Communications in Mathematical Physics

, Volume 266, Issue 1, pp 37–63

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

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

DOI: 10.1007/s00220-006-0034-0

Cite this article as:
Devetak, I., Junge, M., King, C. et al. Commun. Math. Phys. (2006) 266: 37. doi:10.1007/s00220-006-0034-0


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 L1Lp for Lp 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 L1Lp 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.

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Igor Devetak
    • 1
  • 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