# Random coding for sharing bosonic quantum secrets. (arXiv:1808.06870v2 [quant-ph] UPDATED)

We consider a protocol for sharing quantum states using continuous variable
systems. Specifically we introduce an encoding procedure where bosonic modes in
arbitrary secret states are mixed with several ancillary squeezed modes through
a passive interferometer. We derive simple conditions on the interferometer for
this encoding to define a secret sharing protocol and we prove that they are
satisfied by almost any interferometer. This implies that, if the
interferometer is chosen uniformly at random, the probability that it may not
be used to implement a quantum secret sharing protocol is zero. Furthermore, we
show that the decoding operation can be obtained and implemented efficiently
with a Gaussian unitary using a number of single-mode squeezers that is at most
twice the number of modes of the secret, regardless the number of players. We
benchmark the quality of the reconstructed state by computing the fidelity with
the secret state as a function of the input squeezing.