Communications in Mathematical Physics

, Volume 296, Issue 1, pp 111–143 | Cite as

Comments on Hastings’ Additivity Counterexamples

  • Motohisa Fukuda
  • Christopher King
  • David K. Moser


Hastings [12] recently provided a proof of the existence of channels which violate the additivity conjecture for minimal output entropy. In this paper we present an expanded version of Hastings’ proof. In addition to a careful elucidation of the details of the proof, we also present bounds for the minimal dimensions needed to obtain a counterexample.


Quantum Channel Input State Random State Partial Isometry Output Entropy 


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  1. 1.
    Amosov G.G.: Remark on the additivity conjecture for the quantum depolarizing channel. Probl. Inf. Trans. 42(2), 69–76 (2006)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Amosov, G.G.: The strong superadditivity conjecture holds for the quantum depolarizing channel in any dimension. Phys. Rev. A 75(6), P. 060304 (2007)Google Scholar
  3. 3.
    Amosov G.G., Holevo A.S., Werner R.F.: On some additivity problems in quantum information theory. Prob. Inf. Trans. 36, 305–313 (2000)MATHGoogle Scholar
  4. 4.
    Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics 169, Berlin-Heidelberg-NewYork: Springer, 1997Google Scholar
  5. 5.
    Bronk B.V.: Exponential ensemble for random matrices. J. Math. Phys. 6, 228 (1965)CrossRefADSGoogle Scholar
  6. 6.
    Bruss D., Faoro L., Macchiavello C., Palma M.: Quantum entanglement and classical communication through a depolarising channel. J. Mod. Opt. 47, 325 (2000)MathSciNetADSGoogle Scholar
  7. 7.
    Cover T.M., Thomas J.A.: Elements of Information Theory. NewYork, John Wiley and Sons (1991)MATHCrossRefGoogle Scholar
  8. 8.
    Cubitt T., Harrow A.W., Leung D., Montanaro A., Winter A.: Counterexamples to additivity of minimum output p-Renyi entropy for p close to 0. Commun. Math. Phys. 284, 281–290 (2008)MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Fujiwara A., Hashizume T.: Additivity of the capacity of depolarizing channels. Phys. Lett. A 299, 469–475 (2002)MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Fukuda M.: Simplification of additivity conjecture in quantum information theory. Quant. Info. Proc. 6, 179–186 (2007)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fukuda M., Wolf M.M.: Simplifying additivity problems using direct sum constructions. J. Math. Phys. 48, 072101 (2007)CrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Hastings M.B.: A Counterexample to additivity of minimum output entropy. Nature Physics 5, 255–257 (2009)CrossRefADSGoogle Scholar
  13. 13.
    Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265(1), 95–117 (2006)MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Hayden P., Winter A.: Counterexamples to the maximal p-norm multiplicativity conjecture for all p >  1. Commun. Math. Phys. 284(1), 263–280 (2008)MATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Holevo A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory 44(1), 269–273 (1998)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Holevo A.S.: On complementary channels and the additivity problem. Prob. Th. and Appl. 51, 133–143 (2005)Google Scholar
  17. 17.
    King C.: Additivity for unital qubit channels. J. Math. Phys. 43, 4641–4653 (2002)MATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    King C.: The capacity of the quantum depolarizing channel. IEEE Trans. Info. Theory 49, 221–229 (2003)MATHCrossRefGoogle Scholar
  19. 19.
    King C., Nathanson M., Ruskai M.B.: Multiplicativity properties of entrywise positive maps. Lin. Alg. Appl. 404, 367–379 (2005)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    King C., Ruskai M.B.: Minimal entropy of states emerging from noisy quantum channels. IEEE Trans. Info. Theory 47, 192–209 (2001)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lloyd S., Pagels H.: Complexity as thermodynamic depth. Ann. Phys. 188, 186–213 (1988)CrossRefMathSciNetADSGoogle Scholar
  22. 22.
    King C., Matsumoto K., Nathanson M., Ruskai M.B.: Properties of conjugate channels with applications to additivity and multiplicativity. Markov Processes and Related Fields 13(2), 391–423 (2007)MATHMathSciNetGoogle Scholar
  23. 23.
    Page D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291 (1993)MATHCrossRefMathSciNetADSGoogle Scholar
  24. 24.
    Sanchez-Ruiz J.: Simple proof of pages conjecture on the average entropy of a subsystem. Phys. Rev. E 52, 5653 (1995)CrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Schumacher B., Westmoreland M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56(1), 131–138 (1997)CrossRefADSGoogle Scholar
  26. 26.
    Sen S.: Average entropy of a quantum subsystem. Phys. Rev. Lett. 77(1), 13 (1996)CrossRefADSGoogle Scholar
  27. 27.
    Shor P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246(3), 453–472 (2004)MATHCrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Werner R.F., Holevo A.S.: Counterexample to an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43(9), 4353–4357 (2002)MATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Zyczkowski K., Sommers H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A 34, 7111–7125 (2001)MATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Motohisa Fukuda
    • 1
  • Christopher King
    • 2
  • David K. Moser
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA
  2. 2.Department of MathematicsNortheastern UniversityBostonUSA
  3. 3.Department of PhysicsNortheastern UniversityBostonUSA

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