Categories of quantum and classical channels

Quantum Information Processing

Volumn 15, Issue 12, 2016, Pages 5179-5209

  Coecke, Bob ^a   Heunen, Chris ^a   Kissinger, Aleks ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Abstract C* algebras; Categorical quantum mechanics; Completely positive maps; Quantum channel

Indexed keywords

ALGEBRA; INFORMATION THEORY; QUANTUM COMMUNICATION; QUANTUM OPTICS; QUANTUM THEORY;

ABSTRACT NOTIONS; COMPLETELY POSITIVE MAP; FINITE DIMENSIONAL; PURE STATE; QUANTUM CHANNEL; QUANTUM INFORMATION THEORY; SUPEROPERATORS;

COMMUNICATION CHANNELS (INFORMATION THEORY);

EID: 84996590508     PISSN: 15700755     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11128-014-0837-4     Document Type: Article

References (40)

1. Abramsky, S., Coecke, B.: Categorical Quantum Mechanics. Elsevier, Amsterdam (2008)
2. Abramsky, S., Heunen, C.: H*-algebras and nonunital frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics. Clifford Lect. 71, 1–24 (2012)
3. Alicki, R.: Comment on ‘reduced dynamics need not be completely positive’. Phys. Rev. Lett. 75, 3020 (1995)
4. Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)
5. Boixo, S., Heunen, C.: Entangled and sequential quantum protocols with dephasing. Phys. Rev. Lett. 108, 120–402 (2012)
6. Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10(3), 285–290 (1975)
7. Coecke, B.: Axiomatic description of mixed states from Selinger’s CPM-construction. Electron. Notes Theor. Comput. Sci. 210, 3–13 (2008)
8. Coecke, B. (ed.): New Structures for Physics. No. 813 in Lecture Notes in Physics. Springer, Berlin (2009).
9. Coecke, B., Duncan, R.: Interacting quantum observables: categorical algebra and diagrammatics. New J. Phys. 13, 043,016 (2011).
10. Coecke, B., Heunen, C.: Pictures of complete positivity in arbitrary dimension. Electron. Proc. Theor. Comput. Sci. 95, 27–35 (2012)
11. Coecke, B., Heunen, C., Kissinger, A.: Compositional quantum logic. In: Coecke, B., Ong, L., Panangaden, P. (eds.) Computation, Logic, Games, and Quantum Foundationsno. 7860 in Lectures Notes in Computer Science, pp. 21–36. Springer, New York (2013).
12. Coecke, B., Paquette, É.O., Pavlović, D.: Classical and quantum structuralism. In: Gay, S., Mackey, I. (eds.) Semantic Techniques in Quantum Computation, pp. 29–69. Cambridge University Press, Cambridge (2010)
13. Coecke, B., Pavlović, D.: Quantum measurements without sums. Mathematics of Quantum Computing and Technology. Taylor and Francis, New York (2007)
14. Coecke, B., Pavlović, D., Vicary, J.: A new description of orthogonal bases. Math. Struct. Comput. Sci. 23(3), 555–567 (2012)
15. Coecke, B., Perdrix, S.: Environment and classical channels in categorical quantum mechanics. Computer Science Logic, pp. 230–244. Springer, New York (2010).
16. Davidson, K.R.: C*-algebras by Example. American Mathematical Society, Providence (1991)
17. Duncan, R.: Types for Quantum Computing. Ph.D. thesis, Oxford University, Oxford (2006).
18. Heunen, C., Contreras, I., Cattaneo, A.S.: Relative frobenius algebras are groupoids. J. Pure Appl. Algebra 217, 114–124 (2013)
19. Heunen, C., Kissinger, A., Selinger, P.: Completely positive projections and biproducts. In: Proceedings of Quantum Physics and Logic X (2013) (to appear) arxiv:1308.4557
20. Heunen, C., Vicary, J.: Introduction to Categorical Quantum Mechanics. Oxford University Press (to appear)
21. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)
22. Keyl, M.: Fundamentals of quantum information theory. Phys. Rep. 369, 431–548 (2002)
23. Keyl, M., Werner, R.F.: Channels and maps. In: Bruß, D., Leuchs, G. (eds.) Lectures on Quantum Information, pp. 73–86. Wiley, Hoboken (2007)
24. Li, B.: Real Operator Algebras. World Scientific, Singapore (2003)
25. Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1971).
26. Panangaden, P., Paquette, É.O.: New structures for physics. In: Coecke, B. (ed.) New Structures for Physics. No. 813 in Lecture Notes in Physics, pp. 939–979. Springer, New York (2009).
27. Paulsen, V.: Completely Bounded Maps and Operators Algebras. Cambridge University Press, Cambridge (2002)
28. Pechukas, P.: Reduced dynamics need not be completely positive. Phys. Rev. Lett. 74, 1060–1062 (1994)
29. Redei, M.: Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Stud. Hist. Philos. Sci. Part B 27, 493–510 (1996)
30. Rosenthal, K.I.: Quantales and their Applicatoins. Pitman Research Notes in Mathematics. Longman Scientific & Technical, Harlow (1990)
31. Ruan, Z.J.: On real operator spaces. Acta Math. Sinica 19(3), 485–496 (2003)
32. Selinger, P.: Dagger compact closed categories and completely positive maps. Quantum Programming Languages, Electronic Notices in Theoretical Computer Science, pp. 139–163. Elsevier, Amsterdam (2007).
33. Selinger, P.: Idempotents in dagger categories. Quantum Programming Languages, Electronic Notes in Theoretical Computer Science, pp. 107–122. Elsevier, Amsterdam (2008).
34. Selinger, P.: A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics. No. 813 in Lecture Notes in Physics, pp. 289–356. Springer, New York (2009).
35. Shaji, A., Sudarshan, E.C.G.: Who’s afraid of not completely positive maps? Phys. Lett. A 341(1–4), 48–54 (2005)
36. Stinespring, W.F.: Positive functions on C*-algebras. Proc. Am. Math. Soc. 6(2), 211–216 (1955)
37. Størmer, E.: Positive Linear Maps of Operator Algebras. Springer, New York (2013).
38. Vicary, J.: Categorical formulation of finite-dimensional quantum algebras. Commun. Math. Phys. 304(3), 765–796 (2011)
39. Zakrzewski, S.: Quantum and classical pseudogroups I. Commun. Math. Phys. 134, 347–370 (1990)
40. Życzkowski, K., Bengtsson, I.: On duality between quantum states and quantum maps. Open Syst. Inf. Dyn. 11, 3–42 (2004)