Communications in Mathematical Physics

, Volume 284, Issue 1, pp 263–280 | Cite as

Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1

  • Patrick Hayden
  • Andreas WinterEmail author


For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p > 1, the minimum output Rényi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Rényi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p = 1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.


Entropy Entangle State Quantum Channel Quantum Information Theory Classical Capacity 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.Department of MathematicsUniversity of BristolBristolUK
  3. 3.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore

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