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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This work was done at the Isaac Newton Institute (Cambridge) in August–December 2004 and we are grateful for the institute's hospitality. We thank W. Unruh for comments on an earlier draft of this paper. We acknowledge the support of EC grants RESQ, QUPRODIS and PROSECCO. M.H. was additionally supported by the Polish Ministry of Scientific Research and Information Technology, J.O. by the Cambridge-MIT Institute and Newton Trust, and A.W. by EPSRC's “IRC QIP”.
Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests.
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Horodecki, M., Oppenheim, J. & Winter, A. Partial quantum information. Nature 436, 673–676 (2005). https://doi.org/10.1038/nature03909
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