# 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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## 1 Answer

I have luckily found an answer to this question:

The fidelity is concave in both its arguments i.e.

$F(\lambda \rho_1+(1-\lambda)\rho_2,\lambda \sigma_1+(1-\lambda)\sigma_2 )\geq \lambda F(\rho_1,\rho_2)+(1-\lambda)F(\sigma_1,\sigma_2)$.

This means that this maximization problem can be dealt with efficiently with the very well developed tools of convex optimization. The algorithm can be implemented, in particular, with the 'cvx' package for MatLab available in http://cvxr.com/. One has only to be careful and write the fidelity function as

$F(\rho,\sigma)=\text{Tr}(\sqrt{\sqrt{\rho}\sigma \sqrt{\rho}})$.

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