The (square-root) fidelity between a pair of states $\rho,\sigma$ is the quantity $$\sqrt F(\rho,\sigma) = \|\sqrt\rho\sqrt\sigma\|_1 = \operatorname{tr}|\sqrt\rho\sqrt\sigma| =\operatorname{tr}\left[\sqrt{\sqrt\rho\sigma\sqrt\rho}\right].$$ Here $|\sqrt\rho\sqrt\sigma|\equiv \sqrt{\sqrt\rho\sigma\sqrt\rho}$ is a positive semidefinite matrix, and thus $\sqrt F$ equals the sum of its eigenvalues.

Can the individual eigenvalues of $|\sqrt\rho\sqrt\sigma|$ be given any kind of physical (or otherwise) interpretation? This should also be equivalent to asking about the eigenvalues of $\sqrt\rho\sigma\sqrt\rho$, which in turn are equal to the eigenvalues of $\rho\sigma$ (which as discussed here is ensured to be diagonalisable).



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