All Inequalities for the Relative Entropy

BD Ibinson, N Linden, AJ Winter

Research output: Contribution to journalArticle (Academic Journal)peer-review

8 Citations (Scopus)


The relative entropy of two n-party quantum states is an important quantity exhibiting, for example, the extent to which the two states are different. The relative entropy of the states formed by reducing two n-party 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 under 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. 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.
Translated title of the contributionAll Inequalities for the Relative Entropy
Original languageEnglish
Pages (from-to)223 - 238
Number of pages16
JournalCommunications in Mathematical Physics
Volume269 (1)
Publication statusPublished - Jan 2007

Bibliographical note

Publisher: Springer


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