Limits for compression of quantum information carried by ensembles of mixed states

We consider the problem of compression of the quantum information carried by ensemble of mixed states. We prove that for arbitrary coding schemes the least number of quantum bits (qubits) per message needed to convey the signal states asymptotically faithfully is bounded from below by the Holevo function $S\left(\varrho \right)-{\sum }_i{p}_{i}S\left({\varrho }_i\right)$. We also show that a compression protocol can be composed with another one, provided that the latter offers perfect transmission. Such a compound protocol is applied to the case of binary source. It is conjectured to reach the obtained bound. Finally, we point out that in the case of mixed signal states there could be a difference between the maximal compression rates at the coding schemes that are “blind” to the signal and the ones that assume the knowledge about the identities of the signal states.