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Upper bounds for private communication over quantum channels can be derivedby adopting channel simulation, protocol stretching, and relative entropy ofentanglement. All these ingredients have led to single-letter upper bounds tothe secret key capacity which can be directly computed over suitable resourcestates. For bosonic Gaussian channels, the tightest upper bounds have beenderived by employing teleportation simulation over asymptotic resource states,namely the asymptotic Choi matrices of these channels. In this work, we adopt adifferent approach. We show that teleporting over an analytical class offinite-energy resource states allows us to closely approximate the ultimatebounds for increasing energy, so as to provide increasingly tight upper boundsto the secret-key capacity of one-mode phase-insensitive Gaussian channels. Wethen show that an optimization over the same class of resource states can beused to bound the maximum secret key rates that are achievable in a finitenumber of channel uses.

Tight bounds for private communication over bosonic Gaussian channels based on teleportation simulation with optimal finite resources