No Access Submitted: 08 June 2012 Accepted: 12 September 2012 Published Online: 19 October 2012
J. Math. Phys. 53, 102208 (2012);
more...View Affiliations
View Contributors
  • Teiko Heinosaari
  • Maria A. Jivulescu
  • David Reeb
  • Michael M. Wolf
For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian inputs or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem.
The authors thank Francesco Buscemi, Keiji Matsumoto, and David Pérez-García for very valuable discussions. T.H. acknowledges support from the Academy of Finland (Grant No. 138135). M.A.J. would like to thank for financial support the European project QUEVADIS, the Rectorate of University Politehnica Timişoara, and the strategic grant POSDRU/21/1.5/G/13798 co-financed by the European Social Fund - Investing in People, within the Sectorial Operational Programme Human Resources Development 2007–2013. D.R. and M.M.W. acknowledge support from the European projects COQUIT and QUEVADIS, the CHIST-ERA/BMBF project CQC, and from the Alfried Krupp von Bohlen und Halbach-Stiftung.
  1. 1. W. B. Arveson, “Subalgebras of C*-algebras,” Acta Math. 123, 141 (1969). , Google ScholarCrossref
  2. 2. P. M. Alberti and A. Uhlmann, “A problem relating to positive linear maps on matrix algebras,” Rep. Math. Phys. 18, 163 (1980). , Google ScholarCrossref
  3. 3. P. M. Alberti and A. Uhlmann, “Stochasticity and partial order,” in Mathematics and Its Applications (D. Reidel, 1982), Vol. 9. Google Scholar
  4. 4. F. Buscemi, G. M. D'Ariano, M. Keyl, P. Perinotti, and R. F. Werner, “Clean positive operator valued measures,” J. Math. Phys. 46, 082109 (2005). , Google ScholarScitation, ISI
  5. 5. H.-P. Breuer, E.-M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103, 210401 (2009). , Google ScholarCrossref, ISI
  6. 6. Correspondence with and private communication of results from F. Buscemi and K. Matsumoto (2011). Google Scholar
  7. 7. F. Buscemi, “Comparison of quantum statistical models: Equivalent conditions for sufficiency,” Commun. Math. Phys. 310, 625 (2012). , Google ScholarCrossref
  8. 8. S. Boyd and L. Vandenberghe, Convex optimization (Cambridge University Press, 2004). Google ScholarCrossref
  9. 9. A. Chefles, “Quantum state discrimination,” Contemp. Phys. 41, 401 (2000). , Google ScholarCrossref
  10. 10. M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl. 10, 285 (1975). , Google ScholarCrossref, ISI
  11. 11. A. Chefles, R. Jozsa, and A. Winter, “On the existence of physical transformations between sets of quantum states,” Int. J. Quantum Inf. 2, 11 (2004). , Google ScholarCrossref
  12. 12. J. B. Conway, A Course in Functional Analysis (Springer-Verlag, 1990). Google Scholar
  13. 13. E. B. Davies, Quantum Theory of Open Systems (Academic, 1976). Google Scholar
  14. 14. C. W. Hellstrom, Quantum Detection and Estimation Theory (Academic, 1976). Google Scholar
  15. 15. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge University Press, 1952). Google Scholar
  16. 16. A. Jencova, “Generalized channels: Channels for convex subsets of the state space,” J. Math. Phys. 53, 012201 (2012). , Google ScholarScitation
  17. 17. A. Jencova, “Comparison of quantum binary experiments,” Rep. Math. Phys. (to be published), Google Scholar
    e-print arXiv:1110.4792 [quant-ph]. Google Scholar
  18. 18. C.-K. Li and Q.-T. Poon, “Interpolation problems by completely positive maps,” e-print arXiv:1012.1675 [quant-ph]. Google Scholar
  19. 19. K. Matsumoto, “A quantum version of randomization criterion,” e-print arXiv:1012.2650 [quant-ph]. Google Scholar
  20. 20. T. H. Matheiss and D. S. Rubin, “A survey and comparison of methods for finding all vertices of convex polyhedral sets,” Math. Operat. Res. 5, 167 (1980). , Google ScholarCrossref
  21. 21. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000). Google Scholar
  22. 22. V. Paulsen, Completely Bounded Maps and Operator Algebras (Cambridge University Press, 2003). Google ScholarCrossref
  23. 23. D. Reeb, M. J. Kastoryano, and M. M. Wolf, “Hilbert's projective metric in quantum information theory,” J. Math. Phys. 52, 082201 (2011). , Google ScholarScitation
  24. 24. R. T. Rockafellar, Convex Analysis (Princeton University Press, 1970). Google ScholarCrossref
  25. 25. E. Ruch, S. Schranner, and T. H. Seligman, “Generalization of a theorem of Hardy, Littlewood and Polya,” J. Math. Anal. Appl. 76, 222 (1980). , Google ScholarCrossref
  26. 26. M. B. Ruskai, “Beyond strong subadditivity? Improved bounds on the contraction of generalized relative entropy,” Rev. Math. Phys. 6, 1147 (1994). , Google ScholarCrossref, ISI
  27. 27. J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optim. Methods Software 12, 625 (1999). , Google ScholarCrossref
  28. 28. E. Torgersen, Comparison of Statistical Experiments (Cambridge University Press, 1991). Google ScholarCrossref
  29. 29. A. Uhlmann, “The “transition probability” in the state space of a *-algebra,” Rep. Math. Phys. 9, 273 (1976). , Google ScholarCrossref
  30. 30. A. Uhlmann, “The transition probability for states of *-algebras,” Ann. Phys. 42, 524 (1985). , Google ScholarCrossref
  31. 31. M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, “Assessing non-Markovian quantum dynamics,” Phys. Rev. Lett. 101, 150402 (2008). , Google ScholarCrossref, ISI
  32. 32. S. L. Woronowicz, “Positive maps of low dimensional matrix algebras,” Rep. Math. Phys. 10, 165 (1976). , Google ScholarCrossref
  1. © 2012 American Institute of Physics.