Communications in Mathematical Physics

, Volume 266, Issue 1, pp 37–63 | Cite as

Multiplicativity of Completely Bounded p-Norms Implies a New Additivity Result

  • Igor DevetakEmail author
  • Marius Junge
  • Christoper King
  • Mary Beth Ruskai


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 S12) − S1) is restricted to states of the form \((\mathcal{I} \otimes \Phi)\left( | \psi \rangle \langle \psi | \right)\). We show that this follows from multiplicativity of the completely bounded norm of Φ considered as a map from L 1L 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 1L 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.


Banach Space Entangle State Quantum Channel Entropy Inequality Quantum Information Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Igor Devetak
    • 1
    Email author
  • Marius Junge
    • 2
  • Christoper King
    • 3
  • Mary Beth Ruskai
    • 4
  1. 1.Electrical Engineering DepartmentUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Department of MathematicsNortheastern UniversityBostonUSA
  4. 4.Department of MathematicsTufts UniversityMedfordUSA

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