## Abstract

The relative entropy of two distributions of n random variables, and more generally of two n-party quantum states, is an important quantity exhibiting, for example, the extent to which the two distributions/states are different. The relative entropy (if the states formed by restricting to a smaller number m of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy.

Using techniques from convex geometry, we prove that monotonicity tinder restrictions is the only general inequality satisfied by relative entropies. In doing so we make a connection to secret sharing schemes with general access structures: indeed, it turns out that the extremal rays of the cone defined by monotonicity are populated by classical secret sharing schemes.

A suprising outcome is that the structure of allowed relative entropy values of subsets of multiparty states is much simpler than the structure of allowed entropy values. And the structure of allowed relative entropy values (unlike that of entropies) is the same for classical probability distributions and quantum states.

Original language | English |
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Title of host publication | 2006 IEEE International Symposium on Information Theory, Vols 1-6, Proceedings |

Place of Publication | NEW YORK |

Publisher | IEEE Computer Society |

Pages | 237-241 |

Number of pages | 5 |

ISBN (Print) | 978-1-4244-0505-3 |

Publication status | Published - 2006 |

Event | IEEE International Symposium on Information Theory - Seattle, United States Duration: 9 Jul 2006 → 14 Jul 2006 |

### Conference

Conference | IEEE International Symposium on Information Theory |
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Country/Territory | United States |

City | Seattle |

Period | 9/07/06 → 14/07/06 |

## Keywords

- SYSTEMS
- QUANTUM INFORMATION-THEORY
- ENTANGLEMENT