Simple construction of quantum universal variable-length source coding

We simply construct a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate $H(rho_p)$, tend to 0. If $H(rho_p)$ is estimated, we can compress the coding rate to the admissible rate $H(rho_p)$ with a probability close to 1. However, when we perform a naive measurement for the estimation of $H(rho_p)$, the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of $H(rho_p)$ and the non-demolition of the input state. Our protocol can be used not only for the Schumacher's scheme but also for the compression of entangled states.