  We show that, if the accessible information is used as a security quantifier,
quantum channels with a certain symmetry can convey private messages at a
tremendously high rate, as high as less than one bit below the rate of
non-private classical communication. This result is obtained by exploiting the
quantum data locking effect. The price to pay to achieve such a high private
communication rate is that accessible information security is in general not
composable. However, composable security holds against an eavesdropper who is
forced to measure her share of the quantum system within a finite time after
she gets it.
