  Given an unknown quantum state distributed over two systems, we determine how
much quantum communication is needed to transfer the full state to one system.
This communication measures the "partial information" one system needs
conditioned on it's prior information. It turns out to be given by an extremely
simple formula, the conditional entropy. In the classical case, partial
information must always be positive, but we find that in the quantum world this
physical quantity can be negative. If the partial information is positive, its
sender needs to communicate this number of quantum bits to the receiver; if it
is negative, the sender and receiver instead gain the corresponding potential
for future quantum communication. We introduce a primitive "quantum state
merging" which optimally transfers partial information. We show how it enables
a systematic understanding of quantum network theory, and discuss several
important applications including distributed compression, multiple access
channels and multipartite assisted entanglement distillation (localizable
entanglement). Negative channel capacities also receive a natural
interpretation.
