Abstract
We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘indivisible’ channels which can not be written as non-trivial products of other channels and study the set of ‘infinitesimal divisible’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Holevo, A.S.: Statistical Structure of Quantum Theory. Springer Lecture Notes in Physics, Berlin- Heidelberg-New York: Springer, 2001
Horn R.A. (1967). Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8: 219
Holevo A.S. (1986). Theor. Probab. Appl. 32: 560
Denisov L.V. (1988). Th. Prob. Appl. 33: 392
Jamiolkowski A. (1972). Rep. Math. Phys. 3: 275
Choi M.D. (1975). Lin. Alg. Appl. 10: 285
Kraus K. (1983). States, Effects and Operations. Springer, Berlin-Heidelberg-New York
Wolf M.M. and Perez-Garcia D. (2007). Phys. Rev. A 75: 012303
Lindbald G. (1976). Commun. Math. Phys. 48: 119
Gorini V., Kossakowski A. and Sudarshan E.C.G. (1976). J. Math. Phys. 17: 821
Davies E.B. (1980). Rep. Math. Phys. 17: 249
Perez-Garcia D., Wolf M.M., Petz D. and Ruskai M.B. (2006). J. Math. Phys. 47: 083506
Schmidt, W.M.: Diophantine Approximation. Lecture Notes in Math. 785, Berlin-Heidelberg-New York: Springer Verlag, 1980
Bhatia, R.: Matrix Analysis. Springer Graduate Texts in Mathematics 169, Berlin-Heidelberg-New York: Springer, 1997
Streater R.F. (1995). Statistical Dynamics. Imperial College Press, London
Wigner, E.P.: Gruppentheorie. Braunschweig: Vieweg 1931; Group Theory. London: Academic Press, 1959
Bargmann V. (1964). J. Math. Phys. 5: 862
Kadison R. (1965). Topology 3(supp. 2): 177
Buscemi F., D’Ariano G.M., Keyl M., Perinotti P. and Werner R. (2005). J. Math. Phys. 46: 082109
Nielsen M.A. and Chuang I.L. (2000). Quantum Computation and Quantum Information. Cambridge University Press, Cambridge
Uhlmann A. (1976). Rep. Math. Phys. 9: 273
Stoermer E. (1963). Acta Math. 110: 233
King C. and Ruskai M.B. (2001). IEEE Trans. Info. Theory 47: 192
Fujiwara A. and Algoet P. (1999). Phys. Rev. A 59: 3290
Ruskai M.B., Szarek S. and Werner E. (2002). Lin. Alg. Appl. 347: 159
Gorini V. and Sudarshan E.C.G. (1976). Commun. Math. Phys. 46: 43
Verstraete, F., Verschelde, H.: http://arxiv.org/list/quant-ph/0202124, 2002; F. Verstraete, J. Dehaene, B. De Moor.: Phys. Rev. A 64, 010101(R) (2001)
Vollbrecht K.G.H. and Werner R.F. (2000). J. Math. Phys. 41: 6772
Bacon D., Childs A.M., Chuang I.L., Kempe J., Leung D.W. and Zhou X. (2001). Phys. Rev. A 64: 062302
Eisert, J., Wolf, M.M.: http://arxiv.org/list/quant-ph/0505151, 2005; ‘Gaussian quantum channels’. In: Quantum Information with continuous variables of atoms and light, N. Cerf, G. Leuchs, E.S. Polzik (eds.) London: Imperial College Press, 2006
Verstraete F., Cirac J.I., Latorre J.I., Rico E. and Wolf M.M. (2005). Phys. Rev. Lett. 94: 140601
Wolf, M.M., Eisert, J., Cubitt, T.S., Cirac, J.I.: arXiv: 0711.3172 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M.B. Ruskai
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Wolf, M.M., Cirac, J.I. Dividing Quantum Channels. Commun. Math. Phys. 279, 147–168 (2008). https://doi.org/10.1007/s00220-008-0411-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0411-y