  We introduce the concept of mutual independence -- correlations shared
between distant parties which are independent of the environment. This notion
is more general than the standard idea of a secret key -- it is a fully quantum
and more general form of privacy. The states which possess mutual independence
also generalize the so called private states -- those that possess private key.
We then show that the problem of distributed compression of quantum information
at distant sources can be solved in terms of mutual independence, if free
entanglement between the senders and the receiver is available. Namely, we
obtain a formula for the sum of rates of qubits needed to transmit a
distributed state between Alice and Bob to a decoder Charlie. We also show that
mutual independence is bounded from above by the relative entropy modulo a
conjecture, saying that if after removal of a single qubit the state becomes
product, its initial entanglement is bounded by 1. We suspect that mutual
independence is a highly singular quantity, i.e. that it is positive only on a
set of measure zero; furthermore, we believe that its presence is seen on the
single copy level. This appears to be born out in the classical case.
