Multiplicativity of completely bounded p-norms implies a new additivity result

Igor Devetak, Marius Junge, Christoper King, Mary Beth Ruskai

Research output: Contribution to journalArticlepeer-review


We prove additivity of the minimal conditional entropy associated with a quantum channel Φ, represented by a completely positive (CP), trace-preserving map, when the infimum of S(γ12) - S(γ1) is restricted to states of the form (I ⊗ Φ)(|ψ » « ψ|). We show that this follows from multiplicativity of the completely bounded norm of Φ considered as a map from L 1 → L p for L p spaces defined by the Schatten p-norm on matrices, and give another proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L 1 → L p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.

Original languageEnglish (US)
Pages (from-to)37-63
Number of pages27
JournalCommunications in Mathematical Physics
Issue number1
StatePublished - Aug 2006

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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