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In the computation of the entanglement of formation(EoF) of a 2 qubits mixed state, $\rho$, according to Wooters, we need to compute the concurrence of the state by computing the eigen values $\{\sigma_1,\sigma_2,\sigma_3,\sigma_4\}$ of the matrix $\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$, with $\tilde{\rho}$ is the spin flipped matrix defined as $\tilde{\rho}=(i\sigma^2\otimes i\sigma^2)\rho^{*}(i\sigma^2\otimes i\sigma^2)$.

My question is that: what's the physical meaning of the eigenvalues $\{\sigma_1,\sigma_2,\sigma_3,\sigma_4\}$?

Obviously it's somehow related with a 2 qubit state $\rho$ and its spinor flipped state $\tilde{\rho}$. For example, $\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$ is related with the Bures distance between $\rho$ and $\tilde{\rho}$. At the same time, the Bures distance between $\rho$ and its closest separable state is a function of the concurrence $C=max(0,\sigma_1-\sigma_2-\sigma_3-\sigma_4)$.

But what exactly those eigenvalues mean in a physically intuitive picture? Why the Bures distance between $\rho$ and its closest separable state is related with $C=max(0,\sigma_1-\sigma_2-\sigma_3-\sigma_4)$? An observation is that the formula of $C$ is so similar with the Minkowski metric. Are there any relations between them?

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