On the Alberti-Uhlmann Condition for Unital Channels

Sagnik Chakraborty; Dariusz Chruściński; Gniewomir Sarbicki; Frederik vom Ende

doi:10.22331/q-2020-11-08-360

On the Alberti-Uhlmann Condition for Unital Channels

Sagnik Chakraborty¹, Dariusz Chruściński¹, Gniewomir Sarbicki¹, and Frederik vom Ende^2,3

¹Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziądzka 5/7, 87-100 Toruń, Poland
²Department of Chemistry, Technische Universität München, 85747 Garching, Germany
³Munich Centre for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany

Published:	2020-11-08, volume 4, page 360
Eprint:	arXiv:2003.07889v3
Doi:	https://doi.org/10.22331/q-2020-11-08-360
Citation:	Quantum 4, 360 (2020).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We address the problem of existence of completely positive trace preserving (CPTP) maps between two sets of density matrices. We refine the result of Alberti and Uhlmann and derive a necessary and sufficient condition for the existence of a unital channel between two pairs of qubit states which ultimately boils down to three simple inequalities.

► BibTeX data

@article{Chakraborty2020albertiuhlmann,
  doi = {10.22331/q-2020-11-08-360},
  url = {https://doi.org/10.22331/q-2020-11-08-360},
  title = {On the {A}lberti-{U}hlmann {C}ondition for {U}nital {C}hannels},
  author = {Chakraborty, Sagnik and Chru{\'{s}}ci{\'{n}}ski, Dariusz and Sarbicki, Gniewomir and Ende, Frederik vom},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {360},
  month = nov,
  year = {2020}
}

► References

[1] P.M. Alberti and A. Uhlmann. A Problem Relating to Positive Linear Maps on Matrix Algebras. Rep. Math. Phys., 18 (2): 163–176, 1980. 10.1016/0034-4877(80)90083-X.
https://doi.org/10.1016/0034-4877(80)90083-X

[2] T. Ando. Majorization, doubly stochastic matrices, and comparison of eigenvalues. Linear Algebra Appl., 118: 163–248, 1989. 10.1016/0024-3795(89)90580-6.
https://doi.org/10.1016/0024-3795(89)90580-6

[3] W.B. Arveson. Subalgebras of $C^*$-algebras. Acta Math., 123 (1): 141–224, 1969. 10.1007/BF02392388.
https://doi.org/10.1007/BF02392388

[4] I. Bengtsson and K. Życzkowski. Geometry of Quantum States: an Introduction to Quantum Entanglement. Cambridge University Press, 2017. 10.1017/9781139207010.
https://doi.org/10.1017/9781139207010

[5] F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner. The Second Laws of Quantum Thermodynamics. Proc. Natl. Acad. Sci. USA, 112 (11): 3275–3279, 2015. 10.1073/pnas.1411728112.
https://doi.org/10.1073/pnas.1411728112

[6] F. Buscemi. Fully Quantum Second-law–like Statements From the Theory of Statistical Comparisons. arXiv preprint arXiv:1505.00535, 2015.
arXiv:1505.00535

[7] F. Buscemi, D. Sutter, and M. Tomamichel. An Information-theoretic Treatment of Quantum Dichotomies. Quantum, 3: 209, 2019. 10.22331/q-2019-12-09-209.
https://doi.org/10.22331/q-2019-12-09-209

[8] S. Chakraborty and D. Chruściński. Information Flow Versus Divisibility for Qubit Evolution. Phys. Rev. A, 99 (4): 042105, 2019. 10.1103/PhysRevA.99.042105.
https://doi.org/10.1103/PhysRevA.99.042105

[9] A. Chefles, R. Jozsa, and A. Winter. On the Existence of Physical Transformations Between Sets of Quantum States. Int. J. Quantum Inf, 2 (01): 11–21, 2004. 10.1142/S0219749904000031.
https://doi.org/10.1142/S0219749904000031

[10] M. Dall'Arno, F. Buscemi, and V. Scarani. Extension of the Alberti-Ulhmann Criterion Beyond Qubit Dichotomies. Quantum, 4: 233, 2020. 10.22331/q-2020-02-20-233.
https://doi.org/10.22331/q-2020-02-20-233

[11] A. Fujiwara and P. Algoet. One-to-one Parametrization of Quantum Channels. Phys. Rev. A, 59 (5): 3290, 1999. 10.1103/PhysRevA.59.3290.
https://doi.org/10.1103/PhysRevA.59.3290

[12] G. Gour, M.P. Müller, V. Narasimhachar, R.W. Spekkens, and N.Y. Halpern. The Resource Theory of Informational Nonequilibrium in Thermodynamics. Phys. Rep., 583: 1–58, 2015. 10.1016/j.physrep.2015.04.003.
https://doi.org/10.1016/j.physrep.2015.04.003

[13] T. Heinosaari, M.A. Jivulescu, D. Reeb, and M.M. Wolf. Extending Quantum Operations. J. Math. Phys., 53 (10): 102208, 2012. 10.1063/1.4755845.
https://doi.org/10.1063/1.4755845

[14] C.W. Helstrom. Quantum Detection and Estimation Theory. J. Stat. Phys., 1 (2): 231–252, 1969. 10.1007/BF01007479.
https://doi.org/10.1007/BF01007479

[15] R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 2012. 10.1017/CBO9780511810817.
https://doi.org/10.1017/CBO9780511810817

[16] M. Horodecki and J. Oppenheim. Fundamental Limitations for Quantum and Nanoscale Thermodynamics. Nat. Commun., 4 (1): 1–6, 2013. 10.1038/ncomms3059.
https://doi.org/10.1038/ncomms3059

[17] Z. Huang, C.-K. Li, E. Poon, and N.-S. Sze. Physical Transformations Between Quantum States. J. Math. Phys., 53 (10): 102209, 2012. 10.1063/1.4755846.
https://doi.org/10.1063/1.4755846

[18] A. Jenčová. Generalized Channels: Channels for Convex Subsets of the State Space. J. Math. Phys., 53 (1): 012201, 2012. 10.1063/1.3676294.
https://doi.org/10.1063/1.3676294

[19] C. King and M.B. Ruskai. Minimal Entropy of States Emerging From Noisy Quantum Channels. IEEE Trans. Inform. Theory, 47 (1): 192–209, 2001. 10.1109/18.904522.
https://doi.org/10.1109/18.904522

[20] M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (UK), 2000. 10.1017/CBO9780511976667.
https://doi.org/10.1017/CBO9780511976667

[21] V. Paulsen. Completely Bounded Maps and Operator Algebras, volume 78. Cambridge University Press, 2003. 10.1017/CBO9780511546631.
https://doi.org/10.1017/CBO9780511546631

[22] M.B. Ruskai, S. Szarek, and E. Werner. An Analysis of Completely-positive Trace-preserving Maps on $M_2$. Linear Algebra Appl., 347 (1-3): 159–187, 2002. 10.1016/S0024-3795(01)00547-X.
https://doi.org/10.1016/S0024-3795(01)00547-X

[23] E. Størmer. Generalities for Positive Maps. In Positive Linear Maps of Operator Algebras, pages 1–12. Springer, 2013. 10.1007/978-3-642-34369-8_1.
https://doi.org/10.1007/978-3-642-34369-8_1

[24] F. vom Ende. Exploring the Limits of Open Quantum Dynamics II: Gibbs-Preserving Maps from the Perspective of Majorization. arXiv preprint arXiv:2003.04164, 2020a.
arXiv:2003.04164

[25] F vom Ende. Strict Positivity and $D$-Majorization. arXiv preprint arXiv:2004.05613, 2020b.
arXiv:2004.05613

[26] F. vom Ende and G. Dirr. The $d$-Majorization Polytope. arXiv preprint arXiv:1911.01061, 2019.
arXiv:1911.01061

Cited by

[1] Katarzyna Siudzińska, Sagnik Chakraborty, and Dariusz Chruściński, "Interpolating between Positive and Completely Positive Maps: A New Hierarchy of Entangled States", Entropy 23 5, 625 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2023-06-09 06:49:58). The list may be incomplete as not all publishers provide suitable and complete citation data.

On SAO/NASA ADS no data on citing works was found (last attempt 2023-06-09 06:49:58).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.