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

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Araki H. (1990) On an inequality of Lieb and Thirring. Lett. Math. Phys. 19, 167–170MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Audenaert, K.M.R. A note on the pq norms of completely positive maps. http://arxiv.org/list/math-ph/0505085, 2005Google Scholar
  3. 3.
    Amosov G.G., Holevo A.S., Werner R.F. (2000) On Some Additivity Problems in Quantum Information Theory. Prob. Inf. Trans. 36, 305–313MATHGoogle Scholar
  4. 4.
    Barnum H., Nielsen M.A., Schumacher B. (1998) Information transmission through a noisy quantum channel. Phys. Rev. A 57, 4153–4175CrossRefADSGoogle Scholar
  5. 5.
    Bennett C.H., Shor P.W., Smolin J.A., Thapliyal A.V. (1999) Entanglement-assisted classical capacity of noisy quantum channels. Phys. Rev. Lett. 83, 3081–84CrossRefADSGoogle Scholar
  6. 6.
    Bennett C.H., Shor P.W., Smolin J.A., Thapliyal A.V. (2002) Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Inform. Theory 48, 2637–2655MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Blecher D.P., Paulsen V.I. (1991) Tensor products of operator spaces. J. Funct. Anal. 99, 262–292MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Carlen, E. Lieb, E. A Minkowski type trace inequality and strong subadditivity of quantum entropy. Amer. Math. Soc. Trans. 189, 59–62 (1999), reprinted in [32].Google Scholar
  9. 9.
    Choi M-D (1975) Completely Positive Linear Maps on Complex Matrices. Lin. Alg. Appl. 10, 285–290MATHCrossRefGoogle Scholar
  10. 10.
    Devetak I. (2005) The Private Classical Capacity and Quantum Capacity of a Quantum Channel. IEEE Trans. Inform. Theory 51, 44–55CrossRefMathSciNetGoogle Scholar
  11. 11.
    DiVincenzo D. P., Shor P. W., Smolin J. A. Quantum-channel capacity of very noisy channels. Phys. Rev. A 57, 830–839 (1998); erratum 59, 1717 (1999)Google Scholar
  12. 12.
    Effros E.G., Ruan Z.J. (1991) Self-duality for the Haagerup tensor product and Hilbert space factorizations. J. Funct. Anal. 100, 257–284MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Effros E.G., Ruan Z.J. (2000) Operator Spaces. Oxford Univ. Press, OxfordMATHGoogle Scholar
  14. 14.
    Epstein H. (1973) Remarks on two theorems of E. Lieb. Commun. Math. Phys. 31, 317–325MATHCrossRefADSGoogle Scholar
  15. 15.
    Holevo, A. S. Coding Theorem for Quantum Channels. http://arxiv.org/list/quant-ph/9809023; Quantum coding theorems. Russ. Math. Surv. 53, 1295–1331 (1999)Google Scholar
  16. 16.
    Holevo A.S. (2002) On Entanglement-Assisted Classical Capacity. J. Math. Phys. 43, 4326–4333MATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Holevo A.S., Werner R.F. (2001) Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63, 032312CrossRefADSGoogle Scholar
  18. 18.
    Horodecki M., Oppenheim J. (2005) Andreas Winter Quantum information can be negative. Nature 436, 673–676CrossRefADSGoogle Scholar
  19. 19.
    Horodecki, M., Oppenheim, J., Andreas Winter Quantum state merging and negative information. Commun. Math. Phys. http://arxiv.org/list/quant-ph/0512247, 2005 (in press)Google Scholar
  20. 20.
    Horodecki M., Shor P., Ruskai M.B. (2003) Entanglement Breaking Channels. Rev. Math. Phys. 15, 629–641MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Jencová, A. A relation between completely bounded norms and conjugate channels. Commun. Math. Phys., DOI 10.1007/s00220-006-0035-zGoogle Scholar
  22. 22.
    Junge, M. Factorization theory for spaces of operators. Habilitation thesis, Kiel University, 1996Google Scholar
  23. 23.
    Junge, M. Vector-valued L p spaces for von Neumann algebras with QWEP. In preparationGoogle Scholar
  24. 24.
    Junge M., Ruan Z.-J. (2004) Decomposable Maps on Non-commutative L p spaces. Contemporary Mathematics 365, 355–381MathSciNetGoogle Scholar
  25. 25.
    King C. (2003) Maximal p-norms of entanglement breaking channels. Quantum Inf. and Comput. 3(2): 186–190MathSciNetGoogle Scholar
  26. 26.
    King C., Ruskai M.B. (2001) Minimal Entropy of States Emerging from Noisy Quantum Channels. IEEE Trans. Info. Theory 47, 1–19MathSciNetGoogle Scholar
  27. 27.
    King, C., Ruskai, M. B. Comments on multiplicativity of maximal p-norms when p = 2. In: Quantum Information, Statistics and Probability, ed. by O. Hirota, River Edge, NJ: World Scientific, 2004, pp. 102–114Google Scholar
  28. 28.
    Kitaev, A. Classical and Quantum Computation Providence, RI: AMS, 2002Google Scholar
  29. 29.
    Klein O. (1931) Zur quantenmechanischen begründung des des zweiten haupsatzes der wärmelehre. Zeit. für Physik. 72, 767–775MATHCrossRefADSGoogle Scholar
  30. 30.
    Kraus K. (1971) General state changes in quantum theory. Ann. Phys. 64, 311–335CrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Kraus K. (1983) States, Effects and Operations: Fundamental Notions of Quantum Theory. Springer-Verlag, Berlin-Heidelberg: New YorkMATHGoogle Scholar
  32. 32.
    Inequalities Selecta of E. Lieb. M. Loss, M.B. Ruskai, eds., Berlin-Heidelberg: New York: Springer, 2002Google Scholar
  33. 33.
    Lieb E.H., Ruskai M.B. (1973), Proof of the Strong Subadditivity of Quantum Mechanical Entropy. J. Math. Phys. 14, 1938–1941, reprinted in [32]CrossRefADSMathSciNetGoogle Scholar
  34. 34.
    Lieb, E., Thirring, W. Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities. In Studies in Mathematical Physics, E. Lieb, B. Simon, A. Wightman, eds., pp. 269–303 (Princeton University Press, 1976) pp. 269–303, reprinted in [32]Google Scholar
  35. 35.
    Lloyd S. (1997) The capacity of a noisy quantum channel. Phys. Rev. A 55, 1613–1622CrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Nelson E. (1974) Notes on non-commutative integration. J. Func. Anal. 15, 103–116MATHCrossRefGoogle Scholar
  37. 37.
    Nielsen, M. Quantum Information Theory, PhD thesis. University of New Mexico, 1998Google Scholar
  38. 38.
    Nielsen M., Chuang I. (2000) Quantum Computation and Quantum Information. Cambridge University Press, CambridgeMATHGoogle Scholar
  39. 39.
    Paulsen V. (2000) Completely Bounded Maps and Operator Algebras. Cambridge University Press, CambridgeGoogle Scholar
  40. 40.
    Pisier, G. The Operator Hilbert Space OH, Complex Interpolation and Tensor Norms. Memoirs AMS. 122, Providence, RI: Amer. Math. Soc. 1996Google Scholar
  41. 41.
    Pisier, G. Non-Commutative Vector Valued L p-spaces and Completely p-summing Maps. Paris: Société Mathématique de France, 1998Google Scholar
  42. 42.
    Pisier G. (2003) Introduction to Operator Space Theory. Cambridge University Press, CambridgeMATHGoogle Scholar
  43. 43.
    Ruskai, M.B. Inequalities for Quantum Entropy: A Review with Conditions for Equality. J. Math. Phys. 43, 4358–4375 (2002); erratum 46, 019901 (2005)Google Scholar
  44. 44.
    Ruskai M.B. (2003) Qubit Entanglement Breaking Channels. Rev. Math. Phys. 15, 643–662MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Ruan Z-J. (1988) Subspaces of C *-algebras. J. Funct. Anal. 76, 217–230MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Segal I.E. (1953) A non-commutative extension of abstract integration. Ann. of Math. 57, 401–457CrossRefMathSciNetGoogle Scholar
  47. 47.
    Shor, P. W. Announced at MSRI workshop, (November, 2002). Notes are available at www.msri.org/publications/ln/msri/2002/quantumcrypto/shor/1/index.htmlGoogle Scholar
  48. 48.
    Shor P.W. (2004) Equivalence of Additivity Questions in Quantum Information Theory. Commun. Math. Phys. 246, 453–472MATHCrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Simon B. (1993) The Statistical Mechanics of Lattice Gases. Princeton Univ. Press, Princetion, NJMATHGoogle Scholar
  50. 50.
    Smolin J., Verstraete F., Winter A. (2005) Entanglement of assistance and multipartite state distillation. Phys. Rev. A 72, 052317CrossRefADSGoogle Scholar
  51. 51.
    Stinespring W.F. (1955) Positive functions on C *-algebras. Proc. Amer. Math. Soc. 6, 211–216MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Watrous J. (2005) Notes on super-operator norms induced by Schatten norms. Quantum Inf. Comput. 5, 57–67MathSciNetGoogle Scholar
  53. 53.
    Werner R.F., Holevo A.S. (2002) Counterexample to an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43(9): 4353–4357MATHCrossRefADSMathSciNetGoogle Scholar

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

Personalised recommendations