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We define an EPR-like uncertainty by using the Duan et al.'s inequality whichgives a necessary and sufficient condition for the separability of any two-modeGaussian state. We show that for a given amount of entanglement, theuncertainty is minimized by pure two-mode squeezed states. Using this fact, wewrite the optimal pure-state decomposition and derive a compact form for theentanglement of formation of two-mode Gaussian states. For illustrationpurposes, we consider symmetric and squeezed thermal states as special casesand evaluate their entanglement of formation explicitly. For the symmetricstates, our result is in agreement with the Giedke et al.'s one. To ourknowledge, our work is the first one which gives the exact entanglement offormation of two-mode squeezed thermal states explicitly.

On the entanglement of formation of two-mode Gaussian states: a compact form