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We determine the maximum squashed entanglement achievable between sender andreceiver of the noiseless quantum Gaussian attenuators and amplifiers and weprove that it is achieved sending half of an infinitely squeezed two-modevacuum state. The key ingredient of the proof is a lower bound to the squashedentanglement of the quantum Gaussian states obtained applying a two-modesqueezing operation to a quantum thermal Gaussian state tensored with thevacuum state. This is the first lower bound to the squashed entanglement of aquantum Gaussian state and opens the way to determine the squashed entanglementof all quantum Gaussian channels. Moreover, we determine the classical squashedentanglement of the quantum Gaussian states above and show that it is strictlylarger than their squashed entanglement. This is the first time that theclassical squashed entanglement of a mixed quantum Gaussian state isdetermined.

The squashed entanglement of the noiseless quantum Gaussian attenuator and amplifier