sigma_2 )\geq \lambda F(\rho_1,\rho_2)+(1-\lambda)F(\sigma_1,\sigma_2)$.
$F(\rho,\sigma)=\text{Tr}(\sqrt{\sqrt{\rho}\sigma \sqrt{\rho}})$.