  It is known that continuous variable quantum information cannot be protected
against naturally occurring noise using Gaussian states and operations only.
Noh et al. (PRL 125:080503, 2020) proposed bosonic oscillator-to-oscillator
codes relying on non-Gaussian resource states as an alternative, and showed
that these encodings can lead to a reduction of the effective error strength at
the logical level as measured by the variance of the classical displacement
noise channel. An oscillator-to-oscillator code embeds K logical bosonic modes
(in an arbitrary state) into N physical modes by means of a Gaussian N-mode
unitary and N-K auxiliary one-mode Gottesman-Kitaev-Preskill-states.
  Here we ask if - in analogy to qubit error-correcting codes - there are
families of oscillator-to-oscillator codes with the following threshold
property: They allow to convert physical displacement noise with variance below
some threshold value to logical noise with variance upper bounded by any
(arbitrary) constant. We find that this is not the case if encoding unitaries
involving a constant amount of squeezing and maximum likelihood error decoding
are used. We show a general lower bound on the logical error probability which
is only a function of the amount of squeezing and independent of the number of
modes. As a consequence, any physically implementable family of
oscillator-to-oscillator codes combined with maximum likelihood error decoding
does not admit a threshold.
