Continuous-variable private quantum channel

In this paper we introduce the concept of quantum private channel within the continuous variables framework (CVPQC) and investigate its properties. In terms of CVPQC we naturally define a 'maximally' mixed state in phase space together with its explicit construction and show that for increasing number of encryption operations (which sets the length of a shared key between Alice and Bob) the encrypted state is arbitrarily close to the maximally mixed state in the sense of the Hilbert-Schmidt distance. We bring the exact solution for the distance dependence and give also a rough estimate of the necessary number of bits of the shared secret key (i.e., how much classical resources are needed for an approximate encryption of a generally unknown continuous-variable state). The definition of the CVPQC is analyzed from the Holevo bound point of view which determines an upper bound of information about an incoming state an eavesdropper is able to get from his optimal measurement