Communications in Mathematical Physics

, Volume 284, Issue 1, pp 281–290 | Cite as

Counterexamples to Additivity of Minimum Output p-Rényi Entropy for p Close to 0

  • Toby Cubitt
  • Aram W. Harrow
  • Debbie Leung
  • Ashley Montanaro
  • Andreas Winter


Complementing recent progress on the additivity conjecture of quantum information theory, showing that the minimum output p-Rényi entropies of channels are not generally additive for p > 1, we demonstrate here by a careful random selection argument that also at p = 0, and consequently for sufficiently small p, there exist counterexamples.

An explicit construction of two channels from 4 to 3 dimensions is given, which have non-multiplicative minimum output rank; for this pair of channels, numerics strongly suggest that the p-Rényi entropy is non-additive for all p ≲ 0.11. We conjecture however that violations of additivity exist for all p < 1.


Entangle State Quantum Channel Minimum Output Random Subspace Schmidt Rank 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Toby Cubitt
    • 1
  • Aram W. Harrow
    • 2
  • Debbie Leung
    • 3
  • Ashley Montanaro
    • 2
  • Andreas Winter
    • 1
    • 4
  1. 1.Department of MathematicsUniversity of BristolBristolUK
  2. 2.Department of Computer ScienceUniversity of BristolBristolUK
  3. 3.Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada
  4. 4.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore

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