Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access July 15, 2019

Infinite dimensional generalizations of Choi’s Theorem

  • Shmuel Friedland EMAIL logo
From the journal Special Matrices


In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

MSC 2010: 46N50; 81Q10; 94A40


[1] M.D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285–290.10.1016/0024-3795(75)90075-0Search in Google Scholar

[2] K.R. Davidson, C*-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996.10.1090/fim/006Search in Google Scholar

[3] R.S. Doran and V.A. Belfi, Characterizations of C*-algebras: The Gelfand-Naimark Theorems, CRC Press, 1986.Search in Google Scholar

[4] S. Friedland and R. Loewy, On the extreme points of quantum channels, Linear Algebra Appl. 498 (2016), 553–573.10.1016/j.laa.2016.02.001Search in Google Scholar

[5] K.-E. Hellwig and K. Kraus, Pure operations and measurements, Commun. Math. Phys., 11 (1969), 214–220.10.1007/BF01645807Search in Google Scholar

[6] A. S. Holevo, Entropy gain and the Choi-Jamiolkowski correspondence for infinite-dimensional quantum evolutions, Theor. Math. Phys. 166 (2011), no. 1, 123–138.10.1007/s11232-011-0010-5Search in Google Scholar

[7] S. Karlin, Positive operators, J. Math. Mech. 8 (1959), 907–937.10.1512/iumj.1959.8.58058Search in Google Scholar

[8] K. Kraus, General state changes in quantum theory, Ann. Phys. 64 (1971), 311–335.10.1016/0003-4916(71)90108-4Search in Google Scholar

[9] M. Reed and B. Simon, Methods of Modern Mathematical Physics: Functional Analysis I, Academic Press, 1998.Search in Google Scholar

[10] M. Rørdam, Classification of Nuclear, Simple C*-algebras, Classification of Nuclear C*- Algebras. Entropy in Operator Algebras (J. Cuntz and V. Jones, eds.), vol. 126, Encyclopaedia ofMathematical Sciences. Subseries: Operator Algebras and Noncommutative Geometry, no. VII, Springer Verlag, Berlin, Heidelberg, 2001, pp. 1–145.10.1007/978-3-662-04825-2_1Search in Google Scholar

[11] S. Sakai, C*-algebras and W*-algebras, Springer 1971.Search in Google Scholar

[12] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer-Verlag, Berlin, 1960.10.1007/978-3-642-87652-3Search in Google Scholar

[13] B. Simon, Trace ideals and their applications, Second Edition, Amer. Math. Soc. 2005.Search in Google Scholar

[14] W. F. Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216.10.2307/2032342Search in Google Scholar

[15] E. Størmer, The analogue of Choi matrices for a class of linear maps on Von Neumann algebras, Internat. J. Math. 26 (2015), no. 2, 1550018, 7 pp.10.1142/S0129167X15500184Search in Google Scholar

[16] J. Watrous, The theory of quantum information, Cambridge University Press, 2018.10.1017/9781316848142Search in Google Scholar

Received: 2019-03-08
Accepted: 2019-06-15
Published Online: 2019-07-15

© 2019 Shmuel Friedland, by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 3.6.2023 from
Scroll to top button