All inequalities for the relative entropy

Ben Ibinson*, Noah Linden, Andreas Winter

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding)

1 Citation (Scopus)


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 languageEnglish
Title of host publication2006 IEEE International Symposium on Information Theory, Vols 1-6, Proceedings
Place of PublicationNEW YORK
PublisherIEEE Computer Society
Number of pages5
ISBN (Print)978-1-4244-0505-3
Publication statusPublished - 2006
EventIEEE International Symposium on Information Theory - Seattle, United States
Duration: 9 Jul 200614 Jul 2006


ConferenceIEEE International Symposium on Information Theory
Country/TerritoryUnited States




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