No Access

On the Use of Total State Decompositions for the Study of Reduced Dynamics

Andrea Smirne

Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, via Celoria 16, 20133 Milan, Italy

Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milan, Italy

E-mail Address: [email protected]

Search for more papers by this author

Nina Megier

Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, via Celoria 16, 20133 Milan, Italy

Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milan, Italy

International Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk, 80–308 Gdańsk, Poland

E-mail Address: [email protected]

Search for more papers by this author

, and

Bassano Vacchini

Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, via Celoria 16, 20133 Milan, Italy

Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milan, Italy

E-mail Address: [email protected]

Search for more papers by this author

https://doi.org/10.1142/S1230161222500081Cited by:0 (Source: Crossref)

Abstract

The description of the dynamics of an open quantum system in the presence of initial correlations with the environment needs different mathematical tools than the standard approach to reduced dynamics, which is based on the use of a time-dependent completely positive trace preserving (CPTP) map. Here, we take into account an approach that is based on a decomposition of any possibly correlated bipartite state as a conical combination involving statistical operators on the environment and general linear operators on the system, which allows one to fix the reduced-system evolution via a finite set of time-dependent CPTP maps. In particular, we show that such a decomposition always exists, also for infinite dimensional Hilbert spaces, and that the number of resulting CPTP maps is bounded by the Schmidt rank of the initial global state. We further investigate the case where the CPTP maps are semigroups with generators in the Gorini-Kossakowski-Lindblad-Sudarshan form; for two simple qubit models, we identify the positivity domain defined by the initial states that are mapped into proper states at any time of the evolution fixed by the CPTP semigroups.

Keywords:

References

1. S. T. Ali, J.-P. Antoine, and J.-P. Gazeau, Coherent States, Wavelets, and Their Generalizations. Theor. Math. Physics, Springer, 2000. Crossref, Google Scholar
2. R. Alicki, Phys. Rev. Lett. 75, 3020 (1995). Crossref, Web of Science, Google Scholar
3. I. Bengtsson and K. Życzkowski, Geometry of quantum states: an introduction to quantum entanglement, Cambridge Univ. Press, 2006. Crossref, Google Scholar
4. H.-P. Breuer, Phys. Rev. A 75, 022103 (2007). Crossref, Web of Science, Google Scholar
5. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems. Oxford Univ. Press, 2002. Google Scholar
6. A. Brodutch, A. Datta, K. Modi, A. Rivas, and C. A. Rodríguez-Rosario, Phys. Rev. A 87, 042301 (2013). Crossref, Web of Science, Google Scholar
7. F. Buscemi, Phys. Rev. Lett. 113, 140502 (2014). Crossref, Web of Science, Google Scholar
8. D. Cariello, Quantum Inf. Comput. 14, 1308 (2014). Web of Science, Google Scholar
9. D. Cariello, Quantum Inf. Comput. 15, 812 (2015). Web of Science, Google Scholar
10. T. Chalermpusitarak, B. Tonekaboni, Y. Wang, L. M. Norris, L. Viola, and G. A. Paz-Silva, PRX Quantum 2, 030315 (2021). Crossref, Google Scholar
11. D. Chruściński and A. Kossakowski, Phys. Rev. Lett. 104, 070406 (2010). Crossref, Web of Science, Google Scholar
12. D. Chruściński and A. Kossakowski, Int. J. Quant. Inf. 09, 129 (2011). Link, Web of Science, Google Scholar
13. D. Chruściński and A. Kossakowski, EPL 97, 20005 (2012). Crossref, Web of Science, Google Scholar
14. D. Chruściński and F. A. Wudarski, Phys. Lett. A 377, 1425 (2013). Crossref, Web of Science, Google Scholar
15. J. M. Dominy and D. A. Lidar, Quant. Inf. Proc. 15, 1349 (2016). Crossref, Web of Science, Google Scholar
16. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976). Crossref, Web of Science, Google Scholar
17. H. Grabert, P. Schramm, and G.-L. Ingold, Phys. Rep. 168, 115 (1988). Crossref, Web of Science, Google Scholar
18. S. Hamedani Raja, K. P. Athulya, A. Shaji, and J. Piilo, Phys. Rev. A 101, 042127 (2020). Crossref, Web of Science, Google Scholar
19. L. Henderson and V. Vedral, J. Phys. A: Math. and Gen. 34, 6899 (2001). Crossref, Google Scholar
20. P. Štelmachovič and V. Bužek, Phys. Rev. A 64, 062106 (2001). Crossref, Web of Science, Google Scholar
21. T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 70, 052110 (2004). Crossref, Web of Science, Google Scholar
22. T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 73, 012106 (2006). Crossref, Web of Science, Google Scholar
23. G. Kimura, Phys. Lett. A 314, 339 (2003). Crossref, Web of Science, Google Scholar
24. G. Lindblad, Commun. Math. Phys. 48, 119 (1976). Crossref, Web of Science, Google Scholar
25. G. Lindblad, J. Phys. A: Math. Gen. 29, 4197 (1996). Crossref, Google Scholar
26. M. Majeed and A. Z. Chaudhry, Eur. Phys. J. D 73, 16 (2019). Crossref, Web of Science, Google Scholar
27. N. Megier, D. Chruściński, J. Piilo, and W. T. Strunz, Sci. Rep. 7, 6379 (2017). Crossref, Web of Science, Google Scholar
28. M. Merkli, arXiv:2107.02515 (2021). Google Scholar
29. K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, Rev. Mod. Phys. 84, 1655 (2012). Crossref, Web of Science, Google Scholar
30. P. Myöhänen, A. Stan, G. Stefanucci, and R. van Leeuwen, EPL 84, 67001 (2008). Crossref, Web of Science, Google Scholar
31. H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001). Crossref, Web of Science, Google Scholar
32. G. A. Paz-Silva, M. J. W. Hall, and H. M. Wiseman, Phys. Rev. A 100, 042120 (2019). Crossref, Web of Science, Google Scholar
33. P. Pechukas, Phys. Rev. Lett. 73, 1060 (1994). Crossref, Web of Science, Google Scholar
34. A. Pomyalov, C. Meier, and D. Tannor, Chem. Phys. 370, 98 (2010). Crossref, Web of Science, Google Scholar
35. J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, J. Math. Phys. 45, 2171 (2004). Crossref, Web of Science, Google Scholar
36. Á. Rivas, S. F. Huelga, and M. B. Plenio, Rep. Progr. Phys. 77, 094001 (2014). Crossref, Web of Science, Google Scholar
37. C. A. Rodríguez-Rosario, K. Modi, A. Kuah, A. Shaji, and E. C. G. Sudarshan, J. Phys. A: Math. Theor. 41, 205301 (2008). Crossref, Web of Science, Google Scholar
38. D. Schmid, K. Ried, and R. W. Spekkens, Phys. Rev. A 100, 022112 (2019). Crossref, Web of Science, Google Scholar
39. A. Shabani and D. A. Lidar, Phys. Rev. Lett. 102, 100402 (2009). Crossref, Web of Science, Google Scholar
40. A. Trevisan, A. Smirne, N. Megier, and B. Vacchini, Phys. Rev. A 104, 052215 (2021). Crossref, Web of Science, Google Scholar
41. B. Vacchini, J. Phys. B: Atomic, Molecular and Optical Physics 45, 154007 (2012). Crossref, Web of Science, Google Scholar
42. B. Vacchini and G. Amato, Sci. Rep. 6, 37328 (2016). Crossref, Web of Science, Google Scholar