Oscillator-To-Oscillator Codes Do Not Have a Threshold

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