State Evolution and Trace-Preserving Completely Positive Maps

  • Masahito HayashiEmail author
Part of the Graduate Texts in Physics book series (GTP)


Until now, we have considered only quantum states and quantum measurement as quantum concepts. In order to prefer information processing with quantum systems, we should manipulate a wider class of state operations. This chapter examines what kinds of operations are allowed on quantum systems. The properties of these operations will also be examined.


State Evolution Conditional Entropy Partial Trace Strong Subadditivity Quantum Relative Entropy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    W.F. Stinespring, Positive functions on C-algebras. Proc. Am. Math. Soc. 6, 211 (1955)Google Scholar
  2. 2.
    K. Kraus, in States, Effects, and Operations, vol. 190, Lecture Notes in Physics (Springer, Berlin Heidelberg New York, 1983)Google Scholar
  3. 3.
    M.-D. Choi, Completely positive linear maps on complex matrices. Lin. Alg. Appl. 10, 285–290 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Fujiwara, P. Algoet, One-to-one parametrization of quantum channels. Phys. Rev. A 59, 3290–3294 (1999)ADSCrossRefGoogle Scholar
  5. 5.
    A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3, 275–278 (1972)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Fujiwara, Mathematics of quantum channels. Suurikagaku 474, 28–35 (2002). (in Japanese)Google Scholar
  7. 7.
    G.M. D’Ariano, P.L. Presti, Imprinting complete information about a quantum channel on its output state. Phys. Rev. Lett. 91, 047902 (2003)ADSCrossRefGoogle Scholar
  8. 8.
    D. Aharonov, A. Kitaev, N. Nisan, Quantum Circuits with Mixed States Proceedings of the 30th Annual ACM Symposium on Theory of Computation (STOC), 20–30 (1997)Google Scholar
  9. 9.
    M. Horodecki, P. Shor, M.B. Ruskai, Entanglement breaking channels. Rev. Math. Phys. 15, 1–13 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996)Google Scholar
  11. 11.
    P. Horodecki, Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    N. Datta, A.S. Holevo, Y. Suhov, Additivity for transpose depolarizing channels. Int. J. Quantum Inform. 4, 85 (2006)CrossRefzbMATHGoogle Scholar
  13. 13.
    K. Matsumoto, F. Yura, Entanglement cost of antisymmetric states and additivity of capacity of some quantum channel. Jhys. A: Math. Gen. 37, L167–L171 (2004)ADSMathSciNetzbMATHGoogle Scholar
  14. 14.
    R.F. Werner, A.S. Holevo, Counterexample to an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43, 4353 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M.B. Ruskai, S. Szarek, E. Werner, An analysis of completely-positive trace-preserving maps on 2\(\times \)2 matrices. Lin. Alg. Appl. 347, 159–187 (2002)CrossRefzbMATHGoogle Scholar
  16. 16.
    M.B. Ruskai, Qubit entanglement breaking channels. Rev. Math. Phys. 15, 643–662 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    A. Fujiwara, H. Nagaoka, Operational capacity and pseudoclassicality of a quantum channel. IEEE Trans. Inf. Theory 44, 1071–1086 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    G. Lindblad, Completely positive maps and entropy inequalities. Comm. Math. Phys. 40, 147–151 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Comm. Math. Phys. 54, 21–32 (1977)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    G. Lindblad, Expectations and entropy inequalities for finite quantum systems. Comm. Math. Phys. 39, 111–119 (1974)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    F. Hiai, D. Petz, The golden-thompson trace inequality is complemented. Lin. Alg. Appl. 181, 153–185 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S. Golden, Lower bounds for Helmholtz function. Phys. Rev. 137, B1127–B1128 (1965)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    K. Symanzik, Proof and refinements of an inequality of Feynman. J. Math. Phys. 6, 1155–1156 (1965)ADSCrossRefGoogle Scholar
  24. 24.
    C.J. Thompson, Inequality with applications in statistical mechanics. J. Math. Phys. 6, 1812–1813 (1965)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    A. Uhlmann, The ‘transition probability’ in the state space of *-algebra. Rep. Math. Phys. 9, 273–279 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    R. Jozsa, Fidelity for mixed quantum states. J. Mod. Opt. 41(12), 2315–2323 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    H. Barnum, C.A. Fuchs, R. Jozsa, B. Schumacher, A general fidelity limit for quantum channels. Phys. Rev. A 54, 4707–4711 (1996)ADSCrossRefGoogle Scholar
  28. 28.
    M.B. Ruskai, Beyond strong subadditivity? improved bounds on the contraction of generalized relative entropy. Rev. Math. Phys. 6, 1147–1161 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    R.L. Frank, E.H. Lieb, Monotonicity of a relative Renyi entropy. J. Math. Phys. 54, 122201 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    F. Hiai, D. Petz, The proper formula for relative entropy and its asymptotics in quantum probability. Comm. Math. Phys. 143, 99–114 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    E. Lieb, M.B. Ruskai, A fundamental property of quantum mechanical entropy. Phys. Rev. Lett. 30, 434–436 (1973)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    E. Lieb, M.B. Ruskai, Proof of the strong subadditivity of quantum mechanical entropy. J. Math. Phys. 14, 1938–1941 (1973)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    P. Hayden, R. Jozsa, D. Petz, A. Winter, Structure of states which satisfy strong subadditivity of quantum entropy with equality. Comm. Math. Phys. 246, 359–374 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    M. Fannes, A continuity property of the entropy density for spin lattice systems. Comm. Math. Phys. 31, 291–294 (1973)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    W. Ochs, A new axiomatic characterization of the von Neumann entropy. Rep. Math. Phys. 8(1), 109–120 (1975)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    H. Araki, E. Lieb, Entropy inequalities. Comm. Math. Phys. 18, 160–170 (1970)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    E. Lieb, Bull. Am. Math. Soc. 81, 1–13 (1975)CrossRefGoogle Scholar
  38. 38.
    R. Alicki, M. Fannes, Continuity of quantum mutual information, quant-ph/0312081 (2003)Google Scholar
  39. 39.
    M. Christandl, A. Winter, Squashed entanglement"-an additive entanglement measure. J. Math. Phys. 45, 829–840 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    H. Fan, A note on quantum entropy inequalities and channel capacities. J. Phys. A Math. Gen. 36, 12081–12088 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    R. König, R. Renner, C. Schaffner, The operational meaning of min- and max-entropy. IEEE Trans. Inf. Theory 55(9), 4337–4347 (2009)MathSciNetCrossRefGoogle Scholar
  42. 42.
    M. Tomamichel, R. Colbeck, R. Renner, A fully quantum asymptotic equipartition property. IEEE Trans. Inf. Theory 55(12), 5840–5847 (2009)MathSciNetCrossRefGoogle Scholar
  43. 43.
    S. Beigi, Sandwiched Rènyi divergence satisfies data processing inequality. J. Math. Phys. 54(12), 122202 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, M. Tomamichel, On quantum Renyi entropies: a new generalization and some properties. J. Math. Phys. 54, 122203 (2013)Google Scholar
  45. 45.
    M. Tomamichel, M. Berta, M. Hayashi, Relating different quantum generalizations of the conditional Rényi entropy. J. Math. Phys. 55, 082206 (2014)Google Scholar
  46. 46.
    A.S. Holevo, Bounds for the quantity of information transmitted by a quantum communication channel. Problemy Peredachi Informatsii, 9, 3–11 (1973) (in Russian). (English translation: Probl. Inf. Transm., 9, 177–183 (1975))Google Scholar
  47. 47.
    E. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math. 11, 267–288 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    D. Petz, Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23, 57–65 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    M.M. Wilde, A. Winter, D. Yang, Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy. Comm. Math. Phys. 331(2), 593 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    T. Ando, Convexity of certain maps on positive definite matrices and applications to Hadamard products. Lin. Alg. and Appl. 26, 203–241 (1979)CrossRefzbMATHGoogle Scholar
  51. 51.
    J. Bergh, J. Löfström, Interpolation Spaces. (Springer-Verlag, New York, 1976)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan

Personalised recommendations