Comparing quantum dynamical entropies

P. Tuyls

Banach Center Publications (1998)

  • Volume: 43, Issue: 1, page 411-420
  • ISSN: 0137-6934

Abstract

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Last years, the search for a good theory of quantum dynamical entropy has been very much intensified. This is not only due to its usefulness in quantum probability but mainly because it is a very promising tool for the theory of quantum chaos. Nowadays, there are several constructions which try to fulfill this need, some of which are more mathematically inspired such as CNT (Connes, Narnhofer, Thirring), and the one proposed by Voiculescu, others are more inspired by physics such as ALF (Alicki, Lindblad, Fannes). Therefore, a natural question arises whether there is a relation between all these different notions. In this paper we will indicate that the CNT entropy turns out to be smaller than the ALF dynamical entropy.

How to cite

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Tuyls, P.. "Comparing quantum dynamical entropies." Banach Center Publications 43.1 (1998): 411-420. <http://eudml.org/doc/208861>.

@article{Tuyls1998,
abstract = {Last years, the search for a good theory of quantum dynamical entropy has been very much intensified. This is not only due to its usefulness in quantum probability but mainly because it is a very promising tool for the theory of quantum chaos. Nowadays, there are several constructions which try to fulfill this need, some of which are more mathematically inspired such as CNT (Connes, Narnhofer, Thirring), and the one proposed by Voiculescu, others are more inspired by physics such as ALF (Alicki, Lindblad, Fannes). Therefore, a natural question arises whether there is a relation between all these different notions. In this paper we will indicate that the CNT entropy turns out to be smaller than the ALF dynamical entropy.},
author = {Tuyls, P.},
journal = {Banach Center Publications},
keywords = {quantum dynamical entropies; ALF dynamical entropy; CNT entropy; speed of information transmission; Holevo-Levitin inequality},
language = {eng},
number = {1},
pages = {411-420},
title = {Comparing quantum dynamical entropies},
url = {http://eudml.org/doc/208861},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Tuyls, P.
TI - Comparing quantum dynamical entropies
JO - Banach Center Publications
PY - 1998
VL - 43
IS - 1
SP - 411
EP - 420
AB - Last years, the search for a good theory of quantum dynamical entropy has been very much intensified. This is not only due to its usefulness in quantum probability but mainly because it is a very promising tool for the theory of quantum chaos. Nowadays, there are several constructions which try to fulfill this need, some of which are more mathematically inspired such as CNT (Connes, Narnhofer, Thirring), and the one proposed by Voiculescu, others are more inspired by physics such as ALF (Alicki, Lindblad, Fannes). Therefore, a natural question arises whether there is a relation between all these different notions. In this paper we will indicate that the CNT entropy turns out to be smaller than the ALF dynamical entropy.
LA - eng
KW - quantum dynamical entropies; ALF dynamical entropy; CNT entropy; speed of information transmission; Holevo-Levitin inequality
UR - http://eudml.org/doc/208861
ER -

References

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  1. [AAFT] R. Alicki, J. Andries, M. Fannes and P. Tuyls, An algebraic approach to the Kolmogorov-Sinai entropy, Rev. Math. Phys. 8(2) (1996), 167-184. Zbl0884.46039
  2. [AF] R. Alicki and M. Fannes, Defining quantum dynamical entropy, Lett. Math. Phys. 32 (1994), 75-82. Zbl0814.46055
  3. [AFTA] J. Andries, M. Fannes, P. Tuyls and R. Alicki, The dynamical entropy of the quantum Arnold cat map, Lett. Math. Phys. 35 (1995), 375-383. Zbl0842.58054
  4. [AN] R. Alicki and H. Narnhofer, Comparison of dynamical entropies of non-commutative shifts, Lett. Math. Phys. 33 (1995), 241-247. Zbl0836.46059
  5. [BNS] F. Benatti, H. Narnhofer and G. L. Sewell, A non-commutative version of the Arnold cat map, Lett. Math. Phys. 21 (1991), 157-192. Zbl0722.46033
  6. [FT] M. Fannes and P. Tuyls, On quantum dyamical entropy, in preparation. 
  7. [OP] M. Ohya and D. Petz, Quantum Entropy and its Use, Texts and monographs in physics, Springer-Verlag Heidelberg Berlin, 1993. 
  8. [Sau] J. L. Sauvageot and J. P. Thouvenot, Une nouvelle définition de l'entropie dynamique des systèmes non-commutatifs, Comm. Math. Phys. 133 (1992), 411-423. Zbl0772.46036
  9. [Sto] E. Størmer, Entropy of some automorphisms of the I I 1 factor of the free group in infinite number of generators, Invent. Math. 110 (1992), 63-73. Zbl0807.46079
  10. [TUY] P. Tuyls, Towards Quantum Kolmogorov-Sinai entropy, Ph.D. thesis, 1997. 

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