1. Institute of Theoretical Physics and Astrophysics, University of Gdask, 80-952 Gdask, Poland
2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
3. Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK

Correspondence to: Jonathan Oppenheim² Correspondence and requests for materials should be addressed to J.O. (Email: j.oppenheim@damtp.cam.ac.uk).

Information—be it classical¹ or quantum²—is measured by the amount of communication needed to convey it. In the classical case, if the receiver has some prior information about the messages being conveyed, less communication is needed³. Here we explore the concept of prior quantum information: 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 its prior information. We find that it is given by the conditional entropy—a quantity that was known previously, but lacked an operational meaning. 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, then sender and receiver instead gain the corresponding potential for future quantum communication. We introduce a protocol that we term '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, noiseless coding with side information, multiple access channels and assisted entanglement distillation.

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