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Accessible Maps in a Group of Classical or Quantum Channels

Koorosh Sadri

Department of Physics, Sharif University of Technology, Tehran, Iran

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Fereshte Shahbeigi

Department of Physics, Sharif University of Technology, Tehran, Iran

Center for Theoretical Physics, Polish Academy of Sciences, 02–668 Warsaw, Poland

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Zbigniew Puchała

Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, 44–100 Gliwice, Poland

Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Kraków, Poland

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Karol Życzkowski

Center for Theoretical Physics, Polish Academy of Sciences, 02–668 Warsaw, Poland

Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Kraków, Poland

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https://doi.org/10.1142/S1230161222500020Cited by:0

Abstract

We study the problem of accessibility in a set of classical and quantum channels admitting a group structure. Group properties of the set of channels, and the structure of the closure of the analyzed group $G$ plays a pivotal role in this regard. The set of all convex combinations of the group elements contains a subset of channels that are accessible by a dynamical semigroup. We demonstrate that accessible channels are determined by probability vectors of weights of a convex combination of the group elements, which depend neither on the dimension of the space on which the channels act, nor on the specific representation of the group. Investigating geometric properties of the set $𝒜$ of accessible maps we show that this set is nonconvex, but it enjoys the star-shape property with respect to the uniform mixture of all elements of the group. We demonstrate that the set $𝒜$ covers a positive volume in the polytope of all convex combinations of the elements of the group.

Dedicated to the memory of Prof. Andrzej Kossakowski (1938–2021)

Keywords:

References

1. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan , “ Completely positive dynamical semigroups of $N$ -level systems”, J. Math. Phys. 17, 821 (1976). Crossref, ISI, Google Scholar
2. G. Lindblad , “ On the generators of quantum dynamical semigroups”, Commun. Math. Phys. 48, 119 (1976). Crossref, ISI, Google Scholar
3. D. Chruściński and S. Pascazio , “ A brief history of the GKLS Equation”, Open Sys. Information Dyn. 24, 1740001 (2017). Link, ISI, Google Scholar
4. M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac , “ Assessing non-Markovian dynamics”, Phys. Rev. Lett. 101, 150402 (2008). Crossref, ISI, Google Scholar
5. Á. Rivas, and S. F. Huelga , Open Quantum Systems. An Introduction, Springer, 2011. Google Scholar
6. W. F. Stinespring , “ Positive functions on $C^{*}$ -algebras”, Proc. American Math. Soc. 6, 211 (1955). Google Scholar
7. J. F. C. Kingman , “ The imbedding problem for finite Markov chains”, Probab. Theory Relat. Fields 1, 14 (1962). Google Scholar
8. J. T. Runnenburg , “ On Elfving’s problem of imbedding a time-discrete Markov chain in a time-continuous one for finitely many states”, Proc. Kon. Ned. Akad. Wet. Ser. A 65, 536 (1962). Google Scholar
9. J. R. Cuthbert , “ The Logarithm Function for Finite-State Markov Semi-Groups”, J. London Math. Soc. s2-6(3), 524 (1973). Crossref, Google Scholar
10. S. Johansen , J. London Math. Soc. s2-8(2), 345 (1974). Crossref, Google Scholar
11. P. Carette , “ Characterizations of embeddable $3 \times 3$ stochastic matrices with a negative eigenvalue”, New York J. Math 1, 129 (1995). Google Scholar
12. E. B. Davies , “ Embeddable Markov matrices”, Electron. J. Probab. 15, 1474 (2010). Crossref, ISI, Google Scholar
13. M. Casanellas, J. Fernàndez-Sànchez, and J. Roca-Lacostena, “The embedding problem for Markov matrices”, arXiv:2005.00818. Google Scholar
14. K. Korzekwa and M. Lostaglio , “ Quantum advantage in simulating stochastic processes”, Phys. Rev. X 11, 021019 (2021). ISI, Google Scholar
15. L. V. Denisov , “ Infinitely divisible Markov mappings in quantum probability theory”, Theory Probab. Appl. 33, 392 (1989). Crossref, ISI, Google Scholar
16. M. M. Wolf and J. I. Cirac , “ Dividing quantum channels”, Commun. Math. Phys. 279, 147 (2008). Crossref, ISI, Google Scholar
17. T. S. Cubitt, J. Eisert, and M. M. Wolf , “ The complexity of relating quantum channels to master equations”, Commun. Math. Phys. 310, 383 (2012). Crossref, ISI, Google Scholar
18. D. Davalos, M. Ziman, and C. Pineda , “ Divisibility of qubit channels and dynamical maps”, Quantum 3, 144 (2019). Crossref, ISI, Google Scholar
19. Z. Puchała, Ł. Rudnicki, and K. Życzkowski , “ Pauli semigroups and unistochastic quantum channels”, Phys. Lett. A 383, 2376 (2019). Crossref, ISI, Google Scholar
20. S. Chakraborty and D. Chruściński , “ Information flow versus divisibility for qubit evolution”, Phys. Rev. A 99, 042105 (2019). Crossref, ISI, Google Scholar
21. V. Jagadish, R. Srikanth, and F. Petruccione , “ Convex combinations of Pauli semigroups: geometry, measure and an application”, Phys. Rev. A 101, 062304 (2020). Crossref, ISI, Google Scholar
22. K. Siudzińska and D. Chruściński , “ Memory kernel approach to generalized Pauli channels: Markovian, semi-Markov, and beyond”, Phys. Rev. A 96, 022129 (2017). Crossref, ISI, Google Scholar
23. M. C. Caro and B. Graswald , “ Necessary Criteria for Markovian Divisibility of Linear Maps”, J. Math. Phys. 62, 042203 (2021). Crossref, ISI, Google Scholar
24. F. Shahbeigi, D. Amaro-Alcalá, Z. Puchała, and K. Życzkowski , “ Log-Convex set of Lindblad semigroups acting on $N$ -level system”, J. Math. Phys. 62, 072105 (2021). Crossref, ISI, Google Scholar
25. K. Siudzińska , “ Markovian semigroup from mixing non-invertible dynamical maps”, Phys. Rev. A 103, 022605 (2021). Crossref, ISI, Google Scholar
26. H. Weyl , “ Quantenmechanik und Gruppentheorie”, Z. Physik 46, 1 (1927). Crossref, Google Scholar
27. F. Shahbeigi, K. Sadri, M. Moradi, K. Życzkowski, and V. Karimipour , “ Quasi-inversion of quantum and classical channels in finite dimensions”, J. Phys. A: Math. Theor. 54, 34 (2021). Crossref, ISI, Google Scholar
28. P. Lencastre, F. Raischel, T. Rogers, and P. G. Lind , “ From empirical data to time-inhomogeneous continuous Markov processes”, Phys. Rev. E 93, 032135 (2016). Crossref, ISI, Google Scholar
29. K. Korzekwa, S. Czachórski, Z. Puchała, and K. Życzkowski , “ Coherifying quantum channels”, New J. Phys. 20, 043028 (2018). Crossref, ISI, Google Scholar
30. R. Kukulski, I. Nechita, Ł. Pawela, Z. Puchała, and K. Życzkowski , “ Generating random quantum channels”, J. Math. Phys. 62, 062201 (2021). Crossref, ISI, Google Scholar
31. T. Baumgratz, M. Cramer, and M. B. Plenio , “ Quantifying coherence”, Phys. Rev. Lett. 113, 140401 (2014). Crossref, ISI, Google Scholar
32. M. Snamina and E. J. Zak, “Dynamical semigroups in the Birkhoff polytope of order 3 as a tool for analysis of quantum channels”, preprint arXiv:1811.09506. Google Scholar
33. I. Bengtsson, A. Ericsson, M. Kuś, W. Tadej, and K. Życzkowski , “ Birkhoff’s polytope and unistochastic matrices, $N = 3$ and $N = 4$ ”, Commun. Math. Phys. 259, 307 (2005). Crossref, ISI, Google Scholar

Vol. 29, No. 01

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History

Received 28 January 2022

Accepted 18 February 2022

Published: August 31, 2022

Keywords

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Accessible Maps in a Group of Classical or Quantum Channels

Abstract

References

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